eu SIGMA-1-1 WELL-FOUNDED RELATIONS AND SET CHAINS

Arbitrary well-founded relations are considered, generalizing constructions involving WPS’s given in [9]. Some new facts concerning function chains and set chains are given, and a new axiom for set theory. AMS Subject Classification: 03e55


Introduction
In a series of papers ( [2], [3], [4], [5], [6], [7]) the author has "constructed" progressively longer chains of stationary sets of inaccessible cardinals.The existence of such chains is independent of ZFC (indeed the existence of an inaccessible cardinal is), so that the construction in fact yields a chain must be postulated.This gives a quantitative theory, justifying the addition of certain axioms to ZFC, stating that the cumulative hierarchy satisfies certain principles regarding its extendibility.
Throughout these constructions, the fact that the axiom holds in V κ if κ is weakly compact has been observed; the stationary sets are in the enforceable filter.One goal of this research has been to obtain a weakly compact cardinal by making the stationary set chains long enough, using the principle of continuing to extend the cumulative hierarchy.
In [7] the set chains constructed involve the notion of a Σ 1 1 WPS.A WPS is a well-founded binary relation (WF).In this paper set chains are obtained from Σ 1  1 WF's, under a certain hypothesis, and various other facts of interest are proved.Understanding the properties of Σ 1 1 WF's is of interest in the context of set chains, and in general.
For a cardinal κ the three structures L κ , OS κ , and H κ , may be considered.In each the Σ 1 WF's, and various subclasses thereof, may be considered.If V = L these all define the same ranks.It has been shown [12] that it is consistent that they do not.
Classes involving second order objects may be defined.These are the classes relevant to constructing stationary set chains.The classes of the previous paragraph for the cardinal κ + provide characterizations of them.As will be seen, OS κ has advantages over H κ in this role.
In [9] it is claimed that if V = L the ranks of the U Σ 1 1l -WPS's (defined below) are the same as the ranks of the Σ 1  1 WPS's.The proof given is incorrect.As of this paper, this question is still open.Some results given here suggest that the claim may be false.
Let Card denote the class of cardinals, Inac the class of strongly inaccessible cardinals, and write Inac κ for Inac ∩ κ.Let Cf(α) denote the cofinality of an ordinal α.

Well-Founded Relations
A well-founded relation (WF) on a set S is a binary relation ≺ on S such that for any function f : ω → S there is an n such that f (n + 1) ≺ f (n) (i.e., there are no infinite descending chains).The rank or height Ω(x) of an element x ∈ Fld(≺) is an ordinal, defined recursively as sup{Ω(y) + 1 : y ≺ x}.Ω(≺) is defined to be sup{Ω(x) + 1 : x ∈ Fld(≺)}, which is taken as 0 if ≺ is empty.
Let Υ(≺) denote the supremum of the lengths of the ascending chains in ≺.It is readily seen that Υ(≺) ≤ Ω(≺); it is well-known [16] that strict inequality can hold.
Recalling the definition from [7], a WPS on a set S is a binary relation on S satisfying the following axioms: F. For all functions f with domain ω there is an n such that f (n + 1) The strict part is the relation ≺ where A ≺ B iff A B ∧ ¬B A; axiom F implies that it is well-founded.Let Ω( ) = Ω(≺).For x ∈ Fld( ) let x denote ∩(x ≺ × x ≺ ); this is readily seen to be a WPS.
A WOS (well-order on a subset) is a WPS where In a WOS ≺ may be defined by "A B ∧ A = B".For a WOS and an x ∈ Fld( ), x is a WOS.A WP (well-preorder) is a WPS where axioms T2-T4 are replaced by A B ∨ B A. A WO (well-order) is a WPS which is both a WOS and a WP.For a WO the rank is also called the order type.

