Frege’s Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem (Theorem 2.9) shows that there is a model of a fragment of the Grundgesetze which defines a model of all the axioms of Zermelo–Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to Gödel’s constructible universe of sets and to Kripke and Platek’s idea of the projectum, as well as to a weak version of uniformization (which does not involve knowledge of Jensen’s fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension (Theorem 3.5). As an application, we resolve an analogue of the joint consistency problem in the predicative setting.

Introduction. The purpose of this paper is to show how results from the theory of inductive definitions can be used to obtain new compactness theorems for uncountable admissible languages. These will include improvements of the compactness theorem by J. Green [9].

In [2] Barwise studies admissible sets satisfying the Σ1-compactness theorem. Our approach is to consider admissible sets satisfying what could be called the abstract extended completeness theorem, that is, sets where the consequence relation of the admissible fragment LA is Σ1-definable over A. We will call such sets Σ1-complete. For countable admissible sets, Σ1-completeness follows from the completeness theorem for LA .

Having restricted our attention to Σ1-complete sets we are led to a stronger notion also true on countable admissible sets, namely what we shall call uniform Σ1-completeness. We will see that this notion can be viewed as extending to uncountable admissible sets, properties related to both the “recursion theory” and “proof theory” of countable admissible sets.

By following Barwise's recent approach to admissible sets allowing “urelements,” we show that there is a natural connection between certain structures arising from the theory of inductive definability, and uniformly Σ1-complete admissible sets . The structures we have in mind are called uniform Kleene structures.

Answering a question of A. Kanamori, we prove that there is no Jónsson filter over the least Jónsson cardinal. We also notice that, assuming □κ, the successor κ+ cannot be Jónsson.

The purpose of this paper is to present a kind of boundedness lemma for direct limits of coarse structural mice, and to indicate some applications to descriptive set theory. For instance, this allows us to show that under large cardinal or determinacy assumptions there is no prewellorder ≤ of length such that for some formula ψ and parameter z

if and only if

It is a peculiar experience to write up a result in this area. Following the work of Martin, Steel, Woodin, and other inner model theory experts, there is an enormous overhang of theorems and ideas, and it only takes one wandering pebble to restart the avalanche. For this reason I have chosen to center the exposition around the one pebble at 1.7 which I believe to be new. The applications discussed in section 2 involve routine modifications of known methods.

A detailed introduction to many of the techniques related to using the Martin-Steel inner model theory and Woodin's free extender algebra is given in the course of [1]. Certainly a familiarity with the Martin-Steel papers, [5] and [6], is a prerequisite, as is some knowledge of the free extender algebra. Probably anyone interested in this paper will already know the necessary descriptive set theory, most of which can be found in [4]. Discussion of earlier results in this direction can be found in [3] or [2].

The theorem introduced here provides a sufficient condition for an axiomatic class to be elementary. It resulted from consideration of a definability problem in , but is introduced in a less specific context and accompanied by applications to .

As in [La, 1], [La, 2], a reflexive, transitive binary relation R on a nonempty set S will be called a quasi-ordering or q.o. of S. A set of pairwise incomparable elements of S is an antichain and a subset of S linearly ordered by R is a chain. A chain {Si: i ∈ ω} with i > j if and only if (siRsj ∧ ¬ sjRSi) will be called an infinite descending chain. If S has no infinite antichains nor any infinite descending chains, S is well quasi-ordered by R, and is called a well quasi-ordering or w.q.o. Any subset of a w.q.o. is a w.q.o. An element s of a q.o. S is minimal in S if ∀t ∈ S(tRs → sRt). A subset S′ of S is a foundation for S if ∀t ∈ S∃s ∈ S′(sRt).

Independence of axioms was not investigated in [Faris]. We show that this quality is lacking, since 2 can be derived from the rest, which without it constitute an independent set. In the proofs of independence it is clearly more informative to decompose 9, C1abKN2abKN3abN5ab and 10, C3abKN2abN5ab into three and two propositions respectively.

By an algebra is meant an ordered set Γ = 〈V,R1, …, Rn, O1, …, Om〉, where V is a class, Ri (1 ≤ i ≤, n) is a relation on nj elements of V (i.e. Ri ⊆ Vni), and Oj (1 ≤ i ≤ n) is an operation on elements of V such that Oj(x1, … xmj) ∈ V) for all x1, …, xmj ∈ V). A sentence S of the first-order functional calculus is valid in Γ, if it contains just individual variables x1, x2, …, relation variables ϱ1, …,ϱn, where ϱi,- is nj-ary (1 ≤ i ≤ n), and operation variables σ1, …, σm, where σj is mj-ary (1 ≤, j ≤ m), and S holds if the individual variables are interpreted as ranging over V, ϱi is interpreted as Ri, and σi as Oj. If {Γi}i≤α is a (finite or infinite) sequence of algebras Γi, where Γi = 〈Vi, Ri〉 and Ri, is a binary relation, then by the direct productΓ = Πi<αΓi is meant the algebra Γ = 〈V, R〉, where V consists of all (finite or infinite) sequences x = 〈x1, x2, …, xi, …〉 with Xi ∈ Vi and where R is a binary relation such that two elements x and y of V are in the relation R if and only if xi and yi- are in the relation Ri for each i < α.

