Slaman and Woodin's discovery (private communication) of upper and lower ...... Let g Ï(x)[si], 1 ⤠i ⤠k, be a list of all realised strings for RÎ at stage s +1,.
Upper Cones As Automorphism Bases S. Barry Cooper ‡ University of Leeds Leeds LS2 9JT England ABSTRACT. It is shown that the complete Turing degrees do not form an automorphism base.
A class A ⊆ the Turing degrees D is an automorphism base (see Lerman [1983]) if and only if any nontrivial automorphism of D necessarily moves at least one of its elements — or, equivalently, the global action of any such automorphism is completely determined by that on A . Jockusch and Posner [1981] demonstrated the existence of a wide range of automorphism bases, and subsequent work of a number of people led eventually to Slaman and Woodin’s discovery (private communication) of upper and lower cones (in fact singletons) which are automorphism bases for the global structure. In fact, Ambos-Spies [ta] showed every nontrivial ideal of computably enumerable (c.e.) degrees to be an automorphism base, while on the other hand there were many upper cones known to be very far from being automorphism bases in that they were rigid in D — that is, all their members were invariant in D (in the sense of Rogers [1967]). The strongest such result was that of Slaman and Woodin (see Nies, Shore and Slaman [ta]): D (≥ 0′′ ) is rigid in D . Moreover, 0′ turned out to be definable in D and hence invariant (see Cooper [ta1]). Below, a nontrivial Turing automorphism is constructed which only moves degrees within their atomic jump classes. The main consequence of this is that the complete Turing degrees do not form an automorphism base for D . ‡ We
would like to acknowledge helpful conversations with G. E. Sacks concerning the degrees of Turing automorphisms, made possible by E.P.S.R.C. Research Grant no. GR/L63396. 1991 Mathematics Subject Classification. Primary 03D25, 03D30; Secondary 03D35. Typeset by AMS-TEX 1
S. Barry Cooper
2
See Cooper [1997] for historical and background information. For basic terminology and notation see Odifreddi [1989] or Soare [1987].
1. The main theorem and basic definitions Theorem 1.1. There exists a nontrivial Turing automorphism which is the identity on D (≥ 0′ ) . Proof. The construction of the required automorphism closely follows that of Cooper [ta2]. Again, D is considered as being the degree structure derived from considering Turing reducibility over the tree ω s + 1 , with τ or σ on a dis-
S. Barry Cooper
18
b -reflected relative to the ∗ -image of the discontinuity. continuity of ∗s+1 , is only Φ b Such axioms can only be Φ -reflected along branches incompatible with the discontinuities via imprinted axioms involving strings σ ′ , τ ′ ⊃ σ, τ , respectively, with ′ τ ′ = Φσβ¯ and τ ′ , σ ′ incompatible with any discontinuity. The axioms to which this applies are specified to be those implementations of stipulated axioms for hatted functionals whose corresponding J -requirements name strings whose ∗−1 s+1 -images are not weakly restrained above RΘ . b α , say, in relation to discontinuities of ∗−1 . There is similar provision for Ψ s+1 (Recall that in the interests of bijectivity of e ∗ , there is also imprinting of axioms is carried out at the receptive stages to in an attempt to ensure that at no stage t + 1 ≥ s + 1 at which a requirement of priority less than that of RΘ requires b σ∗t+1 ’ attention does one reflect an axiom ‘ τ = Ψσα¯ ’ via an axiom ‘ τ ∗t+1 = Ψ α involving a string σ ∗t+1 on the periphery of a discontinuity of ∗s+1 .) Action for subcase II: One tries to extend our list of realised strings in the hope of a subsequent application of subcase I of case (D) of definition 2.2. Choose minimal strings π, ρ ⊃ fs x − 1, (fs x − 1)∗s , respectively, with π(x) 6= g si Θ (x) or fsi (x) each i , 1 ≤ i ≤ k , and with no extension of π x being in the domain, or of ρ |(fs x)∗s | in the range, of ∗s . Define π ∗s+1 = ρ , fs+1 = π , gs+1 = ρ , and wait for g ϑ(x)[t] to become realised at some stage t + 1 > s + 1 . b At each stage t + 1 ≥ s + 1 , if there is a ∗t -matching of the inner P, Q b′ , . . . , Ψ b ′ ] - and P[ bΦ b′ , . . . , Φ b ′ ], Q - preconfigured ranks of fs +1 and gs +1 [Ψ α1 αl k k β1 βk above RΘ at stage t + 1 , with corresponding augmentations b b Q b ′α ], εP,Q b ′α , . . . , Ψ b ′β ], b ′β , . . . , Φ εP, (RΘ , fsk +1 )[Ψ (RΘ , gsk +1 )[Φ t t 1 1 l k b
b
Q say, of εP, (RΘ , fsk +1 ) , εP,Q (RΘ , gsk +1 ) , then choose such a matching to be opt t timal, and appoint restraints of priority RΘ on ∗t at all relevant matched strings at all stages t′ + 1 ≥ t + 1 prior to some requirement of higher priority than that of RΘ requiring attention at a stage > s + 1 .
Finish off by nontrivially progressing the background activity in the case that Rγ does not require attention via case (D) of the construction at stage s + 1 . b β and Ψ b α : Assume a computable listing of ω j , that b -preconfigured rank above Rγ of fs matches the P, b Qat stage s′ + 1 the P, Q i preconfigured rank of gsj above Rγ (via a matching which respects all relevant restraints on ∗s of no lesser priority than that of Rγ ). Then, by parts (a) and (b) of the consequent action, one defines fs′ +1 , gs′ +1 and ∗s′ +1 in such a way (guided by the appropriate fsi , gsj -matching) that at stage b -preconfigured rank above Rγ of fs′ +1 ( = fs ) ∗s′ +1 -matches s′ + 1 = s the P, Q b Q -preconfigured rank of gs′ +1 ( = gs ) above Rγ . Arguing as previously, one the P, can verify that this ∗s′ +1 -matching is not disturbed by new hatted axioms defined at stage s′ + 1 as a result of implementing some previously stipulated axioms, so b -preconfigured rank of fs above Rγ that one again has a ∗s -matching of the P, Q b Q -preconfigured rank of gs above Rγ at the end of stage s . Again, it by the P, follows that for each RΘ of priority greater than or equal to that of Rγ , this still b β or Ψ bα holds with RΘ in place of Rγ . Also, by the initialising of functionals Φ being built by P - or Q - requirements of lower priority than that of Rγ , and by b β and Ψ b α at stage s′ + 1 , this also holds for lower the routine extension of ∗ , Φ b -preconfigured priority (than Rγ ) RΘ in place of Rγ . So, for each RΘ , the P, Q b Q -preconfigured rank of gs above RΘ at rank above RΘ of fs ∗s -matches the P, the end of stage s , as required. If Rγ requires attention at stage s′ + 1 through case (A) the argument is very b -preconfigured rank above Rγ of fs′ +1 by similar. The ∗s′ +1 -matching of the P, Q b Q -preconfigured rank of gs′ +1 above Rγ is again obtained at stage s′ +1 via the P, an implementation of a matching ψ , which again holds despite the possible addition of axioms via the routine extension of the hatted functionals at stage s′ + 1 . Finally, say Rγ requires attention at stage s′ + 1 through subcase II of case b -preconfigured rank above (D). By assumption, one has that for each RΘ , the P, Q b Q -preconfigured rank of gt above RΘ at stage t + 1 , RΘ of ft ∗t -matches the P, b -preconfigured rank above RΘ of fs′ x − 1 and hence that for each RΘ , the P, Q b Q -preconfigured rank of (fs′ x − 1)∗s′ above RΘ at stage ∗s′ -matches the P, bt + 1 . Then by the choice of fs′ +1 , gs′ +1 , one has that for each RΘ , the P, Q b Q -preconfigured rank preconfigured rank above RΘ of fs′ +1 ∗s′ -matches the P, of gs′ +1 above RΘ at stage t + 1 . And again arguing as above, one can replace ∗s′ -matching at stage t+1 by ∗s -matching at the end of stage s . Since one trivially b -preconfigured rank above RΘ of fs ∗s -matches has that for each RΘ , the P, Q
S. Barry Cooper
24
b Q -preconfigured rank of gs above RΘ at stage s+1 when s = 0 , the lemma the P, inductively follows. ⊓ ⊔ b -preconfigured rank above RΘ of fs Lemma 3.2. For each RΘ , the inner P, Q b Q -preconfigured rank of gs above RΘ at each reactive stage matches the inner P, s + 1. Proof. By lemma 3.1, at the receptive stage s = s′ + 1 , say, for each RΘ the b Q -preconfigured rank b -preconfigured rank above RΘ of fs′ ∗s′ -matches the P, P, Q of gs′ above RΘ where ∗s′ , fs′ , gs′ = ∗s , fs , gs , respectively. This matching can now be adapted to give the required matching of the inb -preconfigured rank above RΘ of fs by the inner P, b Q -preconfigured ner P, Q rank of gs above RΘ at stage s + 1 . Assume that ∗s gives an isomorphism beb P,Q b′ , . . . , Ψ b ′ ] corresponding to the tween the inner stratified rank ε ′ (RΘ , fs )[Ψ augmentation b P,Q
s b P,Q ′ b ′α ] b εs′ (RΘ , fs )[Ψα1 , . . . , Ψ l
of
α1 αl b P,Q εs′ (RΘ , fs ) and
the inner stratified b
b′ , . . . , Φ b ′ ] corresponding to the augmentation εP,Q rank εs′ (RΘ , gs )[Φ β1 βk s′ (RΘ , gs ) b b′ , . . . , Φ b ′ ] of εP,Q [Φ β1 βk s′ (RΘ , gs ) , under which fs is mapped to gs , and where, in adb ′α or Φ b ′ is consistent with its corresponding dition, each functional of the form Ψ βj
i
b α [s] or Φ b β [s] . Assuming that the the inner P, Q b -preconfigured rank above RΘ Ψ i j b Q -preconfigured rank of gs above RΘ at of fs does not ∗s -match the inner P, stage s + 1 (otherwise there is nothing more to prove), there must be an axiom of the form ‘ σ = Φτα ’ or ‘ σ = Ψτβ ’ enumerated at stage s′ + 1 (with α , β configuring) b -preconfigured such that there is a breakdown in the ∗s -matching of the inner P, Q b Q -preconfigured rank of gs above RΘ at rank of fs above RΘ by the inner P, stage s + 1 at some level below that of α ∈ Tf or β ∈ Tg , respectively, due to an already existing hatted axiom at that level. One defines the required matching ψ (respecting any existing restraints on ∗s at stage s + 1 ), with corresponding augmentations b P,Q
b b Q b ′′ ′ , . . . , Ψ b ′′ ′ ], εP,Q b ′′′ , . . . , Φ b ′′′ ] εP, (RΘ , fs )[Ψ (RΘ , gs )[Φ s s α1 α′ β1 β ′ b P,Q
l
k
of εs (RΘ , fs ) , εs (RΘ , gs ) , respectively, as follows. In defining ψ one follows the inductive definition 1.7 of preconfigurations, restricted to the inner preconfigurations relative to that part of these augmentations already inductively defined. Assume that one has defined these augmentations at every index ⊆ some αi′′ or βi′′ , respectively, where one writes b b Q b ′′ ′′ , . . . , Ψ b ′′′′ , . . . , Φ b ′′α′′ ], εP,Q b ′′β ′′ ] (RΘ , fs )[Ψ εP, (RΘ , gs )[Φ s s α1 β1 p q
for those parts of the augmentations so far defined, that one has suitably defined the isomorphism (3.3)
b P,Q
ψ : εs
b
b ′′ ′ , . . . , Ψ b ′′ ′ ] −→ εP,Q b ′′′ , . . . , Φ b ′′′ ] (RΘ , fs )[Ψ (RΘ , gs )[Φ s α1 α′ β1 β ′ l
k
Upper Cones As Automorphism Bases b P,Q
at all nonempty entries in εs and that for some α ˆ either b P,Q
εs or
b ′′ ′′ , . . . , Ψ b ′′ ′′ ] of index ⊆ some α′′ or β¯′′ , (RΘ , fs )[Ψ i i αp α 1
b
P,Q,Φ b ′′ ♮ ](ˆ b ′′ ♮ ] 6= ∅, b ′′ ♮ , . . . , Ψ b ′′ ♮ , . . . , Ψ α) = εs,αˆ αˆ (RΘ , fs )[Ψ (RΘ , fs )[Ψ α α α α 1
b P,Q
εs
25
1
p′
b P,Q,Ψ ¯ α ˆ
¯ˆ ) = ε ¯ b ′′♮ ](α b ′′♮ , . . . , Φ (RΘ , gs )[Φ s,α ˆ β β 1
p′
b ′′♮ ] 6= ∅ b ′′♮ , . . . , Φ (RΘ , gs )[Φ β β 1
q′
q′
b ′′ ♮ , . . . , Ψ b ′′ ♮ ⊇ Ψ b ′′ ′′ , . . . , Ψ b ′′ ′′ or Φ b ′′♮ , . . . , Φ b ′′♮ ⊇ Φ b ′′′′ , . . . , Φ b ′′′′ , respecfor some Ψ αp βq α β α α β β 1
1
p′
tively, where for each b ′′ ♮ = Ψ b ♮ [s] one has Ψ α α i
i
1
1
q′
αi♮
6= αj′′ , some j , or for each βi♮ b ′′♮ = Φ b ♮ [s] , respectively. or Φ βi βi
6=
βj′′ ,
some j , respectively,
Assuming the former applies (the action corresponding to the second possibility is completely analogous), one lets α ˆ be the least (according to ≺ ) such index. b P,Q,Φ
Assume first that α ˆ is configuring, and consider each (σ, τ ) ∈ εs,αˆ αˆ (RΘ , fs ) b ′′ ♮ , . . . , Ψ b ′′ ♮ ] under the following three cases. By the definition of the stratified [Ψ α α 1
p′
rank one may assume that ψ(τ ) ↓ . Case 1: ψ(τ ) = τ ∗s and each pair b
b
Q b ′′ ♯ , . . . , Ψ b ′′ ♯ ], εP,Q (RΘ , gs )[Φ b ′′♯ , . . . , Φ b ′′♯ ] εP, (RΘ , fs )[Ψ s s α α β β 1
1
l′′
k′′
