There are shown to be many ε-Hermitian spaces, and an isometry criterion is stated which holds under MAℵ1 and is false under 2ℵ0< 2ℵ1.

In spite of the fact that true arithmetic reduces to the monadic second-order theory of the real line, Peano arithmetic cannot be interpreted in the monadic second-order theory of the real line.

Using side-by-side Sacks forcing, it is shown that it is consistent that 2ω be large and that there be many types of ultrafilters of character ω1.

Friedberg [3] showed that every degree of unsolvability above 0′ is the jump of some degree, and Sacks [9] showed that the degrees above 0′ which are recursively enumerable (r.e.) in 0′ are the jumps of the r.e. degrees.

In this paper we examine the extent to which the Sacks jump theorem can be combined with the minimal pair theorem of Lachlan [4] and Yates [13]. We prove below that there is a degree c > 0′ which is r.e. in 0′ but which is not the jump of half a minimal pair of r.e. degrees.

This extends Yates' result [13] proving the existence of noncappable degrees (that is, r.e. degrees a < 0′ for which there is no corresponding r.e. b > 0 with a ∩ b = 0).

It also throws more light on the class PS of promptly simple degrees. It was shown by Ambos-Spies, Jockusch, Shore and Soare [1] that PS coincides with the class NC of noncappable degrees, and with the class LC of all low-cuppable degrees, and (using earlier work of Maass, Shore and Stob [5]) that PS splits every class Hn or Ln, n ≥ 0, in the high-low hierarchy of r.e. degrees.

If c > 0′, with c r.e. in 0′, let

and call c−1 the jump class for c. It is easy to see that every jump class contains members of PS (= NC = LC). By Sacks [8] there exists a low a ∈ LC, where of course [a, 0′] (= {br.e. ∣a ≤ b ≤ 0′}) ⊆ LC = PS. But by Robinson [7] [a, 0′] intersects with every jump class.

Though many working mathematicians embrace a rough and ready form of Platonism, that venerable position has suffered a checkered philosophical career. Indeed the three schools of thought with which most of us began our official philosophizing about mathematics—Intuitionism, Formalism, and Logicism—all stand in fundamental disagreement with Platonism. Nevertheless, various versions of Platonistic thinking survive in contemporary philosophical circles. The aim of this paper is to describe these views, and, as my title suggests, to trace their roots.

I'll begin with some preliminary remarks about the big three schools. This seems a reasonable approach to the issues both because most observers are familiar, at least in a general way, with the tenets of Intuitionism, Formalism, and Logicism, and because it is in reaction to these that contemporary Platonism has taken shape.

Suppose that I is a precipitous ideal over a cardinal κ and j is a generic embedding of I. What is the nature of j? If we assume the existence of a supercompact cardinal then, by Foreman, Magidor and Shelah [FMS], it is quite unclear where some of such j's are coming from. On the other hand, if ¬∃κ0(κ) = κ++, then, by Mitchell [Mi], the restriction of j to the core model is its iterated ultrapower by measures of it. A natural question arising here is if each iterated ultrapower of can be obtained as the restriction of a generic embedding of a precipitous ideal. Notice that there are obvious limitations. Thus the ultrapower of by a measure over λ cannot be obtained as a generic embedding by a precipitous ideal over κ ≠ λ. But if we fix κ and use iterated ultrapowers of which are based on κ, then the answer is positive. Namely a stronger statement is true:

Theorem. Let τ be an ordinal and κ a measurable cardinal. There exists a generic extension V* of V so that NSℵ1 (the nonstationary ideal on ℵ1) is precipitous and, for every iterated ultrapower i of V of length ≤ τ by measures of V based on κ, there exists a stationary set forcing “the generic ultrapower restricted to V is i”.

Our aim will be to prove this theorem. We assume that the reader is familiar with the paper [JMMiP] by Jech, Magidor, Mitchell and Prikry. We shall use the method of that paper for constructing precipitous ideals. Ideas of Levinski [L] for blowing up 2ℵ1 preserving precipitousness and of our own earlier paper [Gi] for linking together indiscernibles will be used also.

The aim of this paper is to enrich the algebraic-axiomatic approach to recursion theory developed in [1] by an analogue to the classical arithmetical hierarchy and an abstract notion of degree. The results presented here are rather initial and elementary; indeed, the main problem was the very choice of right abstract concepts.