1. Let X be a statement which is formulated in the lower predicate calculus in terms of the relations of addition, multiplication, and equality, and — possibly — of the elements of a given commutative field M. Suppose moreover that X is in prenex normal form, e.g.

where Z does not include any further quantifiers. In order that X be satisfied by the algebraic closure M* of M, it is necessary and sufficient that there exist Herbrand functions (or choice functions) φ1(x1, x2), φ2(x1, x2), φ3(x1, x2, x3, x4) with arguments ranging over M* and taking values in M* such that

holds for all x1, x2, x3, x4 in M*. In general the definition of these functions is far from being unique, and a priori they bear no relation to the functions which are defined ‘naturally’ in M* i.e. the rational, and more generally the algebraic, functions with coefficients in M or M*. However, we shall show in the present paper that the entire domain of variation of the arguments x1, x2, x3, x4 — regarded as the affine space S4 over M* — can be divided up into a finite number of regions Di such that in each Di the functions ϕk can be chosen as algebraic functions of their arguments. Indeed, our complete result (§ 3) proves rather more than that. In particular, it turns out that the regions Di may be taken as differences of algebraic varieties in M*.

Let denote the ideal of Lebesgue measure zero subsets of the real line. Then add() denotes the minimal cardinality of a subset of whose union is not an element of . In [1] Bartoszynski gave an elegant combinatorial characterization of add(), namely: add() is the least cardinal number κ for which the following assertion fails:

For every family of at mostκ functions from ω to ω there is a function F from ω to the finite subsets of ω such that:

1. For each m, F(m) has at most m + 1 elements, and

2. for each f inthere are only finitely many m such that f(m) is not an element of F(m).

The symbol A(κ) will denote the assertion above about κ. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (Rm: m < ω) be any sequence of integers such that for each m Rm, ≤ Rm+1, and such that limm→∞Rm = ∞. Then the truth of the assertion A(κ) is preserved if in 1 we say instead that

1′. For each m, F(m) has at most Rm elements.

We shall use this observation later on. We now define three more statements, denoted B(κ), C(κ) and D(κ), about cardinal number κ.

In a series of five papers, Fitch has constructed a system of combinatory logic K′ which is adequate for much of classical analysis, and is demonstrably consistent if we assume the validity of transfinite induction up to a certain ordinal (at present undetermined). The system has the following peculiarities:

1.1. It is non-finitary, i.e. the class of Gödel-numbers of its theorems does not form the range of values of any recursive function (nor, as a matter of fact, of any function definable in elementary number theory).

1.2. It does not permit quantification over real numbers; i.e. it contains no theorems of the form

1.3. Because of 1.1, we cannot, except in trivial cases, construct actual proofs in K′; we have to resort to a metalanguage (of undetermined strength) in order to show that certain formulae are theorems of K′. Also because of 1.2, many important theorems of classical mathematics are not forthcoming in K′ itself, but only in the aforementioned metalanguage, e.g. in the form

where ‘x’ is a syntactical rather than a numerical variable, a U-real is an expression of K′ rather than a number, and ‘ … x– – –’ ascribes a syntactical rather than a real number-theoretic property to x.

The purpose of this paper is to construct a finitary metalanguage for K′ in which all of Fitch's important theorems may be proved.

We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. As an application, we introduce a new method to force (weak) measurability of all definable sets with respect to a certain (non-ccc) ideal.

The minimal Gödel class with identity (MGCI) is the class of closed, prenex quantificational formulas whose prefixes have the form ∀x1∀x2∃x3 and whose matrices contain arbitrary predicate letters and the identity sign “=”, but contain no function signs or individual constants. The MGCI was shown undecidable (for satisfiability) in 1983 [Go2]; this both refutes a claim of Gödel's [Gö, p. 443] and settles the decision problem for all prefix-classes of quantification theory with identity.

In this paper, we show the decidability of a natural subclass of the MGCI. The formulas in this subclass can be thought of as exploiting only half of the power of the existential quantifier. That is, since an MGCI formula has prefix ∀x1∀x2∃x3, in general its truth in a model requires for any elements a and b, the existence of both a witness for and a witness for . The formulas we consider demand less: they require, for any elements a and b, a witness for the unordered pair {a, b}, that is, a witness either for or for .

Craig showed in [3] that Beth's theorem fails for 2nd order logic. This is so since the syntax for 2nd order logic can be represented in the structure of natural numbers and itself can be characterized in 2nd order logic. Thus the satisfaction relation S is implicitly definable yet not explicitly definable (by a Tarski diagonal argument) and Beth fails. Kreisel pointed out in a review of Craig's paper that a similar argument shows that Beth's theorem fails for ω-logic or for any logic whose syntax can be represented in and in which can be characterized. In this paper we use the abstract model-theoretic notions of truth adequacy, truth-maximality and truth-completeness developed by Feferman in [4] to prove the following generalization of Craig's result: If L is a truth-complete logic, then no structure in which the syntax is represented is L-characterizable. This is applied via a corollary to give new examples of failures of Beth's theorem. Another way to construe this result is that truth-complete logics L have the desirable model-theoretic property (see [5]) that the structure in which the syntax is represented is not generalized finite (interpreting generalized finite as L-characterizability).

Since the basic notions of [4] are defined in terms of many-sorted logics, we will consider all our logics to be many-sorted.

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1 is speedable over another system S0 on a set of formulae A iff, for each recursive function h, there is a formula α in A such that the length of the shortest proof of α in S0 is larger than h of the shortest proof of α in S1. (Here we equate the length of a proof with something like the number of characters making it up, not its number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1 but unprovable in S0 from the collection A–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.

We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below, ϕ is an acceptable numbering of the partial recursive functions [Rog87] and Φ a (Blum) complexity measure associated with ϕ [Blu67], [DW83].