It is shown that for any cardinal κ, , and if κ is weakly compact then .

In this paper we formulate epsilon substitution method for a theory -FIX for nonmonotonic inductive definitions. Then we give a termination proof of the H-processes based on Ackermann [1].

Equational logic for total functions is a remarkable fragment of first-order logic. Rich enough to lend itself to many uses, it is also quite austere. The only predicate symbol is one for a notion of equality, and there are no logical connectives. Proof theory for equational logic therefore is different from proof theory for other logics and, in some respects, more transparent. The question therefore arises to what extent a logic with a similar proof theory can be constructed when expressive power is increased.

The increase mainly studied here allows one both to consider arbitrary partial functions and to express the condition that a function be total. A further increase taken into account is equivalent to a change to universal Horn sentences for partial and for total functions.

Two ways of increasing expressive power will be considered. In both cases, the notion of equality is modified and nonlogical function symbols are interpreted as ranging over partial functions, instead of ranging only over total functions. In one case, the only further change is the addition of symbols that denote logical functions, such as the binary projection function Ae that maps each pair ‹a0, a1› of elements of a set A into the element a0. An addition of this kind results in a language, and also in a system of logic based on this language, which we call equational In the other case, instead of adding a symbol for Ae, one admits those special universal Horn sentences in which the conditions expressed by the antecedent are, in a sense, pure conditions of existence. Languages and systems of logic that result from a change of this kind will be called near-equational. According to whether the number of existence conditions that one may express in antecedents is finite or arbitrary, the resulting language and logic shall be finite or infinitary, respectively. Each of our finite near-equational languages turns out to be equivalent to one of our equational languages, and vice versa.

We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to show that a rosy group with a þ-generic stable type is stable. In the context of super-rosy theories of finite rank we conclude that non-trivial stable types of Uþ-rank 1 must arise from definable stable sets.

We construct, assuming that there is no inner model with a Woodin cardinal but without any large cardinal assumption, a model Kc which is iterable for set length iterations, which is universal with respect to all weasels with which it can be compared, and (assuming GCH) is universal with respect to set sized premice.

A metric space (X, d) is monotone if there is a linear order < on X and a constant c such that d(x, y) ≤ c d(x, z) for all x < y < z in X. We investigate cardinal invariants of the σ-ideal Mon generated by monotone subsets of the plane. Since there is a strong connection between monotone sets in the plane and porous subsets of the line, plane and the Cantor set, cardinal invariants of these ideals are also investigated. In particular, we show that non(Mon) ≥ mσ-linked, but non(Mon) < mσ-centered is consistent. Also cov(Mon) < c and cof (N) < cov(Mon) are consistent.

In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$ , for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.

In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo–Fraenkel set theory $\mathrm {ZFC}^{-}$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of $\mathrm {ZFC}^{-}$ that are bi-interpretable, but not isomorphic—even $\langle H_{\omega _1},\in \rangle $ and $ \langle H_{\omega _2},\in \rangle $ can be bi-interpretable—and there are distinct bi-interpretable theories extending $\mathrm {ZFC}^{-}$ . Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails.

It will be proved that the Shoenfield cupping conjecture holds in R/M, the quotient of the recursively enumerable degrees modulo the cappable r.e. degrees. Namely, for any [a], [b] ∈ R/M such that [0] ≺ [b] ≺ [a] there exists [c] ∈ R/M such that [c] ≺ [a] and [a] = [b] ∨ [c].

In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, ω-model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences among the introduced subsystems. In particular we prove the equivalence of Σ11 transfinite dependent choice and Π21 reflection on ω-models of Σ11-DC.

Markov's principle is more than a convenience in constructive arithmetic and analysis; it is absolutely essential to significant areas of constructive cardinal arithmetic. In turn, logical relations among intuitively appealing principles of constructive cardinal arithmetic parallel relations between MPS and other “problematic axioms” for constructive mathematics, such as the limited principle of omniscience. Finally, simple closure properties on the Dedekind finite sets provide ready examples of statements which are strictly weaker than Markov's principle and yet are independent of extensions of IZF.

MPS, Markov's principle with variables over sets, is equivalent to each of these elementary properties of ⊿, the class of Dedekind finite sets:

1. ∀A(A Є ⊿ ↔ CP(A)).

2. [*]A Є ⊿ ↔ ∀B (A + B is infinite ↔ B is infinite).

3. [**]A Є ⊿ ↔ ∀B((A + 1) x B is infinite ↔ B is infinite).

CP is Tarski's cancellation property. Consequently, MPS is tantamount, in constructive mathematics, to the standard classical characterizations of ⊿ in terms of cardinality.