OS κ
The structure OS κ is defined in [7]; an earlier version, K s κ , may be found in [2].To review the definition, let L OS be the language with two sorts, Ord and Seq.Variables of Ord sort are denoted α, β, etc., and those of Seq sort s, t, etc.The functions and relations are: 0, 1, α + β, α < β, Dom(s) (of sort Ord), Elem(s, α) (of sort Ord), and equality on Ord and on Seq.As in [2], s(α) may be written for Elem(s, α) and |s| for Dom(s).
The version in [7] has also the function symbol Rstr(s, α, β) (of sort Seq).The graph of this function has a ∆ 0 definition in the smaller language, namely, Bounded quantifiers in L OS are those of the form ∀γ < β or ∃γ < β where β is a term.∆ 0 , Σ 1 , and Π 1 formulas are defined as usual, where free variables and unbounded quantifiers may be of either sort.
For κ ∈ Card OS κ is the structure for L OS where Ord is interpreted as κ and Seq as {f : α → κ : α < κ}.The functions and relations have their self-evident interpretations; Elem has value 0 if the index is out of range.
Various facts of interest can be proved to hold in OS κ for any κ.Indeed, an axiom system A OS is given in [7], and basic facts can be proved in it.For the version above, the axiom for Rstr may be omitted.

Lemma 2.
a.The Σ 1 predicates are closed under bounded quantification.

b. If
Proof.This is lemma 21 of [7].An examination of the proof shows that it can be carried out in A OS .
Note also that α < ω is a ∆ 0 predicate, and it is not difficult to show that Peano Arithmetic is interpretable in A OS .The sort "integer" may be added to the language, without changing the complexity of formulas; n, m, . . .will be used to denote integers.
The rank of a WF on Seq is less than (κ <κ ) + .That of a WF on Ord is less than κ + .When κ = λ + the bounds become ((λ + ) λ ) + = (2 λ ) + and λ ++ The language L f OS is also of interest; some consideration may be found in [9].This adds second order function variables; these take an ordinal argument and produce an ordinal value.OS κ may be considered as a structure for this language by interpreting second order function symbols as elements of N where N denotes κ κ .As above, t = Rstr(F, α, β) is a ∆ 0 0 definable predicate.Say that a formula of 1 formulas over OS κ for κ ∈ Card, multiple second order existential quantifiers may be combined.The Σ 1P 1 predicates are closed under quantification over variables of Ord sort, and existential quantification over variables of Seq sort.
).A formula ∀s∃F φ(α, F ) may be written as The analog of this theorem holds for Σ 1L  1 predicates (defined below); proofs may be found in [13], [9].Proof.Part a follows by theorem 11 of [9].For part b, suppose R ⊆ N k is open.R can be specified by a subset D R of (κ <κ ) k .Since κ <κ = κ, D R can be coded as an element F R of N using a "separator" value.Then G ∈ R iff "there is an Note that part b answers a question noted following theorem 15 of [9], and renders that theorem irrelevant.

Classes over κ
In this section, let κ be a cardinal.Let L ∈ be the language of set theory.Let L κ be the κth level of the constructibility hierarchy, considered as a structure for L ∈ .Let H κ be sets whose transitive closure has cardinality less than κ, considered as a structure for L ∈ .
Classes of relations C will be defined, on a domain D (i.e., subsets of D k for some k).If defined by formulas the free variables are restricted to D; parameters are also in D, unless otherwise specified.
The following classes are defined.
Let C ′ be as C , but with unrestricted free variables and parameters.Because there is a Σ 1 bijection between κ and L κ , these may be used interchangeably.
Let F f , ∈ f , and = f be as in [9].L ∈ may be interpreted in L OS , by interpreting ∈ as ∈ f and = as = f .Denote this interpretation as I f .Lemma 6.For any formula Proof.The first claim follows by induction on φ.The second follows by induction, using the fact, noted in [9], that ∈ f and = f are ∆ 1 ; and lemma 12 of [9].
There is also an interpretation of L OS in L ∈ .Ord is interpreted as the ordinals, and Seq as functions with domain an ordinal and ordinal values.Denote this interpretation as I ∈ .The formulas for the symbols of L OS are ∆ 0 , except the graph of α + β, which is ∆ 1 , and so ∆ 0 formulas translate to ∆ 1 formulas.
Proof.Let φ be a Σ 1 formula defining a WF in L κ .By lemma 6, φ I f defines a Σ 1 WF in OS κ , of the same rank.Further φ I f has ordinal arguments, and ordinal parameters can be chosen.This proves C C .C C follows by interpreting ordinals as length 1 sequences.C C and C C follow using As will be seen below, more can be shown if κ is a successor cardinal.

