For each n > 0, two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are synonymous in the sense of deBouvere. It is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are synonymous with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories.

It has been stated by Tarski [5] and “proved” by Grzegorczyk [3] that:

Proved has been put in quotes, not because Grzegorczyk's proof is faulty but because the system he describes as mereology is in fact not Leśniewski's mereology.

Leśniewski's first attempt at describing the collective class, i.e. mereology, was done in ordinary language with no rigorous logical foundation. In describing the collective class, he needed to use the notion of distributive class. So as to clearly distinguish and expose the interplay between the two notions of class, he introduced his calculus of name (name being the distributive notion), which is also called ontology, since he used the primitive term “is.” At this stage, mereology included ontology. Then, in order to have a logically rigorous system, he developed as a basis, a propositional calculus with quantifiers and semantical categories (types), called protothetic. At this final stage, what is properly called mereology includes both protothetic and ontology.

What Grzegorczyk describes as mereology is even weaker than Leśniewski's initial version. To quote from [3]:

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.

A. H. Lachlan [2] and C. E. M. Yates [4] independently showed that minimal pairs of recursively enumerable (r.e.) degrees exist. Lachlan and Richard Ladner have shown (unpublished) that there is no uniform method for producing a minimal pair of r.e. degrees below a given nonzero r.e. degree. It is not known whether every nonzero r.e. degree bounds a r.e. minimal pair, but in the present paper it is shown (uniformly) that every high r.e. degree bounds a r.e. minimal pair. (A r.e. degree is said to be high if it contains a high set in the sense of Robert W. Robinson [3].)

Theorem. Let a be a recursively enumerable degree for which a′ = 0″. Then there are recursively enumerable degrees b0 and b1 such that0 < bi < a for each i ≤ 1, and b0 ⋂ b1 = 0.

The proof is based on the Lachlan minimal r.e. pair construction. For notation see Lachlan [2] or S. B. Cooper [1].

By Robinson [3] we can choose a r.e. representative A of the degree a, with uniformly recursive tower {As, ∣ s ≥ 0} of finite approximations to A, such that CA dominates every recursive function where

We define, stage by stage, finite sets Bi,s, i ≤ 1, s ≥ 0, in such a way that Bi, s + 1 ⊇ Bi,s for each i, s, and {Bi,s ∣ i ≤ 1, s ≥ 0} is uniformly recursive.

In [1] Diego showed that there are only finitely many nonequivalent formulas in n variables in the positive implicational propositional calculus P. He also gave a recursive construction of the corresponding algebra of formulas, the free Hilbert algebra In on n free generators. In the present paper we give an alternative proof of the finiteness of In, and another construction of free Hilbert algebras, yielding a normal form for implicational formulas. The main new result is that In is built up from n copies of a finite Boolean algebra. The proofs use Kripke models [2] rather than the algebraic techniques of [1].

Let V be a finite set of propositional variables, and let F(V) be the set of all formulas built up from V ⋃ {t} using → alone. The algebra defined on the equivalence classes , by setting

is a free Hilbert algebra I(V) on the free generators . A set T ⊆ F(V) is a theory if ⊦pA implies A ∈ T, and T is closed under modus ponens. For T a theory, T[A] is the theory {B ∣ A → B ∈ T}. A theory T is p-prime, where p ∈ V, if p ∉ T and, for any A ∈ F(V), A ∈ T or A → p ∈ T. A theory is prime if it is p-prime for some p. Pp(V) denotes the set of p-prime theories in F(V), P(V) the set of prime theories. T ∈ P(V) is minimal if there is no theory in P(V) strictly contained in T. Where X = {A1, …, An} is a finite set of formulas, let X → B be A1 →····→·An → B (ϕ → B is B). A formula A is a p-formula if p is the right-most variable occurring in A, i.e. if A is of the form X → p.

A very important example of almost strongly minimal theories are the algebraically closed fields. A. Macintyre has shown [3] that every ω1-categorical field is algebraically closed. Therefore every ω1-categorical field is almost strongly minimal. It will be shown that not every ω1-categorical ring is almost strongly minimal.

Let R0 be the factor ring C[y/(y2), where C[y] is the ring of polynomials in the indeterminate y over the field of complex numbers and (y2) the ideal generated by y2 in C[y].

It is straightforward to prove that R0 has the following properties:

1. R0 is a commutative ring with identity.

2. R0 is of characteristic 0.

3. For every polynomial p(x) = ∑ a1x1 ∈ R0[x] with of ai2 ≠ 0 for some i > 0 there is an a ∈ R0 such that p(a) · p(a) = 0.

4. For all x, y ∈ R0 such that x2 = 0 and y ≠ 0 there exists a z ∈ R0 with y · z = x.

5. There is an x ≠ 0 such that x2 = 0.

These properties can be ∀∃-axiomatised in a countable first order logic (see [4]). Let T be the set of these sentences. With Theorem 7 we get that T is model-complete.