[*] and [**] imply that ⊿ is closed under addition and multiplication. Closure under addition is, in turn, constructively equivalent to the closure of ⊿ under all combinatorial functions and to the fact that ⊿ is closed under each strict combinatorial function individually. Generally speaking, a function on P(ω) is combinatorial whenever it preserves finiteness, respects cardinality and has a “moduluslike” associate function. It is strict when it is strictly increasing relative to the subset ordering, [cf. §6 for precise definitions.] From this, we see that MPS is foundational for cardinal arithmetic on the constructive Dedekind finite sets.

We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than ${\omega _1}$ which measures how complex it is to verify differentiability for that function. We show that for each recursive ordinal $\alpha > 0$, the set of Turing indices of $C[0,1]$ functions that are differentiable with rank at most α is ${{\rm{\Pi }}_{2\alpha + 1}}$-complete. This result is expressed in the notation of Ash and Knight.

In the past sixty years or so, a real forest of intuitionistic models for classical theories has grown. In this paper we will compare intuitionistic models of first order classical theories according to relevant issues, like completeness (w.r.t. first order classical provability), consistency, and relationship between a connective and its interpretation in a model. We briefly consider also intuitionistic models for classical ω-logic.

All results included here, but a part of the proposition (a) below, are new. This work is, ideally, a continuation of a paper by McCarty, who considered intuitionistic completeness mostly for first order intuitionistic logic.

We show that Intuitionistic Open Induction iop is not closed under the rule DNS(Ǝ1). This is established by constructing a Kripke model of iop + ¬Ly(2y > x), where Ly(2y > x) is universally quantified on x. On the other hand, we prove that iop is equivalent with the intuitionistic theory axiomatized by PA− plus the scheme of weak ¬¬ LNP for open formulas, where universal quantification on the parameters precedes double negation. We also show that for any open formula φ(y) having only y free. (PA−)i ⊢ Lyφ(y). We observe that the theories iop, i∀1 and iΠ1 are closed under Friedman's translation by negated formulas and so under VR and IP. We include some remarks on the classical worlds in Kripke models of iop.

Let κℝ be the least ordinal κ such that Lκ (ℝ) is admissible. Let A = {x ϵ ℝ ∣ (∃α < κℝ) such that x is ordinal definable in Lα (ℝ)}. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the theory: ZFC − Replacement + “There exists ω Woodin cardinals which are cofinal in the ordinals.” T has consistency strength weaker than that of the theory ZFC + “There exists ω Woodin cardinals”, but stronger than that of the theory ZFC + “There exists n Woodin Cardinals”, for each n ϵ ω. Let M be the canonical, minimal inner model for the theory T. In this paper we show that A = ℝ ∩ M. Since M is a mouse, we say that A is a mouse set. As an application, we use our characterization of A to give an inner-model-theoretic proof of a theorem of Martin which states that for all n, every real is in A.

§1. In [2], as part of an analogy between the concepts of recursive emimerability and regressiveness, Dekker showed that the intersection of any two regressive sets which are recursively equivalent is a regressive set. In [1], McLaughlin and the author showed that the intersection of two regressive sets, if infinite, has an infinite regressive subset. However, the following theorem shows that the analogy is not complete in this instance.

Theorem. There exist two regressive sets whose intersection is not regressive.

A classical result of descriptive set theory expresses every co-analytic subset of the real line as the union of an increasing sequence of Borel sets, the length of the chain being at most the first uncountable ordinal ℵ1 (see [5], [8]). An effective analog of this theorem, obtained by replacing co-analytic (Π11) and Borel (Δ11) with their lightface analogs, would represent every Π11 subset of the real line as the union of a chain of Δ11 sets. No such analog is true, however, because some Δ11 sets are not the union of their Δ11 subsets. For example, the set W, consisting of those reals which code well-orderings (in some standard coding) is Π11, but, by the boundedness principle ([3], [9]), any Δ11 subset of W contains codes only for well-orderings shorter than ω1, the first nonrecursive ordinal. Accordingly, we define the core of a Π11 set to be the union of its Δ11 subsets; clearly this is the largest subset of the given Π11 set for which an effective version of the classical representation could exist.

In §1, we develop the elementary properties of cores of Π11 sets. For example, such a core is itself Π11 and can be represented as the union of a chain of Δ11 sets in a natural way; the chain will have length at most ω1. We show that the core of a Π11 set is “almost all” of the set, while on the other hand there are uncountable Π11 sets with empty cores.

We analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form $\Sigma _{1 + \alpha + 3}^0 $, for α < ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α + 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of ${\rm{\Sigma }}_{1 + \alpha + 3}^0 $ determinacy, for $\alpha < \omega _1^{CK} $. This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: The least θ so that all winning strategies in ${\rm{\Sigma }}_4^0 $ games belong to Lθ+1 is the least so that $L_\theta \models {\rm{``}}{\cal P}\left( \omega \right)$ exists, and all wellfounded trees are ranked”.

We construct a model in which there is a strongly compact cardinal κ such thai the set S(κ, κ+) ={ a Є Pκκ+: o.t.(a) = (a⋂ κ)+}is non-stationary.