extending b b Q b ′′ ′′ , . . . , Ψ b ′′ ′′ ], εP,Q b ′′′′ , . . . , Φ b ′′′′ ] εP, (RΘ , fs )[Ψ (RΘ , gs )[Φ s αp s βq α1 β1
¯ˆ } ⊆ {β ♯ , . . . , β ♯ ′′ } and σ ∗s = with {α1′′ , . . . , αp′′ , α ˆ } ⊆ {α1♯ , . . . , αl♯′′ } , {β1′′ , . . . , βq′′ , α 1 k ′′ τ ∗s ˆ b must have been chosen in such a way that any index β ⊇ α ˆ at which Φαˆ the corresponding ∗s -matching breaks down cannot be inductively reached in the bΨ b ′′ ♯ , . . . , Ψ b ′′ ♯ ] - or P[ bΦ b ′′♯ , . . . , Φ b ′′♯ ], Q - preconfiguration definition of inner P, Q[ α α β β 1
1
l′′
k′′
¯ˆ , respectively, except via an axiom for a functional Ψ b ′′♯ b ′′ ♯ or Φ passing via α ˆ or α β α
j′
i′
b ♯ [s] or Φ b ♯ [s] . which is not already in the corresponding Ψ α β i′
j′
If ψ(σ) is as yet undefined, one now defines ψ(σ) = σ ∗s . In either case one b ′′¯ the axiom ‘ ψ(σ) = Φ b ′′¯ ψ(τ ) ’. enumerates into Φ α ˆ α ˆ Case 2: ψ(τ ) = τ ∗s , but case 1 does not apply. In this case, if ψ(σ) ↑ , one chooses a string π which is incompatible with any argument or value of a functional (hatted or otherwise) which properly extends an already defined ψ(σ ′ ) , σ ′ maximal ⊆ σ , at stage s + 1 , which is incompatible with b P,Q,Φ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] , for any other already defined ψ(σ ′ ) with (σ ′ , τ ) ∈ εs,αˆ αˆ (RΘ , fs )[Ψ α1
αp
which one can define, consistently with all existing restraints on ∗s of no lesser
S. Barry Cooper
26
priority than that of RΘ , ψ(σ) = π , and for which π ⊇ each ψ(σ ′ ) with σ ′ ⊆ σ . And if such a definition ψ(σ) = π is possible, it is implemented. In either case an b ′′¯ ψ(τ ) ’ is enumerated into Φ b ′′¯ . axiom ‘ ψ(σ) = Φ α ˆ α ˆ
Case 3: ψ(τ ) 6= τ ∗s . Again, if ψ(σ) ↑ one asks for a proper extension π of the longest ψ(σ ′ ) with ′ σ ⊂ σ , for which one can define, consistently with all existing restraints on ∗s of no lesser priority than that of RΘ , ψ(σ) = π , and for which π ⊇ each ψ(σ ′ ) with σ ′ ⊆ σ , and if such a definition ψ(σ) = π is possible, define ψ(σ) = π . Enumerate b ′′¯ an axiom ‘ ψ(σ) = Φ b ′′¯ ψ(τ ) ’. into Φ α ˆ α ˆ b ′′¯ consists of all such axioms enumerated into Φ b ′′¯ via such applications of Φ α ˆ α ˆ ′′ b cases 1, 2 and 3. Define Ψαˆ in the case that the second possibility holds, by acting b
α ˆ (RΘ , gs ) similarly via analogous cases 1 to 3 in relation to pairs (σ, τ ) ∈ εP,Q,Ψ ¯ s,α ˆ b ′′♮ , . . . , Φ b ′′♮ ] . [Φ
β1
βq′
Now complete the definition of the required ψ and its corresponding augmentations b b Q b ′′ ′ , . . . , Ψ b ′′ ′ ], εP,Q b ′′′ , . . . , Φ b ′′′ ] εP, (RΘ , fs )[Ψ (RΘ , gs )[Φ s s α1 α′ β1 β ′ l
b P,Q
b P,Q
k
of εs (RΘ , fs ) , εs (RΘ , gs ) , respectively, in the case that α ˆ is nonconfiguring. Again, the action for the former (that for the latter being similar) is described relb αˆ P,Q,Φ b ′′ ♮ , . . . , Ψ b ′′ ♮ ] under three cases analogous (RΘ , fs )[Ψ ative to each (σ, τ ) ∈ ε s,α ˆ
α1
αp′
to those above. Case 1 ′ : ψ(σ) ↓= σ ∗s and each pair b b Q b ′′ ♯ , . . . , Ψ b ′′ ♯ ], εP,Q b ′′♯ , . . . , Φ b ′′♯ ] εP, (RΘ , fs )[Ψ (RΘ , gs )[Φ s s α α β β 1
1
l′′
k′′
extending b b Q b ′′ ′′ , . . . , Ψ b ′′ ′′ ], εP,Q b ′′′′ , . . . , Φ b ′′′′ ] εP, (RΘ , fs )[Ψ (RΘ , gs )[Φ s αp s βq α1 β1
¯ˆ } ⊆ {β ♯ , . . . , β ♯ ′′ } and τ ∗s = with {α1′′ , . . . , αp′′ , α ˆ } ⊆ {α1♯ , . . . , αl♯′′ } , {β1′′ , . . . , βq′′ , α 1 k b ′′ σ∗s , is such that if ψp is a maximal consistent extension of the form Φ α ˆ b P,Q
ψ ′ : εs
b
b ′′ ♯ ] −→ εP,Q (RΘ , gs )[Φ b ′′ ♯ , . . . , Ψ b ′′♯ ] b ′′♯ , . . . , Φ (RΘ , fs )[Ψ s α α β β 1
1
l′′
′′
k′′
′
of that part ψ of ψ which has already been defined, with ψ ⊆ ∗s ∪ ψ ′′ , then the pair must have been chosen in such a way that any index βˆ ⊆ α ˆ at which the corresponding potential matching ψp breaks down cannot be reached by retracing bΨ b ′′ ♯ , . . . , Ψ b ′′♯ ], Q - prebΦ b ′′♯ , . . . , Φ b ′′ ♯ ] - or P[ the inductive definition of inner P, Q[ β β α α 1
1
l′′
k′′
¯ˆ , respectively, except via an axiom for a functional Ψ b ′′ ♯ configuration from α ˆ or α α
i′
b ′′♯ which is not already in the corresponding Ψ b ♯ [s] or Φ b ♯ [s] . or Φ α β β j′
i′
j′
Upper Cones As Automorphism Bases
27
If ψ(τ ) is as yet undefined, now define ψ(τ ) = τ ∗s . In either case enumerate b ′′¯ the axiom ‘ ψ(τ ) = Φ b ′′¯ ψ(σ) ’. into Φ α ˆ α ˆ
Case 2 ′ : ψ(σ) = σ ∗s , but case 1 does not apply. Similarly to Case 2 above, if ψ(σ) ↑ , choose a string ̺ which is incompatible with any argument or value of a functional (hatted or otherwise) which properly extends an already defined ψ(τ ′ ) , τ ′ maximal ⊆ τ , at stage s + 1 , which is inb P,Q,Φ
compatible with any other already defined ψ(τ ′ ) with (σ, τ ′ ) ∈ εs,αˆ αˆ (RΘ , fs ) b ′′ ♯ , . . . , Ψ b ′′ ♯ ] , for which one can define, consistently with all existing restraints [Ψ α α 1
l′′
on ∗s of no lesser priority than that of RΘ , ψ(τ ) = ̺ , and for which ̺ ⊇ each ψ(τ ′ ) with τ ′ ⊆ τ . If such a definition ψ(σ) = ̺ is possible, it is implemented, and b ′′¯ ψ(σ) ’ is enumerated into Φ b ′′¯ . in either case an axiom ‘ ψ(τ ) = Φ α ˆ α ˆ
Case 3 ′ : ψ(σ) 6= σ ∗s . Again, if ψ(τ ) ↑ ask for a proper extension ̺ of the longest ψ(τ ′ ) with τ ′ ⊂ τ , for which one can define, consistently with all existing restraints on ∗s of no lesser priority than that of RΘ , ψ(τ ) = ̺ , and for which ̺ ⊇ each ψ(τ ′ ) with τ ′ ⊆ τ , b ′′¯ and if such a definition ψ(τ ) = ̺ is possible, define ψ(τ ) = ̺ . Enumerate into Φ α ˆ b ′′¯ ψ(σ) ’. an axiom ‘ ψ(τ ) = Φ α ˆ
b ′′¯ consists of all such axioms enumerated into Φ b ′′¯ via such applications of cases Φ α ˆ α ˆ b ′′ in the case that the second possibility holds, by acting 1 ′ , 2 ′ and 3 ′ . Define Ψ α ˆ b
α ˆ (RΘ , gs ) similarly via analogous cases to 1 ′ to 3 ′ in relation to pairs (σ, τ ) ∈ εP,Q,Ψ ¯ s,α ˆ b ′′♮ ] . b ′′♮ , . . . , Φ [Φ
β1
βq′
One needs to verify (a) that ψ is a bijective mapping in (3.3), (b) that b b ′′′ , . . . , Φ b ′′′ ] are respective aug(RΘ , gs )[Φ and εP,Q s β β ′
b Q b ′′ ′ , . . . , Ψ b ′′ ′ ] εP, (RΘ , fs )[Ψ s α1 α′ l
b Q εP, (RΘ , fs ) s
b εP,Q (RΘ , gs ) , s
1
k
mentations of and and (c) that the mapping ψ constructed does respect the structure of the inner preconfigurations relative to fs and gs . First notice that (a) depends on no such σ in cases 2 or 3 of the definition of bΨ b′ , . . . , Ψ b′ ]ψ , or τ in cases 2 ′ or 3 ′ of the definition of ψ being inner P, Q[ α1 αl bΦ b′ , . . . , Φ b ′ ], Q - preconfigured relative to fs or gs , respectively, above RΘ or P[ β1 βk at stage s + 1 with index previous to that at which the definition of ψ(σ) 6= σ ∗s or ψ(τ ) 6= τ ∗s , respectively, is determined (so that, in particular, σ ∗s or τ ∗s , respectively, is not restrained above RΘ at stage s + 1 ). And this will follow from the fact that the augmentations b b + P,Q b′ , . . . , Ψ b ′ ], + εP,Q b′ b′ εs′ (RΘ , fs )[Ψ α1 αl s′ (RΘ , gs )[Φβ1 , . . . , Φβk ]
agree with the augmentations b + P,Q b ′′α , . . . , Ψ b ′′α ], εs (RΘ , fs )[Ψ 1 l
b + P,Q b ′′β , . . . , Φ b ′′β ] εs (RΘ , gs )[Φ 1 k
S. Barry Cooper
28
at all relevant levels of their inductive definitions, and that ∗s does match the b -preconfigured rank above RΘ of fs and the P, b Q -preconfigured rank of gs P, Q above RΘ at the end of stage s . Specifically, first assume that α ˆ is as in the above definition of ψ , and is such that b P,Q
ψ is a mapping from strings named in members of those components of εs (RΘ , fs ) b ′′ ′′ , . . . , Ψ b ′′ ′′ ] of index ⊂ α [Ψ ˆ onto strings named in members of those components αp α 1
b P,Q
of εs
¯ˆ , but, say, that there is a pair b ′′′′ , . . . , Φ b ′′′′ ] of index ⊂ α (RΘ , gs )[Φ βq β 1
b αˆ P,Q,Ξ
(ˆ σ , τˆ) ∈ εs,αˆ
b ′′ ′′ , . . . , Ψ b ′′ ′′ ], (RΘ , fs )[Ψ αp α1
b ′′ , for which ψ(ˆ with Ξαˆ = Φαˆ or Ψ σ , τˆ) ( = (ψ(ˆ σ ), ψ(ˆ τ )) ) is undefined. α ˆ Assume first that α ˆ is configuring, so that it is case 2 or 3 of the definition of ψ through which ψ(ˆ σ ) fails to become defined. b ′′ , since then, by the choice of α One cannot have Ξαˆ = Ψ ˆ , one obtains α ˆ
(ˆ σ , τˆ) ∈
bΨ b ′′ + P,Q, b ′′ ′′ , . . . , Ψ b ′′ ′′ ], εs,αˆ αˆ (RΘ , fs )[Ψ αp α1
via cases 1, 2 or 3 (above) corresponding to some b P,Q,Ψ ¯ α ˆ b ′′′′ , . . . , Φ b ′′′′ ], (RΘ , gs )[Φ β1 β ′
(σ, τ ) ∈ + εs,αˆ¯
k
−1
−1
where ψ (τ ) ↓= τˆ , in which case one defines ψ (σ) = σ ˆ. On the other hand, if Ξαˆ = Φαˆ and ψ(ˆ τ ) ↓ , then one either defines ψ(ˆ σ ) via σ ) = to an appropriately chosen π according to case 2 or case 1, or one defines ψ(ˆ 3 above. For this, one need only check that if case 2 or 3 applies, then one is able to choose a string π as described therein. If such a π does not exist as in case 2, one must have that for each π either i) π is compatible with an argument or value of some relevant functional at stage s + 1 , which properly extends an already defined ψ(σ ′ ) , σ ′ maximal ⊆ σ , or ii) π is compatible with some other already defined ψ(σ ′ ) with (σ ′ , τ ) ∈ b P,Q,Φ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] , or εs,αˆ αˆ (RΘ , fs )[Ψ αp
α1
∗−1
iii) There is some σ0 ⊆ σ ˆ or π0 ⊆ π for which σ0∗s or π0 s , respectively, is ∗−1
restrained at stage s + 1 with σ0∗s | π or π0 s | σ ˆ , respectively, or iv) π 6⊇ ψ(σ ′ ) , some σ ′ ⊆ σ ˆ. There is no problem in choosing a π which is not subject to i), ii) or iv), and, in addition, to which iii) only applies if some σ0 ⊆ σ ˆ is such that σ0∗s is restrained, but there is no σ ′ ⊂ σ ˆ for which ψ(σ ′ ) ↓ and σ0∗s ⊆ ψ(σ ′ ) . bΨ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] - preconfigured relative to fs above RΘ at stage Let σ ˆ0 be P, Q[ s + 1 due to
α1
αp
b αˆ Q,Φ b ′′ ′′ , . . . , Ψ b ′′α′′ ], (ˆ σ0 , τˆ0 ) ∈ εP, (RΘ , fs )[Ψ α1 s,α ˆ p
Upper Cones As Automorphism Bases
29
say, where σ ˆ0 ⊆ σ ˆ and σ ˆ0 is the least such string ⊇ σ0 . If ψ(ˆ σ0′ ) ↓ , some ¢ ¡ P,Qˆ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] , σ ˆ0′ ≈ σ ˆ0 εs (RΘ , fs )[Ψ α α1 p and hence consistently with the existing restraints, by definition 5 one has ψ(ˆ σ0′ ) = ′ ∗s ∗s σ ˆ0 , contradicting the fact that it was not possible to choose π ⊃ σ0 . On the other hand, if ψ(ˆ σ0′ ) ↑ for all ¢ ¡ P,Qˆ b ′′ ′′ , . . . , Ψ b ′′α′′ ] , σ ˆ0′ ≈ σ ˆ0 εs (RΘ , fs )[Ψ α1 p it must be the case that ¢ ¡ P,Qˆ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] . σ ˆ0 ≈ σ ˆ εs (RΘ , fs )[Ψ αp α1 But since σ0∗s is restrained at stage s + 1 , it must be the case that at some reactive stage u + 1 < s + 1 , some current Rγ -determined (say) fu+1 was defined via case (C) of definition 2.2, resulting in fu ⊆ fv+1 , each v with u ≤ v ≤ s , where σ0 , by b -preconfigured relative to the conditions of case (C) of definition 2.2, is inner P, Q fu above Rγ at stage u + 1 , and hence (since fu+1 is current and the restraint is assumed not to have been cancelled at any stage previous to s + 1 ) relative to fv at each stage v + 1 with u + 1 ≤ v + 1 ≤ s + 1 . Since ψ is required to respect the restraint on σ0∗s , RΘ is of no greater priority than that of Rγ , so that σ0 b -preconfigured relative to fs above RΘ at stage s + 1 . By lemma is inner P, Q b -preconfigured rank above RΘ of fs matches the inner P, b Q3.2, the inner P, Q preconfigured rank of gs above RΘ at stage s + 1 . But in this case there must be some ¢ ¡ P,Qˆ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] (RΘ , fs )[Ψ σ ′ ≈ σ0 ε 0
s
α1
αp
for which ψ(σ0′ ) ↓= σ0′ ∗s , again contradicting the impossibility of choosing π ⊃ σ0∗s . For case 3, the existence of π follows similarly. Moreover, a very similar argument applies in the case that α ˆ is nonconfiguring, via a parallel consideration of cases 1 ′ , 2 ′ and 3 ′ of the definition of ψ . Having ensured that the definition of ψ does not terminate prematurely, and given that the injectiveness of ψ follows trivially from that of ψ and the construction of ψ , (a) follows. For (b), first assume that a definition of a hatted functional during the description b Q b ′′ ′ , . . . , Ψ b ′′ ′ ] , say, from being an augmentation of of ψ prevents εP, (RΘ , fs )[Ψ s α α′