Coding Formulas
Formulas with parameters from Seq can be coded as elements of Seq.Formulas with parameters from Ord can be coded as elements of Ord.Both codings will be given.Each involves a coding of finite sequences.
In Seq a sequence (or list) of elements of ordinal length can readily be coded as an element; this has uses, and finite sequences are a special case.The list s γ : γ < δ may be coded as δ ⌢ η ⌢ 0 . . .⌢ s ⌢ 0 . . ., where η γ = ζ≤γ |s ζ |.The statement that s occurs in a list l is ∆ 0 .This coding may be used for a pairing function on sequences.Let J S (s, t) denote the code for the pair s, t; this function has a ∆ 0 graph.
In Ord a finite sequence of elements can be coded as an element using the Godel pairing function.The sequence α i : i < n may be coded as J 0 (n, J 0 (α 1 , . . ., J 0 (α n−1 , n) . . .)). Functions manipulating these codes are ∆ 1 .
It is well-known that a formula φ without parameters can be coded as an integer φ , so that syntactic functions are ∆ 1 in the language of arithmetic, and hence in L OS over OS κ for κ ∈ Card.
Suppose φ is a formula and x 1 , . . ., x k are its free variables in alphabetic order.Suppose v i is a value of the sort of x i .Let l be the code for v 1 , . . ., v k .The sentence with parameters φ(v 1 , . . ., v k ) can be coded by replacing l(0) = k by φ .This may be done for either the codes over Seq or the codes over Ord.Over Seq, a value v i of sort Ord may be considered a sequence of length 1, and if k = 0 the code is a sequence of length 1.Over Ord, if k = 0 the code may be taken as J 0 ( φ , 0).
The code just described will be denoted φ(v 1 , . . ., v k ) .Let Φ denote the sentences with parameters and I their codes (although as has been observed by some authors the latter can be used for the former).
Theorem 9.Over Seq or Ord, there is a ∆ 1 formula Tru 0 (c) such that for any κ ∈ Card, in OS κ , for any Proof.The proof will use "partial truth assignments"; see definition 1.71 of [11] for a related concept.A partial truth assignment is a list in 4 parts, a list of terms, a list of their values, a list of formulas, and a list of their truth values, satisfying certain restrictions.These restrictions may be stated by a ∆ 1 formula PTA(a).
PTA may be broken into cases; one example will be given, and the rest left to the reader.For all terms t = t 1 + t 2 occurring in a, for i = 1, 2, either t i is a variable or t i (with value list adjusted) occurs in a; in addition v = v 1 + v 2 where v is the value of t and for i = 1, 2 v i is the value of t i .
Tru 0 (c) may be stated in Σ 1 form as "for some a, PTA(a) and c occurs in a and c has value 1 in a".Tru 0 (c) may be stated in Π 1 form as "for all a, if PTA(a) and c occurs in a then c has value 1 in a".
Proof.This follows by a standard argument.
In L κ (or H κ , or any admissible set) an ordered pair of elements may be coded as the ordered pair in the set.Sentences with parameters may be coded as in the case of OS κ with parameters from Ord.It is a standard fact of admissible set theory that there is a Σ 1 predicate Tru(c) which holds for a code c of a Σ 1 sentence with parameters iff the sentence is true (see proposition V.1.8 of [1] for example).Although not the usual method, the method of corollary 10 can be used to prove this fact.It is only necessary to observe that in an admissible set A, any ∆ 0 sentence with parameters has a partial truth assignment in A.
A formula φ( α p , α, s p , s), where the free variables are listed in alphabetic order, with values v Oi for the variables α pi , and values v Si for the variables s pi , may be coded by replacing l(0) by φ in the code l for the list v O , v S .Similar remarks hold over Ord.
In an expression φ a parameter of φ depending on a value α may be denoted α.