The purpose of this paper is to extend the coding method (see Beller, Jensen and Welch [82]) into the context of large cardinals.

Theorem. Suppose μ is a normal measure on κ in V and 〈 V, A〉 ⊨ ZFC. Then there is a 〈V, A〉-definable forcing for producing a real R such that:

(a) V[R] ⊨ ZFC and A is V[R]-definable with parameter R.

(b) V[R] = L[μ*, R], where μ* is a normal measure on κ in V[R] extending μ.

(c) V ⊨ GCH → is cardinal and cofinality preserving.

Corollary. It is consistent that μ is a normal measure, R ⊆ ω is not set-generic over L[μ] and 0+ ∉ L[μ, R].

Some other corollaries will be discussed in §4 of the paper.

The main difficulty in L[μ]-coding lies in the problem of “stationary restraint”.

As in all coding constructions, conditions will be of the form belonging to an initial segment of the cardinals, where p(γ) is a condition for almost disjoint coding into a subset of γ +. In addition for limit cardinals γ in Domain(p), 〈p γ′∣γ′ < γ〉 serves to code pγ .

An important restriction in coding arguments is that for inaccessible for only a nonstationary set of γ′ < γ. The reason is that otherwise there are conflicts between the restraint imposed by the different and the need to code extensions of pγ below γ.

For several years much research has been directed at ideals associated with large cardinals. In this paper we discuss ideal families and use this tool to investigate the saturation properties of various large cardinal ideals.

Much of the work in this paper is a direct outgrowth of material presented in my doctoral dissertation written at the University of Michigan under the patient guidance of Andreas Blass. I would also like to thank the referee for several helpful questions and comments on the original version of this paper.

We discuss saturating ultrafilters on N, relating them to other types of non-principal ultrafilter. (a) There is an (ω, c)-saturating ultrafllter on N iff 2λ ≤ c for every λ < c and there is no cover of R by fewer than c nowhere dense sets, (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafllter on N is (ω, κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω, κ)-saturating, and (ii) the set of (ω, κ)-saturating ultrafilters is invariant under homeomorphisms of βN/N. (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω, c)-saturating.

When equational logic for partial functions is interpreted using Kleene equality as the predicate, the relation of logical consequence may be said to express what identities of partial functions follow from a given set of identities. In the analogous situation for total functions, there is a complete set of inference rules consisting of reflexivity, symmetry, transitivity, replacement, and substitution; in the case of partial functions, unrestricted substitution fails to be a valid inference rule, and there remains the question of how to obtain a complete set of rules. The first part of the present paper shows that completeness cannot be obtained by a mere restriction of the substitution rule, for a counterexample shows that even a rule allowing substitution in all consequentially valid instances fails, in conjunction with the other four rules, to yield a complete set of rules.

The second part of the paper defines a combined rule of transitivity-substitution which, in conjunction with reflexivity, symmetry, replacement, and substitution only of variables, yields a complete set of rules. The new rule is first stated in a form that allows an unbounded number of premises, and then is altered to a three-premise form. In both forms, the rule suffers from the shortcoming that in its formulation an auxiliary notion of conditional existence is involved, which is given by a recursive syntactic definition. As a result, the set of instantiations of the rule is recursively enumerable, but not (apparently) recursive (assuming a recursive set of premises).

As Dekker [3] suggested, certain fragments of the isols can exhibit an arithmetic rather more resembling that of the natural numbers than the general isols do. One such natural fragment is Barback's “tame models” (cf. [2], [6] and [7]), whose roots go back to Nerode [8]. In this paper we study another variety of such fragments: the hyper-torre isols introduced by Ellentuck [4]. Let Y denote an infinite isol with D(Y) the collection of all isols A ≤ f∧(Y) for some recursive and combinational unary function f. (Here, as usual, f ∧ is the Myhill-Nerode extension of f to the isols).

Working in the language of first-order arithmetic we consider models of the base theory P−. Suppose M is a model of P− and let M satisfy induction for Σ1-formulas. First it is shown that the Friedberg-Muchnik finite injury argument can be performed inside M, and then, using a blocking method for the requirements, we prove that the Sacks splitting construction can be done in M. So, the “amount” of induction needed to perform the known finite injury priority arguments is Σ1-induction.