b Q εP, (RΘ , fs ) . s
1
l
b ′′ ′′ be the first such functional which is inconsistent with its Let Ψ αp b b ′′ ′′τ ’ corresponding Ψα′′p [s] . If αp′′ is configuring, one cannot define an axiom ‘ σ = Ψ αp ′′ b b of Ψ ′′ inconsistently with Ψα′′ [s] via case 1, since in that case one would have αp
p
S. Barry Cooper
30
−1
′ b ′′ ∗s some σ ′ = ψ(σ) , τ ′ = ψ(τ ) , with ∗−1 s (σ ) = Ψα′′ p
b P,Q,Ψ α ¯ ′′
(σ ′ , τ ′ ) ∈ εs,α¯ ′′p
p
k
′
And it is easy to see that for any such σ , τ , b P,Q,Ψ α ¯ ′′
(3.4)
p
αp′
one has
b ′′′′ , . . . , Φ b ′′′′ ] ⇔ (RΘ , gs )[Φ β1 β ′′
b P,Q,Ψ α ¯ ′′
(σ ′ , τ ′ ) ∈ εs,α¯ ′′p
due to
b ′′′′ , . . . , Φ b ′′′′ ]. (RΘ , gs )[Φ β1 β ′′ ′
(σ ′ , τ ′ ) ∈ εs,α¯ ′′p
(τ ′ )
p
k
b′ , . . . , Φ b ′ ], (RΘ , gs )[Φ β1 βk
since assuming (3.4) for any σ ′′ , τ ′′ , αp′ ′ , with p′ < p , or p′ = p and σ ′′ , τ ′′ ⊂ σ ′ , τ ′ , respectively, in place of σ ′ , τ ′ , αp′ , respectively, one need only examine case 1 of the definition of ψ to see that since ∗s is an fs , gs -matching of the respective inner preconfigured ranks above RΘ at the end of stage s , with corresponding augmentations ˆ b Q b′ , . . . , Ψ b ′ ], εP,Q b′ , . . . , Φ b ′ ], εP, (RΘ , fs )[Ψ (RΘ , gs )[Φ s α1 αl s β1 βk
one has that (3.4) inductively follows. But then the assumed inconsistency contrab Q b′ , . . . , Ψ b ′ ] being an augmentation of εP,Qb (RΘ , fs ) . dicts εP, (RΘ , fs )[Ψ s s α1 αl b ′′ ′′τ ’ of Ψ b ′′ ′′ So in this case, one must assume that one defines an axiom ‘ σ = Ψ αp
αp
b α′′ [s] via case 2 or 3. But such a definition is not possible via inconsistently with Ψ p b τ ′′′ [s] ↓| Ψ b ′′ ′′τ for case 2, again, since by the conditions of case 2 one cannot have Ψ αp αp any τ ′ compatible with τ . And, again, the definition of such an axiom is impossible via case 3, since then one would have τ = ψ −1 (τ ′ ) defined via case 2 or 3, some b α′′ [s] which properly extends each τ ′ , with τ incompatible with any argument of Ψ p already defined ψ −1 (τ ′′ ) , τ ′′ ⊆ τ ′ , again excluding any such inconsistency with b α′′ [s] . Ψ p b ′′ ′′ contains no axiom inconsistent with Ψ b α′′ [s] , then Ψ b ′′ ′′ is Moreover, if Ψ αp
p
αp
consistent. For axioms defined via case 1, this again follows by the fact that b Q b′ , . . . , Ψ b ′ ] is an augmentation of εP,Qb (RΘ , fs ) . Otherwise, it trivεP, (RΘ , fs )[Ψ s α1 αl s ially follows by the conditions of cases 2 and 3. b ′′′′ , say, with β ′′ configuring, defined via the The argument for a functional Φ q βq b construction of ψ , assumed inconsistent with Φβ ′′ [s] , is similar. q
Finally, notice that the assumption of a definition of a hatted functional during the description of the definition of ψ via cases 1 ′ , 2 ′ or 3 ′ which prevents b b Q Q b ′′ ′ , . . . , Ψ b ′′ ′ ] , say, from being an augmentation of εP, εP, (RΘ , fs )[Ψ (RΘ , fs ) can s s α1 αl′ be ruled out by a very similar argument. For (c), assume first that α ˆ is such that ψ defines an isomorphism from the b P,Q ′′ b ′′ , . . . , Ψ b ′′ ′′ ] to components of index ≺ α ˆ to the restriction of εs (RΘ , fs )[Ψ α1
αl′′
Upper Cones As Automorphism Bases
31
b P,Q
¯ˆ , but that b ′′′′ , . . . , Φ b ′′′′ ] to components of index ≺ α (RΘ , gs )[Φ β1 βk′′ ¯ˆ in the isomorphism breaks down upon inclusion of the components of index α ˆ, α the corresponding stratified preconfigurations. Assume also that α ˆ is a minimal such index (under ≺ ). Let restriction of εs
b αˆ P,Q,Ξ
(ˆ σ , τˆ) ∈ εs,αˆ
b ′′ ′′ , . . . , Ψ b ′′ ′′ ] (RΘ , fs )[Ψ α1 α ′′ l
b ′′ . This be the pair at which the isomorphism breaks down with Ξαˆ = Φαˆ or Ψ α ˆ b P,Q ′′ ′′ b b means that that part of εs (RΘ , fs )[Ψα′′ , . . . , Ψα′′′′ ] defined during the inductive 1 l bΨ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] -preconfigured relative to fs above definition of τˆ being inner P, Q[ α1
αl′′
RΘ at stage s + 1 via indices ≺ α ˆ is isomorphically mapped via ψ to that part of b P,Q ′′ ′′ b ′′ , . . . , Φ b ′′ ] defined during the inductive definition of ψ(ˆ εs (RΘ , gs )[Φ τ ) being inner
β1 βk′′ ′′ ′′ b ′′ ], Q -preconfigured b Q[Φ b ′′ , . . . , Φ P, βk′′ β1
relative to gs above RΘ at stage s + 1
¯ˆ , but that this is no longer the case when τˆ , ψ(ˆ via indices ≺ α τ ) are replaced by σ ˆ , ψ(ˆ σ ) , respectively. By the minimality of α ˆ , one can assume. without loss of generality, that this bΨ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] -preconfigured relative to fs above RΘ implies that σ ˆ is inner P, Q[ α1 αl′′ at stage s + 1 with some index α′ 4 α ˆ , say, due to b α′ P,Q,Ξ
σ , τ ′ ) ∈ εs,α′ (ˆ
b ′′ ′′ , . . . , Ψ b ′′ ′′ ], (RΘ , fs )[Ψ α1 α ′′ l
b α′ , but that ψ(ˆ bΦ b ′′′′ , . . . , Φ b ′′′′ ], σ ) is not inner P[ say, with Ξα′ of the form Φα′ or Ψ β1 βk′′ ′ Q -preconfigured relative to gs above RΘ at stage s + 1 with index α ¯ and b Ξ b α¯ ′ P,Q,
(ψ(ˆ σ ), ψ(τ ′ )) ∈ εs,α¯ ′
b ′′′′ , . . . , Φ b ′′′′ ], (RΘ , gs )[Φ β1 β ′′ k
b α¯ ′ = Φ b α¯ ′ or Ψα¯ ′ . with Ξ ψ(ˆ σ ) cannot be defined via case 1, by the definition of the augmentations b b P,Q b ′′ ′′ , . . . , Ψ b ′′ ′′ ] and εP,Q b ′′′′ , . . . , Φ b ′′′′ ] , and of the mapεs (RΘ , fs )[Ψ (RΘ , gs )[Φ s α1 αl′′ β1 βk′′ ping ψ . But on the other hand, one cannot define ψ(ˆ σ ) via case 2 or 3 by the accompanying conditions put on such a definition. The usual modifications extend the above argument to the whole of ψ . ⊓ ⊔ Say that a requirement Rγ is finitely injuring if and only if it requires attention at most finitely often. A requirement is finitely injured if and only if every requirement of strictly higher priority is finitely injuring. The verification is based on the next lemma. Lemma 3.5. If an R -requirement RΘ is finitely injured, then it is finitely injuring. Proof. Let RΘ be a given R -requirement, and inductively assume s∗ to be a stage such that no higher priority requirement requires attention at a stage s + 1 >
S. Barry Cooper
32
s∗ . This means that no higher priority K -, L - or M - requirement can be linked to a lower priority R -requirement at a reactive stage s + 1 > s∗ via case (C) of the construction, so that the initial segment of the priority ordering of requirements up to and including RΘ is unchanged at stages s + 1 > s∗ . First notice that RΘ can only require attention finitely often in relation to a given potential witness x , say. Such an x may be appointed through part (B) of definition 2.2 at a stage sx + 1 , say, and if cancelled through some requirement of higher priority than RΘ requiring attention or through RΘ requiring attention through part (C), can never be reappointed. In order to get a contradiction, assume that RΘ requires attention at infinitely many stages in relation to x through part (D) of definition 2.2. Part I cannot apply at any such stage s + 1 > s∗ , since in that case at stage s + 1 one would define, and restrain at all later stages, fs+1 ⊇ some π , gs+1 ⊇ some ρ ⊃ g ϑ(x)[sj ] , where π(x) 6= Θgsj (x) (by the action for subcase II, since i > j ), precluding RΘ from requiring attention at any stage > s + 1 . Now assume that RΘ requires attention at infinitely many stages through subcase II of case (D), and that g ϑ(x)[si ] , i ≥ 1 , is a list of all strings realised in relation to x through case (D), where, by choice of x , no such realised string is ever cancelled. By the conditions of subcase II, at no stage s + 1 at which RΘ requires attention through subcase II of case (D) in relation to x do there exist distinct b -preconfigured rank above RΘ of fs matches i, j ≥ 1 , i > j , such that the P, Q i b Q -preconfigured rank of gs above RΘ at stage s + 1 via a matching which the P, j respects all restraints on ∗s of no lesser priority than that of RΘ , and, beyond its core, respects all weak restraints on ∗s of higher priority than that of RΘ . Sublemma 3.6. For all i, j ≥ 1 , i > j , if at the reactive stage s + 1 > sx + b -preconfigured rank above RΘ of fs matches the inner P, b Q1 the inner P, Q i preconfigured rank of gsj above RΘ at stage s + 1 via a matching which respects all restraints on ∗s of no lesser priority than that of RΘ , and, beyond its core, respects all weak restraints on ∗s of higher priority than that of RΘ , then the b -preconfigured rank above RΘ of fs matches the P, b Q -preconfigured rank of P, Q i gsj above RΘ at stage s + 1 via a matching which respects all restraints on ∗s of no lesser priority than that of RΘ , and, beyond its core, respects all weak restraints on ∗s of higher priority than that of RΘ . Proof. Let i > j ≥ 1 , and let ψ be an fsi , gsj -matching of the correspondb - and P, b Q - preconfigured ranks above RΘ at the reactive stage ing inner P, Q s + 1 > sx + 1 , where ψ respects all existing restraints on ∗s of no lesser priority than that of RΘ , and, beyond its core, respects all weak restraints on ∗s of higher priority than that of RΘ , and where one may assume that ψ is an isomorphism from b P,Q b Q b ′α , . . . , Ψ b ′α ] corresponding to an augmentation εP, (RΘ , fs ) some εs (RΘ , fs )[Ψ s i
1
l
b
i
P,Q b Q b ′α , . . . , Ψ b ′α ] of εP, b′ , . . . , Φ b ′ ] correspond[Ψ (RΘ , fsi ) to some εs (RΘ , gsj )[Φ s β1 βk 1 l
Upper Cones As Automorphism Bases
33
b b b′ , . . . , Φ b ′ ] of εP,Q (RΘ , gsj ) , under which (RΘ , gsj )[Φ ing to an augmentation εP,Q s s β1 βk e fsi is mapped to gsj . Inductively define ψ to be an fsi , gsj -matching of the full b - and P, b Q - preconfigured ranks above RΘ at stage s + 1 which also reP, Q spects the appropriate restraints and weak restraints on ∗s , with corresponding augb b b Q Q b ′′ ′ , . . . , Ψ b ′′ ′ ] of + εP, mentations + εP, (RΘ , fsi )[Ψ (RΘ , fsi ) and + εP,Q (RΘ , gsj ) s s s α α′ 1
b
l
b ′′′ , . . . , Φ b ′′′ ] of + εP,Q [Φ (RΘ , gsj ) . The definitions below have certain features in s β1 βk′ common with those of the proof of lemma 3.2. Similarly to before, assume that one has defined the augmentations b b + P,Q b ′′′ ] b ′′ ′ ], + εP,Q b ′′′ , . . . , Φ b ′′ ′ , . . . , Ψ (RΘ , gsj )[Φ εs (RΘ , fsi )[Ψ s βk′ αl′ β1 α1
at every index ⊆ some αi′′ or βi′′ , respectively, where one writes b b + P,Q b ′′ ′′ , . . . , Ψ b ′′ ′′ ], + εP,Q b ′′′′ , . . . , Φ b ′′′′ ] εs (RΘ , fsi )[Ψ (RΘ , gsj )[Φ αp s βq α1 β1
for those parts of the augmentations so far defined, that one has suitably defined the isomorphism (3.7)
ψe :
b + P,Q b ′′ ′ , . . . , Ψ b ′′ ′ ] εs (RΘ , fsi )[Ψ α1 αl′
b P,Q
−→ + εs
b ′′′ , . . . , Φ b ′′′ ] (RΘ , gsj )[Φ β1 β ′ k
b P,Q
b ′′ ′′ , . . . , Ψ b ′′ ′′ ] of index ⊆ some α′′ or at all nonempty entries in + εs (RΘ , fsi )[Ψ i αp α1 ′′ ¯ β , and that for some α ˆ either i
b + P,Q b ′′ ♮ , . . . , Ψ b ′′ ♮ εs (RΘ , fsi )[Ψ α α
](ˆ α) = + εs,αˆ
b + P,Q b ′′♮ , . . . , Φ b ′′♮ εs (RΘ , gsj )[Φ β β
¯ˆ ) = + ε ¯ ](α s,α ˆ
1
p′
b αˆ P,Q,Φ
b ′′ ♮ , . . . , Ψ b ′′ ♮ ] 6= ∅, (RΘ , fsi )[Ψ α α 1
p′
or 1
q′
b P,Q,Ψ α ˆ
b ′′♮ , . . . , Φ b ′′♮ ] 6= ∅ (RΘ , gsj )[Φ β β 1
q′
b ′′ ♮ , . . . , Ψ b ′′ ♮ ⊇ Ψ b ′′ ′′ , . . . , Ψ b ′′ ′′ or Φ b ′′♮ , . . . , Φ b ′′♮ ⊇ Φ b ′′′′ , . . . , Φ b ′′′′ , respecfor some Ψ αp βq α β α α β β 1
p′
tively, where for each b ′′ ♮ = Ψ b ♮ [s] one has Ψ α α i
i
1
1
αi♮
6= αj′′ , some j , or for each βi♮ b ′′♮ = Φ b ♮ [s] , respectively. or Φ βi βi
1
q′
6=
βj′′ ,
some j , respectively,
Assuming the former applies, let α ˆ be the least such index and consider each b P, Q,Φ α ˆ b ′′ ♮ , . . . , Ψ b ′′ ♮ ] under the following two cases. By the (RΘ , fsi )[Ψ (σ, τ ) ∈ + εs,αˆ α α 1
p′
e )↓. definition of the stratified rank one may assume that ψ(τ
b αˆ e ) = ψ(τ ) , ψ(σ) ↓ and (σ, τ ) ∈ εP,Q,Φ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] . Case 1: ψ(τ (RΘ , fsi )[Ψ s,α ˆ αp α 1
e e b ′′¯ the axiom ‘ ψ(σ) One then defines ψ(σ) = ψ(σ) , and enumerates into Φ = α ˆ e e ′′ ψ(τ ) ′′ ψ(σ) e )=Φ b¯ b¯ Φ ’ or ‘ ψ(τ ’ according as α ˆ is configuring or nonconfiguring. α ˆ α ˆ Case 2: Otherwise.