Constructive Ordinals in Seq
This section provides adaptations of various facts about constructive ordinals (as found in [15] for example) to OS κ for κ ∈ Card.Essentially the same development can be carried out for H κ , but this is omitted.
Let Φ 1 denote the Σ 1 formulas with no parameters and a single free variable of sort Seq.Let I 1 ⊆ ω denote the integers which code elements of Φ 1 .For e ∈ I 1 let φ e denote the formula coded by e and let W e denote the subset of Seq defined by φ e .
Let F 0 denote the functions f : Seq → Seq which are total and whose graph is Σ 1 (and hence ∆ 1 ) without parameters in L OS .Let a.For any e 0 ∈ I 1 there is an e 1 ∈ I 1 such that W e 1 (s) ⇔ W e 0 (J S (e 1 , s)) for all s.
Proof.For part a, let Tru 1 (e, s) = Tru(F (e 1 , s)) where for e ∈ I 1 F (e, s) equals φ e (s) , that is, e ⌢ |s| ⌢ s.Let f ∈ F 1 be such that for any e ∈ I 1 f (e) = Tru 1 (N e , J s (N e , s)) where N e is the numeral for e and s is the free variable of φ e ; note that W f (e) (s) ⇔ W e (J s (e, s)).Let e 2 = φ e 0 (J S (f (P 1 (t)), P 2 (t))) where P 1 , P 2 are the "projection functions" for J S ; note that W e 2 (J S (e, s)) ⇔ W e 0 (J S (f (e 0 ), s)).Let e 1 = f (e 2 ).Direct computation verifies that the requirement on e 1 is satisfied.
For part c, if there is an x ∈ Fld(<) such that ¬∃!yW e (J S (x, y))) let x be a minimal such.Then ∃!yW f (e) (J S (x, y)), whence ∃!yW e (J S (x, y)), a contradiction.
Let < O be the predicate on Seq× Seq, which is the least predicate satisfying the following conditions, where O denotes Fld(< O ).
Other facts as in theorem I.3.4 of [15] also follow.Note that for s ∈ O Ω(s) < κ ++ , so in view of the remarks following theorem 8, in some models, this version of O does not represent every Σ 1 WF on Seq.

Constructive Ordinals in Ord
The development of the previous section can be carried out using formulas with free variables and parameters in Ord.For convenience the same notation is used.Following is a list of changes which are needed.The following changes are needed for theorem 11.
• Formulas of Φ 1 have a single free variable of sort Ord.
• F 0 is the f : Ord → Ord which are total and whose graph is Σ 1 .
• F 1 is as before.
• In the proof of theorem 11.a, F (e, s) equals φ e (s) , that is, J 0 (e, s).Let < O be the predicate on Ord×Ord, which is the least predicate satisfying the following conditions, where O denotes Fld(< O ).

Constructive Ordinals in L κ
Because L κ is a "recursively listed" admissible set (see [1]), constructive ordinals in L κ for κ ∈ Card may be taken as elements of either κ or L κ .Choosing them in κ makes the development more similar to that of the preceding section.In particular, the same sentence coding may be used.The following changes are needed for theorem 11.
• I 1 , F 0 , and F 1 are as before.
• In theorem 11.c < ⊆ κ × κ. < O on Ord × Ord is defined as in the previous section.Theorems 13, 14, 15, and the properties of Ω hold as before.
Let I 1p be the codes of the Σ 1 formulas with ordinal parameters and an ordinal free variable.Let φ η be the formula with code η and W η the set defined by φ η .
Let f ∈ F 1 be such that W f (e) = { η, g(t(e, η)) }.Let e 0 be such that W f (e 0 ) = W e 0 .Let h be the function where h(η) = g(t(e 0 , η)).Then W e 0 is the graph of h.
If W e is empty the theorem is readily seen.Otherwise W t(e 0 ,η) = {h(τ (η, γ)) : γ < κ}, and so by theorem 16, Proof.For part a, a O code can be transformed to a O code, and a O code can be transformed to a O code.
Whether Ω(O ) < Ω(C -WO) can be consistent is a question of interest.

Classes over κ +
For κ ∈ Card, a class C of Section 4 over κ + will be denoted Ĉ.Further classes of interest may be defined using second order methods over κ.
As in [8], let L s ∈ denote L ∈ with set variables added, and let L f ∈ denote L ∈ with function variables added.Recall L f OS from Section 3. Let I OS denote the interpretation of L OS in L s ∈ given in [7].Say that a formula is Σ I 1 if it is the translation under I OS of a Σ 1 formula of L OS .As in [9], let N g denote (V κ ) Vκ , let N denote κ κ , and let Σ 1L 1 denote the Lusin class in either N k g or N k for an integer k.
The following class is defined.
Theorem 19.Suppose κ ∈ Card.Then Ĉ H C Ĉ .Thus, Proof.Suppose R ⊆ N k is defined in H κ + by a Σ 1 formula with parameters.