If R is a model of T then I shall denote {a ∈ R ∣ a2 = 0}.

Subcreative sets, introduced by Blum, are known to coincide with the effectively speedable sets. Subcreative sets are shown to be the complete sets with respect to S-reducibility, a special case of Turing reducibility. Thus a set is effectively speedable exactly when it contains the solution to the halting problem in an easily decodable form. Several characterizations of subcreative sets are given, including the solution of an open problem of Blum, and are used to locate the subcreative sets with respect to the complete sets of other reducibilities. It is shown that q-cylindrification is an order-preserving map from the r.e. T-degrees to the r.e. S-degrees. Consequently, T-complete sets are precisely the r.e. sets whose q-cylindrifications are S-complete.

In this sequel to [7] the method of the consistency proof presented there is extended to provide a proof of the ω-consistency of the systems of number theory which were there shown consistent. This proof yields sharp bounds on the ordinal recursions required to establish the κ-consistency of these systems. The main technical innovation of this proof is the extension of what are essentially the methods of Ackermann [1] for handling finite sets of critical formulae of the first and second kinds to apply as well to sets of critical formulae in which the rank ordering is transfinite. The notation, definitions, and results of [7] will be presupposed throughout; we suggest the reader keep a copy of that paper at hand.

In [1] the ω-consistency of arithmetic was proved by a method which yields fine ordinal bounds for κ-consistency, κ ≥ 1. In this paper these bounds are shown to be best possible. The ω-consistency of a number-theoretic system S can be expressed thus: for all sentences ∃xM,

where ProvS is the proof predicate for S, if n is a nonnegative integer then n is the formal numeral (of S) for n, and if G is a formula then ˹G˺ is the Gödel number of G. The κ-consistency of S is the restriction of (1) to Σκ0 sentences ∃xM. The proof in [1] establishes the no-counterexample interpretation of (1), that is, the existence of a constructive functional Φ such that, for all sentences ∃xM, all numbers p, and all functions f,

(see [1, §2]). A functional Φ is an ω-consistency functional for S if it satisfies (2) for all sentences ∃xM, and a κ-consistency functional for S if it satisfies (2) for all Σκ0 sentences ∃xM.

The systems considered in [1] are those obtained from classical first-order arithmetic Z, including the schema for definition of primitive recursive (p.r.) functions, by adjoining, for some p.r. well-founded ordering ≺ of the nonnegative integers, the axiom schema

that is, the least number principle on ≺; it is equivalent to the schema of transfinite induction on ≺.

The purpose of this paper is to investigate continuity properties arising in elementary (i.e., first-order) logic in the hope of illuminating the special status of this logic. The continuity properties turn out to be closely related to conditions which characterize elementary logic uniquely, and lead to various further questions.

The basic setting of nonstandard analysis consists of a set-theoretical structure together with a map * from into another structure * of the same sort. The function * is taken to be an elementary embedding (in an appropriate sense) and is generally assumed to make * into an enlargement of [13]. The structures and * may be type-hierarchies as in [11] and [13] or they may be cumulative structures with ω levels as in [14]. The assumption that * is an enlargement of has been found to be the weakest hypothesis which allows for the familiar applications of nonstandard analysis in calculus, elementary topology, etc. Indeed, practice has shown that a smooth and useful theory can be achieved only by assuming also that * has some stronger properties such as the saturation properties first introduced in nonstandard analysis by Luxemburg [11].

This paper concerns an entirely new family of properties, stronger than the saturation properties. For each cardinal number κ, * satisfies the κ-isomorphism property (as an enlargement of ) if the following condition holds:

For each first order language L with fewer than κ nonlogical symbols, if and are elementarily equivalent structures for L whose domains, relations and functions are all internal (relative to * and ), then and are isomorphic.

In this paper we define by means of a partition property a decreasing sequence N = ‹Nα: α is an ordinal› of classes of ordinals. This property is a generalization of the nonexistence of special Aronszajn trees: the successor cardinal κ+ is in N0 iff there does not exist a special Aronszajn κ+-tree.

The interest in the classes Nα stems from their applicability in model theory, in particular to that aspect of model theory dealing with ordered and two-cardinal models. A model is κ-like iff < is a linear ordering of A of cardinality κ but such that every proper initial segment has cardinality < κ. is α-ordered iff ≼ is a reflexive, linear ordering of some subset of A with order type α. The sequence N can be characterized by a first-order sentence σ in the following manner: The sentence σ has a κ-like α-ordered model iff κ ∉ Nα. This characterization will allow us to translate various independence statements regarding the sequence N to statements about the independence of transfer properties. We say that the transfer property κ → λ holds iff every first-order sentence which has a κ-like model also has a λ-like model. κ ⇸ λ is the negation of κ → λ.