S. Barry Cooper
34
One chooses a string π which is incompatible with any argument or value of a e ′) , functional (hatted or otherwise) which properly extends an already defined ψ(σ σ ′ maximal ⊆ σ , at stage s + 1 , which is incompatible with any other already b αˆ e ′ ) with (σ ′ , τ ) ∈ + εP,Q,Φ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] , for which one can (RΘ , fs )[Ψ defined ψ(σ s,α ˆ
i
αp
α1
define, consistently with all existing restraints and weak restraints on ∗s of no e e ′ ) with lesser priority than that of RΘ , ψ(σ) = π , and for which π ⊇ each ψ(σ e σ ′ ⊆ σ . If such a definition ψ(σ) = π is possible, it is implemented, and an axiom e ′′ ψ(σ) e )=Φ b¯ b ′′¯ . ‘ ψ(τ ’ is enumerated into Φ α ˆ
α ˆ
b ′′¯ consists of all such axioms enumerated into Φ b ′′¯ via such applications of cases Φ α ˆ α ˆ ′′ b 1 and 2. One defines Ψαˆ in the case that the second possibility holds, by acting b P,Q,Ψ α ˆ
similarly via analogous cases 1 and 2 in relation to pairs (σ, τ ) ∈ + εs,αˆ¯ b ′′′′ , . . . , Φ b ′′′′ ] . [Φ β1
(RΘ , gsj )
βq
One needs to verify (a) that ψe is a bijective mapping in (3.7), (b) that b b + P,Q b ′′′ , . . . , Φ b ′′′ ] are respective b ′′ ′ , . . . , Ψ b ′′ ′ ] and + εP,Q (RΘ , gsj )[Φ εs (RΘ , fsi )[Ψ s β β ′ α α′ 1
b
l
b
1
k
Q augmentations of + εP, (RΘ , fsi ) and + εP,Q (RΘ , gsj ) , and (c) that the mapping s s ψe constructed does respect the structure of the preconfigurations relative to fsi and gsj . For (a), assume that α ˆ is as in the definition of ψe , and is such that ψe is a
b P,Q
mapping from strings named in members of those components of + εs (RΘ , fsi ) b ′′ ′′ , . . . , Ψ b ′′ ′′ ] of index ⊂ α [Ψ ˆ onto strings named in members of those components αp α 1
of
b + P,Q b ′′′′ , . . . , Φ b ′′′′ ] εs (RΘ , gsj )[Φ βq β1
(ˆ σ , τˆ) ∈
¯ˆ , but, say, that there is a pair of index ⊂ α
b + P,Q,Ξ b ′′ ′′ , . . . , Ψ b ′′ ′′ ], εs,αˆ αˆ (RΘ , fsi )[Ψ αp α1
e σ , τˆ) ( = (ψ(ˆ e σ ), ψ(ˆ e τ )) ) is undefined. b ′′ , for which ψ(ˆ with Ξαˆ = Φαˆ or Ψ α ˆ b ′′ , since then one obtains One cannot have Ξαˆ = Ψ α ˆ
(ˆ σ , τˆ) ∈
bΨ b ′′ + P,Q, b ′′ ′′ , . . . , Ψ b ′′α′′ ], εs,αˆ αˆ (RΘ , fsi )[Ψ α1 p
via cases 1 or 2 (above) corresponding to some b P,Q,Ψ ¯ α ˆ
(σ, τ ) ∈ + εs,αˆ¯
b ′′′′ , . . . , Φ b ′′′′ ], (RΘ , gsj )[Φ β1 β ′ k
where ψe−1 (τ ) ↓= τˆ , in which case one defines ψe−1 (σ) = σ ˆ. e ) ↓ , then one either defines ψ(σ) e On the other hand, if Ξαˆ = Φαˆ and ψ(τ via case e 1, or one defines ψ(ˆ σ ) = to an appropriately chosen π according to case 2 above. For this, one needs only check that if case 2 applies, then one is able to choose a string π as described therein.
Upper Cones As Automorphism Bases
35
If such a π does not exist, one must have, similarly to before, that for each π either i) π is compatible with an argument or value of some relevant functional at e ′ ) , σ ′ maximal stage s + 1 , which properly extends an already defined ψ(σ ⊆ σ , or e ′ ) with (σ ′ , τ ) ∈ ii) π is compatible with some other already defined ψ(σ b ˆ + P,Q,Φα b ′′ ′′ , . . . , Ψ b ′′ ′′ ] , or ε (RΘ , fs )[Ψ s,α ˆ
i
αp
α1
∗−1
iii) There exists some σ0 ⊆ σ ˆ or π0 ⊆ π for which σ0∗s or π0 s , respectively, ∗−1
is restrained or weakly restrained at stage s + 1 with σ0∗s | π or π0 s | σ ˆ, respectively, or e ′ ) , some σ ′ ⊆ σ iv) π 6⊇ ψ(σ ˆ.
Again, there is no problem in choosing a π which is not subject to i), ii) or iv), and, in addition, to which iii) only applies if some σ0 ⊆ σ ˆ is such that σ0∗s e ′ ) ↓ and is restrained or weakly restrained, but there is no σ ′ ⊂ σ ˆ for which ψ(σ ∗s e ′) . σ0 ⊆ ψ(σ bΨ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] - preconfigured relative to fs above RΘ at stage Let σ ˆ0 be P, Q[ s + 1 due to
αp
α1
(ˆ σ0 , τˆ0 ) ∈
i
b + P,Q,Φ b ′′ ′′ , . . . , Ψ b ′′ ′′ ], εs,αˆ αˆ (RΘ , fsi )[Ψ αp α1
say, where σ ˆ0 ⊆ σ ˆ and σ ˆ0 is the least such string ⊇ σ0 . ′ e σ ) ↓ , some If ψ(ˆ 0
σ ˆ0′ ≈ σ ˆ0
¡+
ˆ P,Q
εs
¢ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] , (RΘ , fsi )[Ψ αp α1
and hence consistently with the existing restraints or weak restraints, by definition e σ′ ) = σ 5 one has ψ(ˆ ˆ0′ ∗s , contradicting the fact that it was not possible to choose 0 π ⊃ σ0∗s . e σ ′ ) ↑ for all On the other hand, if ψ(ˆ 0
σ ˆ0′ ≈ σ ˆ0
¡+
ˆ P,Q
εs
¢ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] , (RΘ , fsi )[Ψ αp α1
it must be the case that σ ˆ0 ≈ σ ˆ
¡+
ˆ P,Q
εs
¢ b ′′ ′′ , . . . , Ψ b ′′ ′′ ] . (RΘ , fsi )[Ψ αp α1
But since then one has τ0 ≈ τˆ
¡+
ˆ P,Q
εs
¢ b ′′ ′′ , . . . , Ψ b ′′α′′ ] , (RΘ , fsi )[Ψ α1 p
e τ ) , so that τ ∗s is not restrained above RΘ at stage s + 1 . one must have τ0∗s | ψ(ˆ 0 b -preconfigured relative to fs above RΘ at stage But in this case, σ0 is inner P, Q s+1 and there is no prepared inner preconfiguration for σ0∗s at RΘ at stage s+1 of index α ˆ , and hence either the conditions of case (C) of definition 2.2 are satisfied,
S. Barry Cooper
36
contradicting the assumption that s + 1 > sx + 1 and x is never cancelled, or τˆ 6= Φσαˆˆ at stage s + 1 due to the cancellation of axioms for Φαˆ at the receptive stage s , contradicting (ˆ σ , τˆ) ∈
b + P,Q,Φ b ′′ ′′ , . . . , Ψ b ′′α′′ ]. εs,αˆ αˆ (RΘ , fsi )[Ψ α1 p
On the other hand, if σ0∗s is weakly restrained at stage s + 1 , one again has b -preconfigured relative to fs above RΘ at (by definition 1.7) that σ0 is inner P, Q stage s + 1 and there is no prepared inner preconfiguration for σ0∗s at RΘ at stage s + 1 of index α ˆ , and the argument is as previously. So similarly to before one gets that that the definition of ψe does not terminate prematurely, and since the injectiveness of ψe follows trivially from that of ψ and the construction of ψe , (a) again follows.
b ′′ ′ be the first such functional which is inconsistent with its correFor (b), let Ψ αp b ′′ ′τ ’ or ‘ τ = Ψ b α′ [s] . One cannot define an axiom ‘ σ = Ψ b ′′ ′σ ’ (according sponding Ψ p
αp αp ′′ b ′ inconsistently Ψ αp ′ ′ e
b α′ [s] with Ψ p e ) , with via case 1, since in that case one would have some σ = ψ(σ) , τ = ψ(τ −1 ′ −1 ′ b ′′ ′ψ (τ ) or ψ −1 (τ ′ ) = Ψ b ′′ ′ψ (σ ) , respectively, due to ψ −1 (σ ′ ) = Ψ to whether
αp′
is configuring or nonconfiguring) of
αp
αp
b P,Q,Ψ α ¯′
(σ ′ , τ ′ ) ∈ εs,α¯ ′p
p
b ′′′ , . . . , Φ b ′′′ ]. (RΘ , gsj )[Φ β1 β ′ k
′
′
And it is easy to see that for any such σ , τ , b P,Q,Ψ α ¯′
(σ ′ , τ ′ ) ∈ εs,α¯ ′p
(3.8)
p
one has
b ′′′ , . . . , Φ b ′′′ ] ⇔ (RΘ , gsj )[Φ β1 β ′
b P,Q,Ψ α ¯′
(σ ′ , τ ′ ) ∈ εs,α¯ ′p
αp′
p
k
b′ , . . . , Φ b ′ ], (RΘ , gsj )[Φ β1 βk
since assuming (3.8) for any σ ′′ , τ ′′ , αp′ ′ , with p′ < p , or p′ = p and σ ′′ , τ ′′ ⊂ σ ′ , τ ′ , respectively, in place of σ ′ , τ ′ , αp′ , respectively, one need only examine case 1 of the definition of ψe to see that since ψ is an fsi , gsj -matching of the respective inner preconfigured ranks above RΘ at stage s + 1 , with corresponding augmentations ˆ
b
Q b ′α , . . . , Ψ b ′α ], εP,Q b ′β , . . . , Φ b ′β ], εP, (RΘ , fsi )[Ψ (RΘ , gsj )[Φ s s 1 1 l k
b
Q (3.8) inductively follows. But then the assumed inconsistency contradicts εP, (RΘ , s b P, Q ′ ′ b ′ ,...,Ψ b ′ ] being an augmentation of εs (RΘ , fs ) . fsi )[Ψ i α α 1
l
b ′′ ′ inconsistently b ′′ ′σ ’ of Ψ So one must assume that one defines an axiom ‘ τ = Ψ αp αp b with Ψα′p [s] via case 2. But, again, this is impossible, since in this case one has b α′ [s] which σ = ψe−1 (σ ′ ) , some σ ′ , with σ incompatible with any argument of Ψ p
properly extends each already defined ψe−1 (σ ′′ ) , σ ′′ ⊆ σ ′ , again excluding any such b α′ [s] . inconsistency with Ψ p
Upper Cones As Automorphism Bases
37
b ′′ ′ is consisb ′′ ′ contains no axiom inconsistent with Ψ b α′ [s] , then Ψ Finally, if Ψ αp αp p b
Q tent. For axioms defined via case 1, this again follows by the fact that εP, (RΘ , fsi ) s b b′ , . . . , Ψ b ′ ] is an augmentation of εP,Q (RΘ , fs ) . Otherwise, it trivially follows [Ψ α1 αl s i by the conditions of case 2. b ′′ , say, inconsistent with Φ b β ′ [s] is similar. The argument for Φ βq′ q
For (c), assume that α ˆ is such that ψe defines an isomorphism from the restricb P,Q b ′′ ′ , . . . , Ψ b ′′ ′ ] to components of index ≺ α tion of + εs (RΘ , fsi )[Ψ ˆ to the restricα α′ 1
l
b + P,Q b ′′′ , . . . , Φ b ′′′ ] εs (RΘ , gsj )[Φ β1 β ′
¯ˆ , but that the to components of index ≺ α ¯ˆ in the isomorphism breaks down upon inclusion of the components of index α ˆ, α corresponding stratified preconfigurations. Assume also that α ˆ is a minimal such index (under ≺ ). Let tion of
k
(ˆ σ , τˆ) ∈
b + P,Q,Ξ b ′′ ′ , . . . , Ψ b ′′ ′ ] εs,αˆ αˆ (RΘ , fsi )[Ψ α1 αl′
b ′′ . This be the pair at which the isomorphism breaks down with Ξαˆ = Φαˆ or Ψ α ˆ b ′′ ′′ + P,Q b b means that that part of εs (RΘ , fsi )[Ψα′ , . . . , Ψα′′ ] defined during the inductive 1 l bΨ b ′′ ′ , . . . , Ψ b ′′ ′ ] - preconfigured relative to fs above RΘ definition of τˆ being P, Q[ α1
αl′
i
at stage s + 1 via indices ≺ α ˆ is isomorphically mapped via ψe to that part of b P,Q + e τ ) being b ′′′ , . . . , Φ b ′′′ ] defined during the inductive definition of ψ(ˆ εs (RΘ , gsj )[Φ β1 βk′ bΦ b ′′′ , . . . , Φ b ′′′ ], Q - preconfigured relative to gs above RΘ at stage s + 1 via P[ β1
βk′
j
¯ˆ , but that this is no longer the case when τˆ , ψ(ˆ e τ ) are replaced by σ indices ≺ α ˆ, e σ ) , respectively. ψ(ˆ By the minimality of α ˆ , one can assume. without loss of generality, that this bΨ b ′′ ′ , . . . , Ψ b ′′ ′ ] - preconfigured relative to fs above RΘ at implies that σ ˆ is P, Q[ α1
αl′
′
i
stage s + 1 with some index α 4 α ˆ , say, due to (ˆ σ, τ ′ ) ∈
b ′ + P,Q,Ξ b ′′ ′ , . . . , Ψ b ′′ ′ ], εs,α′ α (RΘ , fsi )[Ψ α1 αl′
e σ ) is not P[ bΦ b ′′′ , . . . , Φ b α′ , but that ψ(ˆ b ′′′ ], Q say, with Ξα′ of the form Φα′ or Ψ β1 βk′ preconfigured relative to gsj above RΘ at stage s + 1 with index α ¯ ′ and e σ ), ψ(τ e ′ )) ∈ (ψ(ˆ
b Ξ b ′ + P,Q, b ′′′ , . . . , Φ b ′′′ ], εs,α¯ ′ α¯ (RΘ , gsj )[Φ β1 βk′
b α¯ ′ or Ψα¯ ′ . b α¯ ′ = Φ with Ξ bΨ b ′′ ′ , . . . , Ψ b ′′ ′ ] - preconfigured relative to fs above RΘ σ ˆ cannot be inner P, Q[ i α α′ 1
l
at stage s+1 , by the definition of the augmentations and
b + P,Q b ′′′ , . . . , Φ b ′′′ ] , εs (RΘ , gsj )[Φ β1 β ′ k
b + P,Q b ′′ ′ , . . . , Ψ b ′′ ′ ] εs (RΘ , fsi )[Ψ α1 α′ l
and of the mapping ψe . But on the other hand,
S. Barry Cooper
38
e σ ) via case 2 by the accompanying conditions put on such a one cannot define ψ(ˆ definition. ⊓ ⊔ One now needs the following simple facts, from which the required contradiction will immediately follow: (1) There are only finitely many distinct isomorphism types possible for the b -preconfigured and P, b Q -preconfigured ranks of each stratified inner P, Q fsj , gsj (respectively) above RΘ . (2) At all stages s + 1 ≥ sj + 1 at which RΘ requires attention through case b -preconfigured rank (D) of definition 2.2 of the construction, the inner P, Q b Q -preconfigured rank of gs above above RΘ of fsj matches the inner P, j RΘ , each j ≥ 1 , via a matching which respects all relevant restraints and weak restraints on ∗s . b -preconfigured rank of a For (1), remember that a typical stratified inner P, Q b Q string π above RΘ at stage s + 1 , corresponding to an augmentation εP, (RΘ , π) s b ′ ′ P, Q bα , . . . , Ψ b α ] of εs (RΘ , π) , is of the form [Ψ 1 l b P,Q
εs
b′ , . . . , Ψ b′ ] = (RΘ , π)[Ψ α1 αl b α1 P,Q,Φ
(εs,α1
b
P,Q,Φαk b′ , . . . , Ψ b ′ ], . . . , εs,α b′ , . . . , Ψ b ′ ], (RΘ , π)[Ψ (RΘ , π)[Ψ α1 αl α1 αl k
bΨ b ′β P,Q, 1
εs,β1
b b′
b ′α ], . . . , εP,Q,Ψβl (RΘ , π)[Ψ b ′α , . . . , Ψ b ′α ]) b ′α , . . . , Ψ (RΘ , π)[Ψ s,βl 1 1 l l
where b α1 P,Q,Φ
εs,α1
bΨ b ′β P,Q,
1
εs,β1
b
P,Q,Φαk b′ , . . . , Ψ b ′ ], . . . , εs,α b′ , . . . , Ψ b ′ ], (RΘ , π)[Ψ (RΘ , π)[Ψ αl α1 αl α1 k
b b′
b′ , . . . , Ψ b ′ ], . . . , εP,Q,Ψβl (RΘ , π)[Ψ b′ , . . . , Ψ b′ ] (RΘ , π)[Ψ α1 αl α1 αl s,βl
are lists of all such sets of higher priority than RΘ , where b αi P,Q,Φ
εs,αi
bΨ b ′β P,Q,
b ′α , . . . , Ψ b ′α ], ε (RΘ , π)[Ψ s,βj 1 l
j
b ′α , . . . , Ψ b ′α ] (RΘ , π)[Ψ 1 l
are the unions of all representative cross-sections b αi P,Q,Φ
εs,αi
bΨ b ′β P,Q,
b ′α , . . . , Ψ b ′α ], ε (RΘ , π ′ )[Ψ s,βj 1 l
j
b ′α , . . . , Ψ b ′α ] (RΘ , π ′ )[Ψ 1 l
of the corresponding components of and where
b Q b′ , . . . , Ψ b ′ ], εP,Qb (RΘ , π)[Ψ b′ , . . . , Ψ b ′ ], εP, (RΘ , π)[Ψ s α1 αl s α1 αl b αi P,Q,Φ
εs,αi
bΨ b ′β P,Q,
b ′α , . . . , Ψ b ′α ], ε (RΘ , π)[Ψ s,βj 1 l
j
b ′α , . . . , Ψ b ′α ] (RΘ , π)[Ψ 1 l
are the sets of all pairs (σ, τ ) for which (respectively) either σ = Φταi [s] , or σ = b ′ τ [s] , with τ inner P, Q[ bΨ b′ , . . . , Ψ b ′ ] -preconfigured relative to π above R at Ψ α1 αl βj
Upper Cones As Automorphism Bases
39
stage s + 1 according to hαi ≥i′ i or hβj ≥j ′ i , respectively, some i′ or j ′ ∈ ω , with (correspondingly) αi− or βj− Pαi - or Qβj - relevant and configuring, or τ = b σ [s] , with σ inner P, Q[ bΨ b′ , . . . , Ψ b ′ ] -preconfigured relative to π Φσαi [s] , or τ = Ψ α1 αl βj above R at stage s + 1 according to some hαi ≥i′ i or hβj ≥j ′ i , respectively, with (correspondingly) αi− or βj− Pαi - or Qβj - relevant and nonconfiguring. From this, one can verify: Sublemma 3.9. For each γ = some such αi or βj it holds that at any stage hγi(0)◦...◦pγ s + 1 that part of εs (π) contributing to such a typical stratified inner b P, Q -preconfigured rank has a uniform bound determined by the number of M requirements above RΘ and the number of potential outcomes at each node of higher priority than that of RΘ . Proof. By induction. Assume given γ = αi or βj as above. First notice that there are at most finitely many levels of the inductive definition b -preconfiguration via clause 4) of part (i) of definition 1.7 at which of inner P, Q pairs can be contributed to b α P,Q,Φ b′ , . . . , Ψ b ′ ], εs,αi i (RΘ , π)[Ψ α1 αl