C
Ĉ follows because there is an interpretation of L f OS in L OS using the parameter κ which induces such a transformation.Ord is interpreted as κ, Seq is interpreted as {s ∈ Seq : Dom(s) < κ ∧ Ran(s) ⊆ κ}.N is interpreted as {s ∈ Seq : Dom(s) = κ ∧ Ran(s) ⊆ κ}.
The second claim follows by the first claim and theorem 7.
For a Σ 1L 1 version of Ĉ H C see proposition 2.4 of [13].
Theorem 20.Suppose κ ∈ Card.Then Ĉ H Ĉ , and so Υ( Ĉ H ) = Υ( Ĉ ), Proof.The theorem is proved by modifying the composite transformation Ĉ H C Ĉ .For ξ an ordinal in H κ + let P 5 (F ∈ , ξ, α) hold iff α represents ξ in F ∈ .This may be written as Suppose φ( ξ, p) is a Σ 1 formula over H κ + , the ξ i are restricted to range over ordinals, and the p i are ordinals.Then φ holds in The second claim follows from the first, and theorem 7.
For κ ∈ Card, the following classes of relations are defined.
Proof.Suppose R( H) is a relation in C ; by corollary 5 of [8] R is defined by a formula ∃ F ∀ xψ( x, F , G, H) where ψ is a ∆ 0 0 formula of L f ∈ and G are second order parameters.By theorem 9 of [9] and remarks following R( H) is C ⋪ .Hence C C ⋪ .
Recall from [9] the homeomorphism Ê : N → N g derived from a bijection Suppose R( H) is a relation in C ⋫ , being the projection along F of the closed subset K( F , H).As in the proof of theorem 5.b, K c can be specified by a subset D K c of Seq k+l κ .Since κ <κ = κ D K c can be coded as an element G of N using a "separator" value.R may be defined in OS κ by the Σ 1P  1 formula with the parameter G, "there is a F such that no element of G which is a prefix of F , H ". Hence C ⋫ B Ĉ .Proof.This follows by lemma 5.
As seen in [7], the class C is of interest in connection with function and set chains.The class Ĉ provides a first-order characterization.The class Ĉ H has already been considered (in [13] for example).Ĉ has an advantage over Ĉ H , in that the transformation Ĉ Ĉ provides a structured interpretation of the first-order formulas in the second-order ones.
There is an interpretation of H κ + in L f OS .The domain is the set of F ∈ N , which as binary relations are well-founded, extensional, transitive, and have a maximal element.The interpretations of ∈ and = are Σ 1P 1 .

∆ 1 Classes
For any of the classes of relations C of Sections 4 and 9, say that R ∈ C is in class C ∆ if R has a Π 1 definition also, where Π 1 is defined appropriately.

Function Chains
Suppose κ is a regular uncountable cardinal.For f, g : κ → κ say that f ≤ t g if {α < κ : f (α) < g(α)} is in the club filter, and similarly for f < t g and f ≡ t g.As noted in [7], if κ ∈ Inac f, g need only be defined for α ∈ Card.A function chain is a chain in this order.
As also noted in [7], if κ is Mahlo, f, g defined only for α ∈ Inac may be considered.As far as the author knows, it is unknown whether the lengths of the function chains in the order when the domain is Inac are no greater than those when the domain is Card.

Lemma 5 .
a.For any κ ∈ Card, a predicate R ⊆ N k defined in OS κ by a Σ 0 1 formula of L f OS is open.b.If κ is regular uncountable and κ <κ = κ the converse holds.

C
Ĉ is proved in theorem 19.Suppose R( α, s) is a relation in Ĉ , defined by Σ 1 formula φ( β, t, α, s) with parameters β, t.Let φ I be the interpretation under I OS ; this defines a relation on I k Ord × I l Seq for appropriate k, l.In particular Ĉ C .(This stretches the definition of , but WF's transform to WF's and WPS's to WPS's).It follows using results of [7] that C C .It follows using lemma 3 of [8] that C C .That Υ(C ⋫ ) = Υ(C ) for κ ∈ Inac can be improved.Theorem 22. C ⋫ C B C ⋫ , and so for a regular uncountable cardinal κ such that κ <κ = κ, Υ(C ) = Υ(C ⋫ ).
Theorem 23. a.The transformations of theorems 7, 19, 20, and 21 map ∆ relations to ∆ relations.b.For classes C for = , , , , if ∈ C I is a total order then ∈ C ∆ .c. Υ(C ) = Υ(C ∆ ).Proof.Part a follows by additional observations in the proofs of the cited theorems.Part b follows by the usual proof.Part c follows by part b and theorem 18.