In this paper we study the varieties of Stone algebras (S, ∧, ∨, *, 0, 1) and double Stone algebras (D, ∧, ∨, *, +, 0, 1). Our primary interest is to give a structural description of the algebraically and existentially closed members of both classes. Our technique is an application of the natural dualities of Davey [6] and Clark and Krauss [5]. This approach gives a description of the desired models as the algebras of all continuous structure-preserving maps from certain structured Boolean spaces into the generating algebra for the variety. In each case the resulting description can be converted in a natural way into a finite ∀∃-axiomatization for these models. For Stone algebras these axioms appeared earlier in Schmid [20], [21] and in Schmitt [22].

Since both cases we consider satisfy the amalgamation property, the existentially closed members form a model companion for the variety which is also its model completion. Moreover, it is also ℵ0 categorical and its countably infinite member is the unique countable homogeneous universal model for the variety. In the case of Stone algebras, explicit constructions for this model appear in Schmitt [22] and Weispfenning [23]. We give here an explicit construction for double Stone algebras of S. Hayes.

This work was motivated by a problem of Stanley Burris. In [4] Burris and Werner superseded many previous results by showing that for any finite algebra A, the universal Horn class ISP has a model companion. Weispfenning [24], [25] discovered that this model companion is always ℵ0 categorical and has a primitive recursive ∀∃-axiomatization. In spite of these very general theorems, there are few instances in which a structural description of the (any!) existentially closed members of ISP is available. Burris and Werner [4] solve this problem in a special setting.

We prove that the following are consistent with ZFC:

1. 2ω = ℵω1 + #x039A;c = ℵω1 + ΚΒ = ΚU = ω2 (for measure and category simultaneously).

2. .

This concludes the discussion about the cofinality of Κc.

Work in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X′, its Turing jump, is recursive in ∅′ and high if X′ computes ∅″. Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of A ⊕ W is recursive in the jump of A. We prove that there are no deep degrees other than the recursive one.

Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turing functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that (W ⊕ A)′ is forced to disagree with Φ(−;A′). The conversion has some ambiguity; in particular, A cannot be found uniformly from W.

We also show that there is a “moderately” deep degree: There is a low nonzero degree whose join with any other low degree is not high.

The axiom of symmetry (Aℵ0) asserts that for every function F: ω2 → ω2 there is a pair of reals x and y in ω2 so that y is not in the countable set {(F(x))n: n< ω} coded by F(x) and x is not in the set coded by F(y). A(Γ) denotes axiom Aℵ0 with the restriction that graph(F) belongs to the pointclass Γ. In §2 we prove A(). In §3 we show A(), A() and ω2 ⊈ L are equivalent. In §4 several effective versions of A(REC) are examined.

The separation, uniformization, and other properties of the Borel and projective hierarchies over hyperfinite sets are investigated and compared to the corresponding properties in classical descriptive set theory. The techniques used in this investigation also provide some results about countably determined sets and functions, as well as an improvement of an earlier theorem of Kunen and Miller.

We use models of set theory described in [2] and [3] to prove the consistency of several combinatorial principles, for example:

If ℱ is any filter on N containing all the cofinite sets, then there is a finite-to-one function f: N → N such that f(ℱ) is either the filter of cofinite sets or an ultrafilter.

As a consequence of our combinatorial principles, we also obtain the consistency of:

The partial ordering P of slenderness classes of abelian groups, denned and studied in [4], is a four-element chain.

In the remainder of this Introduction, we shall define our terminology and state the combinatorial principles to be considered. In §2, we shall establish some implications between these principles. In §3, we shall prove our consistency results by showing that the strongest of our principles holds in models of set theory constructed in [2] and [3].

A filter on N will always mean a proper filter containing all cofinite sets; in particular, an ultrafilter will necessarily be nonprincipal. We write N ↗ N for the set of nondecreasing functions from the set N of positive integers into itself. A subset ℐ of N ↗ N is called an ideal if it is closed downward (if f(n) ≤ g(n) for all n and if g ∈ ℐ, then f ∈ ℐ) and closed under binary maximum (if f(n) = max(g(n), h(n)) for all n and if g, h ∈ ℐ then f ∈ ℐ).