bΨ b ′β P,Q,
εs,βj
j
b′ , . . . , Ψ b ′ ]. (RΘ , π)[Ψ α1 αl
b -configured This is because pairs contributed at level 1 are via the strings P, Q ′ relative to π above RΘ at stage s + 1 according to some α of higher priority than that of RΘ . And those newly contributed at level n + 1 , say, are via the strings b -preconfigured above RΘ at stage s + 1 according to hγ − ≥i′ i or hγ >i′ i , P, Q say, with γ − or γ ≤i′ , respectively, Pαi - or Qβj - relevant, say, relative to π , where either τ has been at level n of the inductive definition defined to be inner b -preconfigured relative to π above RΘ at stage s + 1 according to hˆ P, Q α ≥j ′ i , say, where req( γ ) is of higher priority than that of req( α ˆ ), there is some σ ′ ⊇ the string in question and a requirement Mx′ of higher priority than that of RΘ at which ∗ is weakly restrained on σ ′ at stage s + 1 . One can also see that the bound on the number of relevant levels of the inductive b -preconfigured need only be dependent on the number and definition of inner P, Q character of the requirements of higher priority than that of RΘ . It is then straightforward to verify that the number of strings figuring in the relevant stratifications at any one stage is uniformly bounded. This is because the strings added at level n + 1 of the induction can only arise from downward Turing relatinships from previously included strings, those new strings corresponding to entries taken from the finite number of strings of outcomes of higher priority than that of RΘ , any one such entry giving rise to a finite number of stratified strings whose bound is again dependent on the number and character of the requirements of priority greater than that of RΘ . Specifically, assume that α′ is of priority greater than that of RΘ , and that b -preconfigured rank up all strings contributing to the relevant stratified inner P, Q
S. Barry Cooper
40
to level n of the inductive definition, together with all those contributing to the b -preconfigured rank at level n + 1 which are associated with an stratified inner P, Q entry in a string of outcomes of higher priority than that of α′ , have been found to be bounded strictly in accordance with the requirements of priority greater than that of RΘ . One may then consider the set of strings arising from the indexed functional corresponding to the final entry of α′ at level n + 1 of the inductive definition. The subset of such strings which survive to the stratified level is strictly related to their possible combined relationships, all via downward Turing relationships from those b -preconfigured rank strings already contributing to the relevant stratified inner P, Q up to level n + 1 . ⊓ ⊔ But sublemma 3.9 implies (by the consistency of all functionals, by definition 5, and noting the restriction to not more than one occurrence of a given instance of a functional along a particular branch of the inductive definition of inner bΨ b′ , . . . , Ψ b ′ ] -preconfiguration relative to a given string) that the possible isoP, Q[ α1 αl b -configured ranks of each fs above RΘ (relative to cormorphism types of the P, Q j b responding augmentations), and hence of the inner P, Q -preconfigured ranks of each b β ′ }1≤j ′ ≤l fsj above RΘ , are limited by the corresponding sets {Φαj′ }1≤j ′ ≤k , {Ψ j of functionals and the relationships between these functionals relative to fsj (as deb Q -preconfigured scribed by the tree Tf above RΘ ). The argument for the inner P, ranks is just the same. For (2), first notice that lemma 3.2 gives the required matching for each s = sj , each j ≥ 1 . Then: bSublemma 3.10. At the end of each reactive stage s + 1 > sj + 1 , the P, Q b Q -preconfigured rank of gs preconfigured rank above RΘ of fsj matches the P, j above RΘ , each j ≥ 1 , via a matching which respects all restraints on ∗s+1 of no lesser priority than that of RΘ and, beyond its core, respects all weak restraints on ∗s+1 of higher priority than that of RΘ . Proof. In order to get a contradiction, assume that at the end of some least b -preconfigured rank above RΘ reactive stage s + 1 > sj + 1 , some j ≥ 1 , the P, Q b Q -preconfigured rank of gs above RΘ , where of fsj fails to suitably match the P, j j is chosen to be minimal. Assume the sublemma at the end of the reactive stage t = s′ + 1 (say) ≥ sj + 1 , where s = t+1 , say, and assume that fs+1 , gs+1 are Rγ -determined at stage s+1 , b - and P, b Q -preconfigured some Rγ . Let ψ ′ be a matching of the respective P, Q ranks of fsj and gsj above RΘ at stage t + 1 , with corresponding augmentations b
b b + P,Q b ′′ , . . . , Ψ b ′′ ], + εP,Q b ′′ , . . . , Φ b ′′ ] εt (RΘ , fsj )[Ψ (RΘ , gsj )[Φ t α1 αl′ β1 βk′
Q of + εP, (RΘ , fsj ) , t straints on ∗t .
b + P,Q εt (RΘ , gsj ) ,
respectively, which respects all relevant re-
Upper Cones As Automorphism Bases
41
As for lemma 3.1, if Rγ requires attention through case (B) or (C) of definition 2.2, or through subcase II of case (D), at stage s + 1 , then ∗s+1 is a trivial b β or Ψ b α is reorganisation of ∗s , and any cancellation of axioms for functionals Φ via initialisation of such functionals of lower priority than that of Rγ , or via the bselective cancellation of such axioms at stage s + 1 . So any failure of the P, Q b Q -preconfigured rank of gs preconfigured rank above RΘ of fsj to match the P, j above RΘ at the end of stage s + 1 must be due to 1. The enumeration of new axioms for functionals of the form Φα , Ψβ , at b β or Ψ b α at stage stage t + 1 , or of new axioms for functionals of the form Φ s + 1 , or 2. The inclusion of new strings in dom (∗s+1 ) or range (∗s+1 ) , or 3. The definition of new restraints at stage s + 1 during the action for case (C) of the construction at stage s + 1 , or a combination of such events. −1
Since any restraint on a value σ ∗u or σ ∗u of ∗u or ∗−1 u , respectively, at stages u + 1 > s + 1 , set up via case (C) of the construction at stage s + 1 must involve a b - or P, b Q - preconfigured relative to fs or string σ already (respectively) inner P, Q gs , respectively, above Rγ at stage s + 1 , one can discount 3 as a factor in isolation from 1. Also, as for the proof of lemma 3.1, one can eliminate 2 as a sole factor. This means one again considers factor 1 leading to the disruption of any given such matching ψ ′ by either some axiom of the form ‘ σ = Φτα ’ for Φα , say, enumerb τ ’ for Ψ b α , say, defined at stage s+1 , causing the correated at stage t+1 , or ‘ σ = Ψ α b b ′′ , . . . , Ψ b ′′ ] of + εP,Qb (RΘ , fs ) sponding potential augmentation + εP,Q (RΘ , fs )[Ψ α1 αl′ j j s+1 b P, Q b ′′α , . . . , Ψ b ′′α ] with no to be different from + εt (RΘ , fsj )[Ψ 1 l′ b P,Q possible in any augmentation of + εs+1 (RΘ , gsj ) whose cors+1
to be inconsistent, or
matching amendment b Q -preconfigured rank of gs above RΘ at the end of stage responding stratified P, j b -preconfigured rank of fs above RΘ at the s+1 is isomorphic to the stratified P, Q j b
Q end of stage s + 1 corresponding to the amended augmentation of + εP, s+1 (RΘ , fsj ) , ∗s+1 or to some axiom of the form ‘ σ ∗s+1 = Ψτβ ’ enumerated for Ψβ at stage t + 1 , ∗s+1 b β at stage s + 1 , some Ψβ , Φ b β , causing the correbτ ’ defined for Φ or ‘ σ ∗s+1 = Φ β
b b + P,Q b ′′ , . . . , Φ b ′′ ] of + εP,Q εs+1 (RΘ , gsj )[Φ s+1 (RΘ , gsj ) β1 βk′ b b ′′ , . . . , Φ b ′′ ] with no (RΘ , gsj )[Φ to be different from + εP,Q t β1 βk′ b + P,Q possible in any augmentation of εs+1 (RΘ , fsj ) whose cor-
sponding potential augmentation to be inconsistent, or
matching amendment b -preconfigured rank of fs above RΘ at the end of stage responding stratified P, Q j b s+1 is isomorphic to the stratified P, Q -preconfigured rank of gsj above RΘ at the
end of stage s + 1 corresponding to the amended augmentation of
First assume that such an axiom ‘ σ = Φτα ’ occurs, where, if ξ
b + P,Q εs+1 (RΘ , gsj ) . = {α1′ , . . . , αp′ } ,
S. Barry Cooper
42
′ ′ ′ ′ ′ ′ , . . . , βr(q) } , βr(i αi′ ′ 4 αj′ ′ each i′ ≤ j ′ , and η = {βr(1) ′ ) 4 βr(j ′ ) each i ≤ j , are the respective sets of indices of all axioms for functionals of the form Φα′ , α′ ∈ Tf , or of all axioms for functionals of the form Ψβ ′ , β ′ ∈ Tg , enumerated at stage ′ ′′ ′ ′ t + 1 , where {β ′ ∈ η | β ′ 4 α ¯ i′ ′ } = {βr(i ′′ ) | r(i ) < i } , one has that α ( = αj ′ , say) b -preconfigured is the ( 4 ) least member of ξ for which the P[Φα′ [s], . . . , Φα′ [s]], Q 1
j′
b Q[Ψβ ′ [s], . . . , Ψβ ′ [s]] -preconfigured rank above RΘ of fsj fails to match the P, r(1) r(j ′ ) rank above RΘ of gsj at stage t + 1 via a matching which respects all restraints on ∗s+1 of no lesser priority than that of RΘ and, beyond its core, respects all weak restraints on ∗s+1 of higher priority than that of RΘ . This means that the obstruction to the required matching appears by the end of stage s′ + 1 in the form of a failure of isomorphism between any pair b b + P,Q b ♯ ′′ , . . . , Ψ b ♯ ′′ ], + εP,Q b ♯ ′′ , . . . , Φ b ♯ ′′ ] εs (RΘ , fsj )[Ψ (RΘ , gsj )[Φ s α1 α ′′ β1 β ′′ l
k
derived from respective augmentations b b + P,Q b ♯ ′′ , . . . , Ψ b ♯ ′′ ], + εP,Q b ♯ ′′ , . . . , Φ b ♯ ′′ ] εs (RΘ , fsj )[Ψ (RΘ , gsj )[Φ s α1 α ′′ β1 β ′′ l
k
b P,Q
b P,Q
of + εs (RΘ , fsj ) , + εs (RΘ , gsj ) , respectively. This failure of isomorphism canb α¯ already enumerated at a stage < s + 1 , and so, not be due to an axiom for Φ by choice of Φα and its consistency, necessarily < t + 1 . Otherwise, by the inductive hypothesis one has fs , gs = ft , gt , respectively, current and the latter Rγ ′ -determined, say, and by lemma 3.2 a ∗t -matching ψ ′ , say, of the respective b - and P, b Q -preconfigured ranks of fs and gs above Rγ at stage s , with P, Q corresponding augmentations b b + P,Q b ′′ , . . . , Ψ b ′′ ], + εP,Q b ′′ , . . . , Φ b ′′ ], εt (Rγ , fs )[Ψ (Rγ , gs )[Φ t α1 αl β1 βk b
b
Q say, of + εP, (Rγ , fs ) , + εP,Q (Rγ , gs ) , respectively. Since the assumption is that t t Rγ ′ does not require attention through case (A) of definition 2.2 at stage s′ + 1 , b - and P, b Q - preconfigured ranks of fs , there must be a ∗s -matching of the P, Q gs , respectively, at stage s + 1 . But then, again by the inductive hypothesis, one has fsj , gsj , current and Rγ ′ -determined, say, at a stage t′ + 1 , say, and now a b - and P, b Q -preconfigured ranks of ft′ +1 matching ψ ′ , say, of the respective P, Q and gt′ +1 above Rγ at stage s , with corresponding augmentations
b b + P,Q b ′′ , . . . , Ψ b ′′ ], + εP,Q b ′′ , . . . , Φ b ′′ ], εt (Rγ , fsj )[Ψ (Rγ , gsj )[Φ t α1 αl β1 βk b
b
Q (Rγ , fsj ) , + εP,Q (Rγ , gsj ) , respectively, and, by the above, with the say, of + εP, t t b - and P, b Q - preconfigured ranks of fs , gs , respectively, ∗s -matching ψ , of the P, Q above Rγ at stage s + 1 , with corresponding augmentations
b b + P,Q b ′α , . . . , Ψ b ′α ], + εP,Q b ′β , . . . , Φ b ′β ] εs (Rγ , fs )[Ψ (Rγ , gs )[Φ s 1 1 l k
Upper Cones As Automorphism Bases
43
b
b
Q (Rγ , gs ) , respectively, such that for each such ψ ′ there exof + εP, (Rγ , fs ) , + εP,Q s s b ′′ , Ψ b ′ or Φ b ′′ , Φ b ′ . But this contradicts ists an inconsistent pair of the form Ψ αj ′ αj ′ βj ′ βj ′ the assumption that Rγ required attention through case (B) or (C) at stage s′ + 1 , since the conditions for Rγ ′ to require attention through case (A) of definition 2.2 at stage s′ + 1 are satisfied. Otherwise, one must assume that the required matching is precluded by the b τ ’ for functionals Ψ b α , α ∈ Tf , enumeration of axioms of the form ‘ σ = Ψ α via routine extension at stage s + 1 , which depends on the ∗s+1 -matching b -preconfigured rank of fs+1 and the P, b Q -preconfigured rank of of the P, Q gs+1 above Rγ , at stage s + 1 . Then the definition of these new axioms for the hatted functionals at stage s + 1 is determined by the isomorphism b P,Q b ′′ , . . . , Φ b ′′ ] , say, corresponding to some augmen∗s+1 between + εs (Rγ , gs+1 )[Φ β1
b P,Q
βk
b
P,Q b b ′′ , . . . , Φ b ′′ ] of + εP,Q tation + εs (Rγ , gs+1 )[Φ (Rγ , gs+1 ) and + εs (Rγ , fs+1 ) s β1 βk b Q b ′′α , . . . , Ψ b ′′α ] , say, corresponding to an augmentation + εP, b ′′α , . . . [Ψ (Rγ , fs+1 )[Ψ s 1 1 l b Q b ′′α ] of + εP, f R ...,Ψ (R , g , f ) . But then, one again has , current and s γ s+1 γ′ s s j j l ′ ′ b - and determined, at a stage t + 1 , and a matching ψ , of the respective P, Q b P, Q -preconfigured ranks of ft′ +1 and gt′ +1 above Rγ at stage s , with corresponding augmentations
b b + P,Q b ′′α ], + εP,Q b ′′α , . . . , Ψ b ′′β ], b ′′β , . . . , Φ εt (Rγ , fsj )[Ψ (Rγ , gsj )[Φ t 1 1 l k b
b
Q (Rγ , gsj ) , respectively. And by the above, and by the say, of + εP, (Rγ , fsj ) , + εP,Q t t b - and P, b Q - preconaction for cases (B) and (C), one has a ∗s -matching of the P, Q figured ranks of fs , gs , respectively, above Rγ at stage s + 1 , with corresponding augmentations
b
b b + P,Q b ′′ , . . . , Ψ b ′′ ], + εP,Q b ′′ , . . . , Φ b ′′ ] εs (Rγ , fs )[Ψ (Rγ , gs )[Φ α1 αl s β1 βk b
Q of + εP, (Rγ , fs ) , + εP,Q (Rγ , gs ) , respectively, such that for each such ψ ′ there s s b ′′α , Ψ b ′α or Φ b ′′ , Φ b ′ . Again, the exists an inconsistent pair of the form Ψ βj ′ βj ′ j′ j′ conditions for Rγ ′ to require attention through case (A) of definition 2.2 at stage s′ + 1 are satisfied, another contradiction.
Assume now that Rγ requires attention at stage s + 1 through case (A) or through subcase I or II of case (D), where, by choice of RΘ , Rγ is of strictly lower b -preconfigured rank above RΘ priority than that of RΘ . Once again, if the P, Q b Q -preconfigured rank of gs above RΘ at the end of fsj fails to match the P, j of stage s + 1 , one can assume this to be due to the enumeration of some axiom of the form ‘ σ = Φτα ’ for Φα , say, at stage t + 1 , or of some axiom of the form b τα ’ for Ψ b α , say, at stage s + 1 , where the possibility of any matching axiom ‘σ = Ψ ∗s+1 τ τ ∗s+1 bα ‘ σ ∗s+1 = Φ ’ or ‘ σ ∗s+1 = Ψα ’ is excluded by the need for consistency of ¯ ¯ bτ ’ the relevant functionals (the argument for a similarly functioning axiom ‘ σ = Φ β
S. Barry Cooper
44
or ‘ σ = Ψτβ ’ being seen to be analogous). But the latter possibility is not possible according to the action for case (D) at stage s + 1 . And for the former, one can b Q b′ , . . . , Ψ b ′ ] as before, assume that the relevant axiom is choose + εP, (RΘ , fs )[Ψ t j
α1 αl b + P,Q b ′α ] at the least possible b ′α , . . . , Ψ εs+1 (RΘ , fsj )[Ψ 1 l b ′ ] -preconfigured relative to fs bΨ b′ , . . . , Ψ of P, Q[ αl α1 j
level of chosen to contribute to (above the inductive definition RΘ at stage s + 1 ), and argue exactly as for cases (B) and (C) above to obtain the required contradiction. ⊓ ⊔
One only needs to observe now that any restraints on ∗s preventing part I applying at some stage s + 1 > s∗ at which case (D) of definition 2.2 applies must in fact ◦ obstruct the core ψ of any relevant potential matching ψ . And, arguing as previously, any such restraints must emanate from a prepared inner preconfiguration above some Rγ , and associated restraints on f , g and ∗ of priority that of some M -requirement above RΘ . Due to the choice of s∗ and the promotion of P - and Q - requirements during the linking process, this prepared inner preconfiguration must at all stages subsequent to its appointment via case (C) be associated with an inner preconfiguration above RΘ maintained by restraints which are not injured. ◦ It follows that for any such ψ , ψ must respect all relevant restraints on ∗ , and the required contradiction follows. Now show that potential witnesses can be appointed for RΘ at at most finitely many stages. In order for RΘ to require attention through clause (B) of definition 2.2 at infinitely many stages, it is necessary that each potential witness x appointed for RΘ is eventually cancelled, either 1. Through some R -requirement of higher priority than that of RΘ requiring attention at some stage s + 1 after that at which x was appointed, or 2. Through RΘ requiring attention through (C) at such a stage s + 1 . By the choice of s∗ , 1 cannot apply at any stage s + 1 > s∗ . For 2, first notice that if RΘ requires attention through (C) at a stage s+1 > s∗ , resulting in fs , say, being restrained ⊆ each ft+1 ( t + 1 ≥ s + 1 , and stage t + 1 being prior to the restraint being injured by activity on a requirement of higher priority than that of RΘ ), then fs ⊆ ft+1 for all t + 1 ≥ s + 1 . This is because one can only define ft+1 | fs at a stage t + 1 ≥ s + 1 if either 1′ . Some R -requirement of higher priority than that of RΘ requires attention at stage t + 1 , or 2′ . RΘ requires attention through part II of case (D) at stage t + 1 , or 3′ . RΘ requires attention through part I of case (D) at stage t + 1 . Then 1′ is impossible by the choice of s∗ . And 2′ cannot happen since at stage s + 1 the existing potential witness for RΘ is cancelled, along with all associated realised strings, and any new potential witness x for RΘ appointed at a stage u+1 , s + 1 < u + 1 < t + 1 , is (by the action for case (B)) greater than any existing
Upper Cones As Automorphism Bases
45
restraint (including that on fs ⊆ ft+1 ), so that the action for case (D) at stage t + 1 resulting in ft+1 ⊃ ft x − 1 must retain any restraint imposed at stage s + 1 which is still in existence at stage t + 1 . While for 3′ , the discussion of possibility 2′ shows that all strings fsi corresponding to realised strings g ϑ(x)[si ] for RΘ at stage t + 1 also respect such restraints, and hence fs ⊆ ft+1 for all t + 1 ≥ s + 1 , as required. It follows immediately that RΘ requires attention through part (C) at at most finitely many stages. This is because, by the above, each prepared inner preconfigb Q uration for a functional Φα involved in the definition of εP, (RΘ , fs ) defined at a s ∗ stage s + 1 > s is retained at each stage t + 1 > s + 1 , and hence, by the conditions of case (C), no other prepared inner preconfiguration for such a Φα can be defined at such a stage t + 1 . This means that the number of stages t + 1 > s∗ at which RΘ requires attention through part (C) of definition 2.2 is bounded by the number b Q of components of εP, (RΘ , fs ) , each s ≥ 0 . s It follows, as required, that RΘ can require attention at at most finitely many stages via case (B) or (D). It remains to show that RΘ requires attention at at most finitely many stages via case (A) of definition 2.2. Sublemma 3.11. If a requirement Rγ is finitely injured, and it requires attention at most finitely often through cases (B), (C) or (D) of definition 2.2, then Rγ requires attention at most finitely often via case (A) of that definition. Proof. Let Rγ be given, and assume t∗ to be a stage such that no higher priority R - requirement requires attention at a stage s + 1 > t∗ , and that if Rγ requires attention at a stage s + 1 > t∗ , then it is via case (A) of definition 2.2. If Rγ does require attention through case (A) at a reactive stage s + 1 > t∗ , then by definition 2.2 there exists a current Rγ -determined pair ft+1 , gt+1 , with b - and t + 1 < s + 1 , for which there exists a matching ψ ′ , say, of the respective P, Q b Q -preconfigured ranks of ft+1 and gt+1 above Rγ at the end of stage s = s′ +1 , P, with corresponding augmentations b b + P,Q b ′′ b ′′ b ′′ , . . . , Ψ b ′′ ], + εP,Q εs′ (Rγ , ft+1 )[Ψ α1 αl′ s′ (Rγ , gt+1 )[Φβ1 , . . . , Φβk′ ]
of
b b + P,Q εs′ (Rγ , ft+1 ) , + εP,Q s′ (Rγ , gt+1 ) ,
respectively, where either b - and P, b Q - preconfigured (1) There exists a ∗s -matching ψ , say, of the P, Q ranks of fs , gs , respectively, above Rγ at stage s + 1 , with corresponding augmentations b b + P,Q b ′α ], + εP,Q b ′α , . . . , Ψ b ′β ] b ′β , . . . , Φ εs (Rγ , fs )[Ψ (Rγ , gs )[Φ s 1 1 l k b
b
Q (Rγ , gs ) , respectively, where for each such ψ ′ there of + εP, (Rγ , fs ) , + εP,Q s s b ′′ , Ψ b ′ or Φ b ′′ , Φ b ′ , or exists an inconsistent pair of the form Ψ αj αj βj βj
S. Barry Cooper
46
b - and P, b Q(2) t + 1 = s′ , and there is no ∗s -matching above Rγ of the P, Q preconfigured ranks of fs , gs , respectively, at stage s + 1 . b - and P, b Q - preconfigured By lemma 3.2, there is a matching of the inner P, Q ranks of fs and gs , respectively, above Rγ , at stage s + 1 (which respects any existing restraints on ∗s and is consistent with ∗s at stage s + 1 ). Arguing as for sublemma 3.6, one can extend this matching to a full fs , gs -matching above Rγ at stage s + 1 . This means that in either of subcases (1) or (2) of case (A) of definition 2.2 one can assume the existence of a full ( ∗s -consistent) fs , gs -matching ψ above Rγ at stage s + 1 , where in subcase (1), ψ is a ∗s -matching at stages b ′′α , Ψ b ′α or s+1 and for each such ψ ′ there exists an inconsistent pair of the form Ψ j j ′′ ′ b b b b Φ , Φ , and in subcase (2) ψ fails to be a ∗s -matching of the P, Q - and P, Q βj
βj
preconfigured ranks of fs and gs , respectively, above Rγ at stage s + 1 , where one can assume that this breakdown of ∗s -matching occurs at index αj ∈ Tf or βj ∈ Tg , say. Inductively assume that there exists some α ˆ , say, such that at each stage > t∗αˆ ∗ (say) > t , no α ⊂ α ˆ occurs as such an αj in (1) or (2) as a result of Rγ requiring attention through case (A) of definition 2.2, and that stage s + 1 > t∗αˆ is such that ˆ . So the action at stage s + 1 Rγ requires attention through case (A) with αj = α depends on the answer to the question: Does there exist a matching ψ ′ as in part (A) of definition 2.2, and an fs , gs matching ψ , say, above Rγ at stage s + 1 , such that ψ , ψ ′ are consistent?
To see that such ψ , ψ ′ do exist, assume first that subcase (1) applies. By b - and P, b Q - preconfigured ranks assumption, matchings ψ ′ of the respective P, Q of ft+1 and gt+1 above Rγ at the end of stage s do exist. This means that the b ′′ and Ψ b ′ must be due to the enumeration of a new axiom inconsistency between Ψ α ˆ α ˆ into Ψαˆ¯ at stage s + 1 . It follows that the required consistency between ψ and ψ ′ can be achieved in the context of the relaxation of the requirement for ψ to be a ∗s -matching at stage s + 1 . Similarly, in subcase (2) the existence of the fs , gs matching ψ above Rγ at stage s + 1 and lemma 3.1 again ensures the existence of the required consistent ψ , ψ ′ . Consequently, the action at stage s + 1 is to non-trivially perseverate the matchb - and P, b Q - preconfigured ranks of ft+1 and gt+1 ing between the respective P, Q above Rγ . This involves choosing ψ to maximally respect ∗s , and implementing ψ in such a way as to reorganise ∗ along the lines of part (b) of the action for subcase I of the action for subcase (D) of the construction. The result is a ∗s+1 -matching b - and P, b Q - preconfigured ranks of ft+1 , gt+1 , respectively, above Rγ of the P, Q at stage s + 1 in which the matching at all indices above α ˆ is derived from ψ ′ , and which now extends to level α ˆ itself. Now notice that in either of cases (1) or (2) one may assume that the matching for no Rγ ′ -determined ft′ +1 , gt′ +1 , with Rγ ′ of higher priority than that of Rγ , is ever perseverated via case (A) of definition 2.2 at a stage s′ + 1 > t∗ . It inductively
Upper Cones As Automorphism Bases
47
follows that α ˆ cannot appear as an αj in (1) or (2) at a stage s′ + 1 > s + 1 as a result of Rγ requiring attention through case (A), in relation to ft+1 , gt+1 , or (due to the definition of fs′ +1 , gs′ +1 at stages s′ + 1 > s + 1 ) in relation to any ft′ +1 , gt′ +1 with t′ + 1 > s + 1 . Hence, by a straightforward sub-induction, there exists a stage t♮ > t∗ such that α ˆ cannot appear as an αj in (1) or (2) at a stage ′ ♮ s + 1 > t as a result of Rγ requiring attention through case (A) in relation to any ft′ +1 , gt′ +1 . Since there are at most finitely many such possible indices α ˆ , the sublemma follows. ⊓ ⊔ This completes the proof of lemma 3.5.
⊓ ⊔
Lemma 3.12. If an M -requirement is finitely injured, then it is finitely injuring. Proof. As for lemma 3.5, assume Mx , say, to be a given M -requirement, and inductively assume s∗ to be a stage such that no higher priority requirement requires attention at a stage s + 1 > s∗ . And as before, no higher priority K -, L -, M or N - requirement can be linked to a lower priority R -requirement at a reactive stage s + 1 > s∗ , and once again the initial segment of the priority ordering of requirements up to and including Mx is unchanged at stages s + 1 > s∗ . To see that such a requirement requires attention at most finitely often via case (C) of the construction, the argument is similarly to that for the R -requirements. One need only note that the revision of the given prepared preconfiguration of index α , say, associated with the restraint on σ ∗s , say, with ∗ weakly restrained on σ at Mx , via case (C) can only occur if the new index α′ , say, is ≺ α . And then Mx can only require attention finitely often via case (A) of the construction by sublemma 3.11. ⊓ ⊔ Lemma 3.13. If a K - or L - requirement is finitely injured, then it is finitely injuring. Proof. Assume Kσ , say, to be a given K -requirement, and inductively assume s∗ to be a stage such that no higher priority requirement requires attention at a stage s + 1 > s∗ . As for lemmas 3.5 and 3.12, no higher priority K -, L - or M requirement can be linked to a lower priority R -requirement at a reactive stage s + 1 > s∗ , and as before the initial segment of the priority ordering of requirements up to and including Kσ is unchanged at stages s + 1 > s∗ . One must first verify that Kσ requires attention at at most finitely many receptive stages. So assume that Kσ requires attention at stage s + 1 through definition 2.1, and let IΞ′ ,σ′ ,τ ′ , say, be a I -requirement of lower priority than that of Kσ with Ξ′ = Φ , say, where one may assume that there exists a σ ′′ ⊇ σ such that σ ′′ ∗s ↓ , and there is a path through σ ′′ as yet unprotected from IΞ′ ,σ′ ,τ ′ at stage s + 1 , with σ ′′ a minimal such string. For each such requirement IΞ′ ,σ′ ,τ ′ and each such corresponding string σ ′′ one either has an axiom for Ξ′ arising from the implementation of an axiom ‘ σ = Ξ′τ ’,
48
S. Barry Cooper
stipulated for Ξ′ at a previous stage, which names some τ ′′ with σ ⊆ τ ′′ ⊆ σ ′′ as its argument or value, leading to the setting up of a weak restraint of priority that of Kσ to preserve τ ′ ∗t at all stages t + 1 ≥ the first reactive stage ≥ s + 1 prior to a stage > s + 1 at which a requirement of higher priority than that of Kσ requires attention through definition 2.2, or σ ′′ becomes unavailable for the ′ implementing of the axiom ‘ σ ′ = Ξτ ’ at each stage t + 1 ≥ s + 1 prior to a stage > s+1 at which a requirement of higher priority than that of Kσ requires attention through definition 2.2. As a result, by the inductive assumptions, each path through such a σ ′′ becomes protected from each such requirement IΞ′ ,σ′ ,τ ′ at every stage t + 1 > s + 1 . And by the choice of s∗ , each such path retains its protection at each subsequent stage. Similarly, each path through such a string σ ′′ becomes protected from each requirement JΞb ′ ,σ′ ,τ ′ of lower priority than that of Kσ at every stage t + 1 > s + 1 , and retains its protection at each subsequent stage, which precludes Kσ from requiring attention through definition 2.1 at any such stage. The proof in relation to the reactive stages now proceeds exactly as in the pre⊔ vious lemma. The argument in relation to a typical L -requirement is similar. ⊓ It immediately follows from lemmas 3.5, 3.12 and 3.13 that every requirement is finitely injured. Lemma 3.14. The respective limits ∗ , f , g of {∗s }s∈ω , {fs }s∈ω , {gs }s∈ω exist, ∗ is low, and there is a bijective e ∗ : ω ω −→ ω ω induced by ∗ . Proof. One first needs to verify the satisfaction of each M -requirement. Let Mx be a given M -requirement, and let sˆ be a stage after which no requirement of priority greater than or equal to that of Mx requires attention. Now assume that at some stage s+1 > sˆ one has J(∗)[s] = 1 , but that some requirement Rγ requires attention, as a result of which, for some (σ, τ ) ∈ ∗s used in computing x ∈ J(∗)[s] , either σ ∗s+1 ↑ or σ ∗s+1 is defined 6= σ ∗s . By the routine extension of ∗ at stage s + 1 , if (σ, τ ) ∈ ∗s , but this is not subject to revision at stage s + 1 , then (σ, τ ) ∈ ∗s+1 . And if (σ, τ ) ∈ ∗s is subject to revision at stage s + 1 , this can only be via case (A), or part I of case (D), of the construction. Assume that σ ∗s+1 is revised as a result of Rγ requiring attention through case (A) of definition 2.2. By the choice of stage s + 1 , Rγ must be of lower priority than that of Mx , and by the conditions of case (A) there can be no restraint on σ ∗s of priority greater than that of Rγ . Also, by the choice of the matching ψ which is implemented during the action for case (A) at stage s + 1 , σ ∗s+1 must be ◦ subject to revision by the core ψ of ψ . This means that if b b + P,Q b ′′ , . . . , Ψ b ′′ ], + εP,Q b ′′ b ′′ εs′ (Rγ , ft+1 )[Ψ α1 αl′ s′ (Rγ , gt+1 )[Φβ1 , . . . , Φβk′ ] b
b
Q + P,Q are the corresponding augmentations of + εP, εs′ (Rγ , gt+1 ) , respecs′ (Rγ , ft+1 ) , bΨ b ′′α , . . . , Ψ b ′′α ] -preconfigured relative to fs above Rγ tively, then σ is inner P, Q[ 1 l′ at stage s + 1 .
Upper Cones As Automorphism Bases
49
This means that following the implementation of ψ at stage s + 1 there is a b -preconfigured rank of fs+1 above Rγ by the P, b Q∗s+1 -matching of the P, Q preconfigured rank of gs+1 above Rγ at the end of stage s + 1 , with corresponding augmentations b b + P,Q b ′′ , . . . , Ψ b ′′ ], + εP,Q b ′′ , . . . , Φ b ′′ ] εs (Rγ , fs+1 )[Ψ (Rγ , gs+1 )[Φ α1 αl′ s β1 βk′
b b Q b of + εP, (Rγ , fs+1 ) , + εP,Q (Rγ , gs+1 ) , respectively, under which σ is inner P, Q s s b ′′ , . . . , Ψ b ′′ ] -preconfigured relative to fs+1 above Rγ at the end of stage s + 1 . [Ψ α1 αl′ As a result, there will, through the routine extension of the hatted functionals at stage s + 1 , be new axioms enumerated for hatted functionals being built by Q bα ′ , requirements of higher priority than that of Rγ , where all such functionals Ψ j b ′′ , but not for Ψ b α ′ [s] , occur in the inductive 1 ≤ j ≤ l′ , for which axioms for Ψ αj ′
j
b ′′α ] -preconfigured relative to fs+1 above bΨ b ′′α , . . . , Ψ definition of σ being inner P, Q[ 1 l′ Rγ at the end of stage s + 1 , are so enlarged, with the consequence that σ is inner b -preconfigured relative to fs+1 above Rγ at the end of stage s + 1 . P, Q But in this case, at the next reactive stage either Mx requires attention through case (C) of definition 2.2, or some requirement of higher priority than that of Mx requires attention, in either case contradicting the choice of sˆ . The argument relating to part I of case (D) is similar. It follows that all the M -requirements are satisfied, giving the lowness of ∗ and the existence of the limit ∗ of {∗s }s∈ω , as required.
To see that Lims fs (z) exists for each z , say that s + 1 is a non deficiency stage (for Rγ ) if and only if Rγ requires attention at stage s + 1 , and no Rγ ′ of greater priority than that of Rγ requires attention at a stage s′ + 1 > s + 1 . Let s + 1 be a non deficiency stage for some R -requirement RΘ . Then by definition 2.2 and lemmas 3.5, 3.12, at each sufficiently large stage > s + 1 RΘ has a fixed potential witness x , say (the eventual witness for RΘ ), and by the construction fs′ +1 ⊃ fs x − 1 at each stage s” + 1 ≥ s + 1 . Since (again by definition 2.2 and lemmas 3.5, 3.12), there exists a non deficiency stage for each RΘ , and the corresponding eventual witnesses are all distinct, the limit of {fs }s∈ω exists, as required. Given the bijective e ∗ : ω ω −→ ω ω induced by ∗ , it will easily follow that Lims gs exists = f e∗ . By definition 1.2, it remains to be shown that there is a bijective mapping e ∗ : ω ω −→ ω ω which satisfies (3.15)
f e∗ = g ⇔ (∃{(σi , τi )}i∈ω ⊂ ∗)(∀i)[σi ⊂ σi+1 ⊂ f & τi ⊂ τi+1 ⊂ g],
for all f, g ∈ ω ω . {(σi , τi )}i∈ω ⊂ ∗ is said to be a ∗ -tower if and only if (∀i)[σi ⊂ σi+1 & τi ⊂ τi+1 ] . If {(σi , τi )}i∈ω is a ∗ -tower, let f(σ,τ ) = ∪{σi | i ∈ ω},
g(σ,τ ) = ∪{τi | i ∈ ω}.
S. Barry Cooper
50
One needs to verify that: (1) If {(σi , τi )}i∈ω , {(σi′ , τi′ )}i∈ω are ∗ -towers, then f(σ,τ ) = f(σ′ ,τ ′ ) ⇔ g(σ,τ ) = g(σ′ ,τ ′ ) , and (2) For each fˆ ∈ ω ω ( gˆ ∈ ω ω ) there exists a ∗ -tower {(σi , τi )}i∈ω ⊂ ∗ with f(σ,τ ) = fˆ ( g(σ,τ ) = gˆ , respectively). For (1), let {(σi , τi )}i∈ω , {(σi′ , τi′ )}i∈ω be ∗ -towers, and assume that g(σ,τ ) = g(σ′ ,τ ′ ) , but f(σ,τ ) 6= f(σ′ ,τ ′ ) . So for all i, j ∈ ω τi ≈ τj′ , but for some i, j ∈ ω one has σi | σj′ , where τi ⊆ τj′ , ∗
′∗
say. Let si + 1 , tj + 1 be stages for which σi s +1 = σi∗ , σj t +1 = σj′∗ , respectively, for all s′ + 1 > s , t′ + 1 > t , respectively. Show that one must have si + 1 > tj + 1 . ′
′
′∗t
∗t
+1
+1
Assume otherwise. So at stage si +1 ≤ tj +1 one defines σj j = σj′∗ ⊆ σi j = ∗s +1 σi i = σi∗ with σi | σj′ . By the minimality of tj + 1 and the routine extension of ∗ at stage tj + 1 , there must be some requirement Rγ which requires attention via case (A) or case (D), part I, of the construction at stage tj +1 , leading to a nontrivial ∗−1 t
reorganisation ∗tj +1 of ∗tj . But in that case, since τi
j
=
∗−1 s +1 τi i
= σi |
σj′
′∗−1 t
= τj
j
∗−1 t +1
one must also define τi j compatibly with σj′ , contradicting the choice of si + 1 . It inductively follows that there must be infinitely many pairs (σi , τi ) such that no tj + 1 ≤ sj + 1 , and hence for all but a finite number of pairs (σj′ , τj′ ) one has σj′ ≈ σi , either contradicting one of {(σi , τi )}i∈ω , {(σi′ , τi′ )}i∈ω being ∗ -towers, or contradicting f(σ,τ ) 6= f(σ′ ,τ ′ ) . A similar argument excludes the possibility of f(σ,τ ) = f(σ′ ,τ ′ ) 6⇒ g(σ,τ ) = g(σ′ ,τ ′ ) for ∗ -towers {(σi , τi )}i∈ω , {(σi′ , τi′ )}i∈ω , and (1) follows. For (2), let fˆ ∈ ω ω , and assume that one has defined {(σ0 , τ0 ), . . . , (σi , τi )} ⊂ ∗ with σ0 ⊂ . . . ⊂ σi ⊂ fˆ and τ0 ⊂ . . . ⊂ τi . Let se be a stage such that for all s + 1 > se one has σj∗s = τj , each j between 0 and i . Assume that for each pair (σ, τ ) with σi ⊂ σ ⊂ fˆ and τi ⊂ τ one has ∗ σ 6= τ (otherwise the inductive step in the definition of an appropriate ∗ -tower {(σi , τi )}i∈ω immediately follows). This means that either there exists some σ with σi ⊂ σ ⊂ fˆ such that for each σ ′ with σ ⊂ σ ′ ⊂ fˆ one has σ ′∗ ↑ , or σ ∗ ↓ for infinitely many σ with σi ⊂ σ ⊂ fˆ, but for all such σ one defines σ ∗ | τi . In order to exclude the first possibility, one need only verify that Kσ is satisfied. Sublemma 3.16. All the K - and L - requirements are eventually satisfied. Proof. First remember that at each stage s + 1 ∗s+1 is a reorganisation of ∗s . (By the routine extension of ∗ at stage s + 1 , this is trivially the case unless Rγ
Upper Cones As Automorphism Bases
51
requires attention via case (A) or subcase I of case (D) at stage s + 1 . And in these latter cases the scope of the redefined values of ∗s+1 is explicitly chosen so as to be consistent with ∗s+1 being a reorganisation of ∗s .) In order to get a contradiction, assume that Kσ fails to be satisfied at the end of the construction, and that h ∈ ω ω and σ ⊂ h are such that for all τ with σ ⊆ τ ⊂ h one has Lims τ ∗s ↑ . Now let sˆ be a stage after which no requirement of priority greater than or equal to that of Kσ requires attention. By the construction and the existence of sˆ , Kσ requires attention at some non deficient receptive stage. Let u + 1 ≤ sˆ be the greatest such stage. Say one sets up a weak restraint of priority that of Kσ to preserve τ ∗t , some τ ⊇ σ , at all stages t + 1 ≥ the first reactive stage ≥ u + 1 . One shows that τ ∗u+1 = τ ∗ , and hence τ 6⊂ h . Since the weak restraint is never cancelled, τ ∗t+1 can only be defined 6= τ ∗t at bΨ b ′′ , . . . , Ψ b ′′ ] -preconfigured some stage t + 1 > u + 1 if and only if τ is inner P, Q[ α1 αl′ relative to ft+1 above some RΘ at a reactive stage t + 1 at which ∗t is reorganised b ′′α derived from the b ′′α , . . . , Ψ via part (A) or (D) part I of the construction, with Ψ 1 l′ appropriate augmentation. But as before, one can then argue that τ becomes inner b -preconfigured relative to fs+1 above Rγ at the end of stage t + 1 , leading to P, Q some requirement of priority greater than or equal to that of Kσ requiring attention, a contradiction. So one can assume that no such τ ∗t is restrained, or weakly restrained, at any stage t + 1 > u + 1 . But by the construction, at stage u + 1 each path through σ is unprotected from each lower priority I - or J - requirement. Hence, at stage u + 1 one has some minimal ∗s -independent string σ ˆ ′′ with h ⊃ ′′ ′′ σ ˆ ⊇ σ for which one defines σ ˆu+1 , where every string ⊆ σ ˆ becomes unavailable for the implementing of each axiom corresponding to a lower priority I - or J requirement at each stage t + 1 ≥ u + 1 , and where at each such stage t + 1 one has each path through σ is protected from each such I - or J - requirement. ′′
By the choice of sˆ and the fact that some requirement Mx′ above Kσ guarantees the existence of Lims σ ∗s , for no σ ′ ⊆ σ is σ ′ ∗t is redefined at a stage t + 1 > sˆ . But this means that any reorganisation of ∗t via case (A) or via case (D) part I of definition 2.2 at a stage t + 1 > sˆ must leave unchanged some σ ′′ ∗t with σ ⊆ σ ′′ ⊆ σ ′ . This is because any such reorganisation is determined by a b -preconfigured rank of ft+1 and the P, b Q -preconfigured matching ψ of the P, Q rank of gt+1 above some appropriate requirement RΘ at stage t + 1 , giving an b P,Q b′ , . . . , Φ b ′ ] , say, correisomorphism between the stratified rank + εt (RΘ , gt+1 )[Φ sponding to some augmentation b P,Q
β1 b + P,Q ′ b ,...,Φ b′ ] εt (RΘ , gt+1 )[Φ β1 βk
of
βk b + P,Q εt (RΘ , gt+1 )
b ′α , . . . , Ψ b ′α ] , say, corresponding to an (RΘ , ft+1 )[Ψ 1 l b b + P,Q ′ ′ + P,Q b b augmentation εt (RΘ , ft+1 )[Ψα1 , . . . , Ψαl ] of εt (RΘ , ft+1 ) , where such augmentations are chosen to maximise the agreement between the corresponding isoand the stratified rank
+
εt
52
S. Barry Cooper
morphism and ∗t , by the fact that, as a result of the protection of the paths through σ from the implementation of axioms corresponding to lower priority I - and J requirements, one will have b Q b ′α , . . . , Ψ b ′α ]) σ0 ≈ σ1 (+ εP, (RΘ , ft+1 )[Ψ t 1 l
for each σ0 , σ1 with σ ⊆ σ0 , σ1 ⊆ σ ˆ ′′ , and by definition 2.3. It follows that one can inductively obtain some such σ ′′ for which σ ′′ ∗ = Lims σ ′′ ∗s ↓ . The required contradiction immediately follows. The argument corresponding to a typical L -requirement Lσ is similar. ⊓ ⊔ In relation to the second possibility, let se be as above. Since σ ∗ ↓ for infinitely many strings σ with σi ⊂ σ ⊂ fˆ, one can assume σ to be such a string with σ ∗s ↑ for all s + 1 ≤ se. Assume that one defines σ ∗s+1 | τi at some (least) stage s + 1 > se. By the construction and the choice of e ∗ , this cannot occur via routine extension of ∗ , and so can only occur through reorganisation of ∗s at stage s + 1 , where σ ∗s ↑ or σ ∗s ↓| σ ∗s+1 . But since σi ⊂ σ , if σ ∗s+1 | τi one must have that σ ∗s+1 is also affected by the reorganisation, leading to σ ∗s+1 | σ ∗s , contrary to assumption. On the other hand, let gˆ ∈ ω ω , and assume that one has defined {(σ0 , τ0 ), . . . . . . , (σi , τi )} ⊂ ∗ with σ0 ⊂ . . . ⊂ σi and τ0 ⊂ . . . ⊂ τi ⊂ gˆ . One can choose se as before. And similarly to before, the only problem arises if either for all but finitely many −1 −1 strings τ with τi ⊂ τ ⊂ gˆ one has τ ∗ ↑ , or τ ∗ ↓ for infinitely many τ with −1 τi ⊂ τ ⊂ gˆ , but for all such τ one defines τ ∗ | σi . Then one can eliminate each case by just the same argument as before. ⊓ ⊔ Lemma 3.17. All the R -requirements are eventually satisfied. Proof. Let RΘ be a typical R -requirement, and let sˆ be a stage after which the priority ordering of requirements above RΘ is fixed and no requirement of priority greater than or equal to that of RΘ requires attention. In order to get a contradiction, assume that f = Θg . Since RΘ cannot require attention via case (B) of definition 2.2 at any stage s + 1 > sˆ , and since no higher priority requirement requires attention at such a stage, there must be a fixed potential witness x , say, at every such stage. But then, by assumption, there must be some reactive stage s + 1 > sˆ at which Θg (x) ↓s = fs (x) , so that RΘ requires attention via case (D) of definition 2.2 at stage s + 1 , contrary to the choice of sˆ . ⊓ ⊔ Lemma 3.18. All the P - and Q - requirements are eventually satisfied. Proof. Let PΦ be a typical P -requirement, and let sˆ be a stage after which the priority ordering of requirements above PΦ is fixed and no requirement of priority greater than that of PΦ requires attention.
Upper Cones As Automorphism Bases
53
b Let X, Y ∈ ω ω be such that X = ΦY . One needs to extract a computable Φ e ∗ b Y (where one can assume that X e∗ , Y e∗ ↓ , by lemma 3.14). for which X e∗ = Φ One need only consider the case in which there are infinitely many reactive stages s + 1 at which reorganisation of ∗s involves a nontrivial redefinition of some σ ∗s or b to be a straightforward τ ∗s for some σ ⊂ X or τ ⊂ Y (otherwise one can take Φ adaptation of Φ ). In such a case one verifies that for appropriate α ∈ Tf it is b to be an appropriate modification of Φ b α¯ . possible to take Φ Sublemma 3.19. Given X and Y , it is possible to choose the instance of PΦ , and the stage sˆ such that at no stage s + 1 > sˆ does a requirement of higher priority than that of PΦ require attention, in such a way that at no such stage is an axiom b τ ’ cancelled with σ, τ ⊂ X, Y , respectively. ‘σ = Φ α ¯ b -configured , or persistently P, Q bProof. Say that h0 ∈ ω ω is persistently P, Q preconfigured , relative to h1 ∈ ω ω above Rγ according to hαi if and only if there bexist infinitely many beginnings σ, π of h0 , h1 , respectively, such that σ is P, Q b -preconfigured, respectively, relative to π above Rγ according configured, or P, Q to hαi at some stage s + 1 . (There similar definitions of h0 ∈ ω ω persistently b Q -configured , or persistently P, b Q -preconfigured , relative to h1 ∈ ω ω above Rγ P, according to hβi .) As noted in section 1, given any h ∈ ω ω there exists an α such that h is b -preconfigured relative to f above each sufficiently low priority persistently P, Q Rγ according to hαi . Let α be such an index of Φ (relative to h = X or Y ) of highest priority, with no other such index α′ ≺ α , for which pα = Φ or Φ−1 . e α¯ is cancelled at stage s + 1 , then either: One notices that if an axiom for Φ
b τα¯ ’ for Φ b α¯ at stage s+1 , with (1) s+1 is receptive, and there is an axiom ‘ σ = Φ b Q -preconfigured relative to gs above α ¯ nonconfiguring, with τ inner P, some RΘ at stage s + 1 , via a path inductively defined via part 5) of definition 1.7 which does not involve a string restrained or weakly restrained above RΘ , with corresponding index α ˆ of higher priority than that of α , b τ ’ is removed from Φ b α¯ [s + 1] , or and ‘ σ = Φ α ¯ (2) s + 1 is reactive, and some requirement Rγ of higher priority than that of b α¯ being initialised by PΦ requires attention at stage s + 1 , resulting in Φ b α¯ [s + 1] = ∅ . the defining of Φ
The existence of sˆ follows immediately.
⊓ ⊔
Let sˆ be such a stage given by sublemma 3.19, and inductively define a functional e Φα¯ by taking e α¯ [ˆ b α¯ [ˆ Φ s + 1] = Φ s + 1], and for each s + 1 > sˆ letting
54
S. Barry Cooper e α¯ [s + 1] = Φ e α¯ [s] ∪ {‘σ = Φ e τα¯ ’ | ‘σ = Φ b τα¯ ’ is an axiom for Φ
b α¯ [s + 1] such that ‘σ = Φ e τ ’ is consistent with Φ e α¯ [s]}. Φ α ¯
e Yα¯ e∗ . e α¯ is a computable, consistent functional for which X e∗ = Φ Sublemma 3.20. Φ e α¯ follows immediately from the Proof. The computability and consistency of Φ e α¯ . construction and from the definition of Φ One needs to verify that for infinitely many pairs of strings σ, τ ⊂ X, Y , respece τ ∗ . This is achieved by taking arbitrary strings σ tively, one has σ ∗ = Φ ˆ , τˆ ⊂ X, Y , α ¯ respectively, and finding such a pair σ, τ ⊇ σ ˆ , τˆ respectively. Let σ0 , τ0 be such that X, Y ⊃ σ0 , τ0 ⊇ σ ˆ , τˆ , respectively, and an axiom ‘ σ0 = Φτα0 ’ is stipulated for Φα at some stage s+1 > sˆ , and such that (using lemma 3.14) there are strings π, ρ with σ0 , τ0 ⊇ π, ρ ⊇ σ ˆ , τˆ , respectively, and π, ρ ∈ Dom(∗u ) at all sufficiently large stages u + 1 . One shows that ‘ σ0 = Φτα0 ’ is implemented along the paths X, Y . Say that σ1 , τ1 ⊂ X, Y , respectively, are permanently available for the imple′ menting of an axiom ‘ σ ′ = Φτα ’ if and only if there is no receptive stage at which some K -requirement of higher priority than that of IΦα ,σ′ ,τ ′ requires attention ′ resulting in σ1 , τ1 becoming unavailable for the implementing of ‘ σ ′ = Φτα ’. It follows immediately from the construction that there are infinitely many pairs σ1 , τ1 ⊂ X, Y , respectively, permanently available for the implementing of ‘ σ0 = Φτα0 ’ along X, Y . Specifically, notice that there is such a pair with σ0 , τ0 ⊆ σ1 , τ1 , respectively. This is because, by choice of sˆ , each relevant string which at some stage becomes unavailable for the implementing of ‘ σ0 = Φτα0 ’ must have arisen with some K requirement of higher priority than that of IΦα ,σ0 ,τ0 requiring attention, leading to each string ⊆ some σ ˆ ′′ ⊂ X or Y becoming unavailable for the implementing of τ0 ‘ σ0 = Φα ’. It follows that at most finitely many such strings ever unavailable for the implementing of ‘ σ0 = Φτα0 ’. By lemma 3.14, one may choose such σ1 , τ1 to be in Dom(∗s′ ) at all sufficiently large stages s′ + 1 . Since X = ΦYα , there must at some stage t + 1 ≥ s + 1 be τ′ stipulated for Φα an axiom ‘ σ0′ = Φα0 ’ with σ1 , τ1 ⊆ σ0′ , τ0′ , respectively. It must then occur that as a result of the stipulation of ‘ σ0 = Φτα0 ’ for Φα at stage s + 1 , that at some stage u + 1 ≥ s + 1 ‘ σ0 = Φτα0 ’ is implemented along X, Y by the choosing of σ ˆ1 , τˆ1 , say, with σ0 , τ0 ⊆ σ ˆ1 , τˆ1 ⊆ σ1 , τ1 , respectively, to be the least ′ such strings with σ0 ⊆ σ ˆ1 ⊆ σ0 and τˆ1 ⊇ τ1′ for which both τˆ1∗u and σ ˆ1∗u ↓ , and τ0 for which both τˆ1 and σ ˆ1 are available for the implementing of ‘ σ0 = Φα ’ at stage s + 1 , and the adding of ‘ σ ˆ1 = Φταˆ1 ’ to the list of axioms for Φα . The prerequisite ∗s+1 -matching and corresponding augmentations necessary for any stipulation of new axioms at the end of any reactive stage s + 1 are guaranteed by lemma 3.1. By the choice of α , there will be at most finitely many stages u + 1 at which σ ˆ1 = Φταˆ1 is not on a true path through Pα at stage u + 1 , or
Upper Cones As Automorphism Bases
55
(by the choice of σ0 , τ0 ) at which there is no pair π, ρ with σ0 , τ0 ⊇ π, ρ ⊇ σ ˆ , τˆ , respectively, and π, ρ ∈ Dom(∗u ) . This means that at some reactive stage u + 1 ∗ b α¯ is progressed by the stipulation of an axiom ‘ π ∗u+1 = Φ b ρα¯ u+1 ’ for Φ b α¯ with Φ e e ∗ ∗ ∗u+1 ∗u+1 ∗ ∗ X, Y ⊃ π, ρ ⊇ σ ˆ , τˆ , respectively, and π ,ρ = π , ρ ⊂ X , Y , respectively, this stipulation being retained at all later stages. ∗ b ρα¯ u+1 ’ is implemented along X e∗ , Y e∗ , and It remains to show: (a) That ‘ π ∗u+1 = Φ b α¯ , with σ, τ ⊂ X, Y , b τ ∗ ’ for Φ (b) That this implementation is via an axiom ‘ σ ∗ = Φ α ¯ ∗ e α¯ . e τ ∗ ’ for Φ respectively, which gives rise to a corresponding axiom ‘ σ = Φ α ¯
For (a), one can argue as before that there are infinitely many pairs π1 , ρ1 ⊂
∗ b ρα¯ ’ X e∗ , Y e∗ , respectively, permanently available for the implementing of ‘ π ∗ = Φ along X e∗ , Y e∗ , giving such a pair with π ∗ , ρ∗ ⊆ π1 , ρ1 .
Again, by lemma 3.14 one may choose such π1 , ρ1 to be in Range(∗s′ ) at all sufficiently large stages s′ + 1 . And by the same argument, there must at some ′ b α¯ an axiom ‘ π ′ = Φ b ρα¯0 ’ with π1 , ρ1 ⊆ π ′ , ρ′ , stage t + 1 ≥ u + 1 be stipulated for Φ 0 0 0 ρ∗ ∗ b respectively. It will then occur that as a result of the stipulation of ‘ π = Φα¯ ’ for ∗ b α¯ at stage u+1 , that at some stage u′ +1 ≥ u+1 ‘ π ∗ = Φ b ρα¯ ’ is implemented along Φ ˆ1 , ρˆ1 , say, with σ ∗ , τ ∗ ⊆ π ˆ1 , ρˆ1 ⊆ π1 , ρ1 , respectively, X e∗ , Y e∗ by the choosing of π ∗ ∗ ′ ˆ1 ⊆ π0 and ρˆ1 ⊇ ρ′1 for which both ρˆ1u′ to be the least such strings with σ ⊆ π ∗ and π ˆ1 u′ ↓ , and for which both ρˆ1 and π ˆ1 are available for the implementing of ∗ ρ b α¯ ’ at stage u′ + 1 , and the adding of ‘ π b ραˆ¯1 ’ to the list of axioms for ‘ π∗ = Φ ˆ1 = Φ b α¯ , so long as such an axiom is consistent with both all existing axioms for Φ b α¯ , Φ and with all existing imprinting requirements. The imprinting requirements cannot prevent such an implementation, since there are at most finitely many R -requirements of priority higher than that of some weak restraint relevant to π ∗ or ρ∗ , and hence finitely many strings on the periphery of a relevant discontinuity. And by the conditions put on the stipulation by the b -preconfigured rank above RΘ , say, of fs+1 at the end of construction, the P, Q b Q -preconfigured rank above RΘ each reactive stage s + 1 , say, matches the P, of gs+1 at the end of stage s + 1 , the existence of the required augmentations (satisfying definition 1.7) ensuring the consistency of any such axiom with existing b α¯ . axioms for Φ b α¯ at some stage, Finally, (b) can only fail if there is some axiom cancelled from Φ ρˆ1 e α¯ , which is inconsistent with the axiom ‘ π bα b¯. but eventually in Φ ˆ1 = Φ ¯ ’ for Φα e α¯ , and by the But this cannot happen by the choice of sˆ , by the definition of Φ b consistency of Φα¯ . ⊓ ⊔ The argument is similar in relation to a typical Q -requirement QΨ .
⊓ ⊔
This completes the verification, and the proof of theorem 1.1.
⊓ ⊔
56
S. Barry Cooper References
K. Ambos-Spies [ta], Automorphism bases, to appear. S. B. Cooper [1997], Beyond G¨ odel’s Theorem: The failure to capture information content, in “Complexity, Logic and Recursion Theory” (A. Sorbi, ed.), Lecture Notes in Pure and Applied Mathematics, vol. 187, Marcel Dekker, New York, pp. 93–122. S. B. Cooper [ta1], On a conjecture of Kleene and Post, to appear. S. B. Cooper [ta2], The Turing universe is not rigid, to appear. R. L. Epstein [1979], Degrees of Unsolvability: Structure and Theory, Lecture Notes in Mathematics No. 759, Springer-Verlag, Berlin, Heidelberg, New York. C. G. Jockusch, Jr. and D. Posner [1981], Automorphism bases for degrees of unsolvability, Israel J. Math. 40, 150–164. M. Lerman [1983], Degrees of Unsolvability, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidelberg, London, New York, Tokyo. K. McEvoy and S. B. Cooper [1985], On minimal pairs of enumeration degrees, J. Symbolic Logic 50, 839–848. A. Nies, R. A. Shore, T. A. Slaman [ta], Interpretability and definability in the recursively enumerable Turing degrees, to appear. P. Odifreddi [1989], Classical Recursion Theory, North-Holland, Amsterdam, New York, Oxford. H. Rogers, Jr. [1967], Some problems of definability in recursive function theory, in “Sets, Models and Recursion Theory” (J. N. Crossley, ed.), Proceedings of the Summer School in Mathematical Logic and Tenth Logic Colloquium, Leicester, August–September, 1965, North Holland, Amsterdam, pp. 183–201. R. I. Soare [1987], Recursively Enumerable Sets and Degrees, Springer-Verlag, Berlin, Heidelberg, London, New York.
