Archive for Mathematical Logic最新文献

We investigate the problem of punctual (fully primitive recursive) presentability of algebraic structures up to primitive recursive and computable isomorphism. We show that for mono-unary structures and undirected graphs, if a structure is not punctually categorical then it has infinitely many punctually non-isomorphic punctual presentations. We also show that the punctual degrees of any computably almost rigid structure as well as the order ((mathbb {Z},<)) are dense. Finally we characterise the Boolean algebras which have a punctually 1-decidable presentation that is computably isomorphic to a 1-decidable presentation.

We formalize various counting principles and compare their strengths over (V^{0}). In particular, we conjecture the following mutual independence between:

• a uniform version of modular counting principles and the pigeonhole principle for injections,

• a version of the oddtown theorem and modular counting principles of modulus p, where p is any natural number which is not a power of 2,

• and a version of Fisher’s inequality and modular counting principles.

Then, we give sufficient conditions to prove them. We give a variation of the notion of PHP-tree and k-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have o(n)-evaluations. As for the remaining two, we utilize well-known notions of p-tree and k-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.

On the set of all first-order complete theories (T(sigma )) of a language (sigma ) we define a binary operation ({cdot }) by the rule: (Tcdot S= {{,textrm{Th},}}({Atimes Bmid Amodels T ,,text {and},, Bmodels S})) for any complete theories (T, Sin T(sigma )). The structure (langle T(sigma );cdot rangle ) forms a commutative semigroup. A subsemigroup S of (langle T(sigma );cdot rangle ) is called an absorption’s formula definable semigroup if there is a complete theory (Tin T(sigma )) such that (S=langle {Xin T(sigma )mid Xcdot T=T};cdot rangle ). In this event we say that a theory T absorbs S. In the article we show that for any absorption’s formula definable semigroup S the class ({{,textrm{Mod},}}(S)={Ain {{,textrm{Mod},}}(sigma )mid Amodels T_0,,text {for some},, T_0in S}) is axiomatizable, and there is an idempotent element (Tin S) that absorbs S. Moreover, ({{,textrm{Mod},}}(S)) is finitely axiomatizable provided T is finitely axiomatizable. We also prove that ({{,textrm{Mod},}}(S)) is a quasivariety (variety) provided T is an universal (a positive universal) theory. Some examples are provided.

It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic HA, in a sense to be made precise. Our main target theory, the intuitionistic small set theory SST is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally.

We study the class of all strongly constructivizable models having (omega )-stable theories in a fixed finite rich signature. It is proved that the Tarski–Lindenbaum algebra of this class considered together with a Gödel numbering of the sentences is a Boolean (Sigma ^1_1)-algebra whose computable ultrafilters form a dense subset in the set of all ultrafilters; moreover, this algebra is universal with respect to the class of all Boolean (Sigma ^1_1)-algebras. This gives a characterization to the Tarski-Lindenbaum algebra of the class of all strongly constructivizable models with (omega )-stable theories.

In the context of a weak formal theory called Basic Intuitionistic Mathematics (textsf{BIM}), we study Brouwer’s Fan Theorem and a strong negation of the Fan Theorem, Kleene’s Alternative (to the Fan Theorem). We prove that the Fan Theorem is equivalent to contrapositions of a number of intuitionistically accepted axioms of countable choice and that Kleene’s Alternative is equivalent to strong negations of these statements. We discuss finite and infinite games and introduce a constructively useful notion of determinacy. We prove that the Fan Theorem is equivalent to the Intuitionistic Determinacy Theorem. This theorem says that every subset of Cantor space (2^omega ) is, in our constructively meaningful sense, determinate. Kleene’s Alternative is equivalent to a strong negation of a special case of this theorem. We also consider a uniform intermediate value theorem and a compactness theorem for classical propositional logic. The Fan Theorem is equivalent to each of these theorems and Kleene’s Alternative is equivalent to strong negations of them. We end with a note on ‘stronger’ Fan Theorems. The paper is a sequel to Veldman (Arch Math Logic 53:621–693, 2014).

We give several new equivalences of NIP for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the NIP context), in an analytic sense. Among other things, we show that for a first order theory T and a formula (phi (x,y)), the following are equivalent:

1. (i)

   (phi ) has NIP with respect to T.

2. (ii)

   For any global (phi )-type p(x) and any model M, if p is finitely satisfiable in M, then p is generalized DBSC definable over M. In particular, if M is countable, then p is DBSC definable over M. (Cf. Definition 3.7, Fact 3.8.)

3. (iii)

   For any global Keisler (phi )-measure (mu (x)) and any model M, if (mu ) is finitely satisfiable in M, then (mu ) is generalized Baire-1/2 definable over M. In particular, if M is countable, (mu ) is Baire-1/2 definable over M. (Cf. Definition 3.9.)

4. (iv)

   For any model M and any Keisler (phi )-measure (mu (x)) over M,

   $$begin{aligned} sup _{bin M}Big |frac{1}{k}sum _{i=1}^kphi (p_i,b)-mu (phi (x,b))Big |rightarrow 0, end{aligned}$$

   for almost every ((p_i)in S_{phi }(M)^{mathbb N}) with the product measure (mu ^{mathbb N}). (Cf. Theorem 4.4.)

5. (v)

   Suppose moreover that T is countable and NIP, then for any countable model M, the space of global M-finitely satisfied types/measures is a Rosenthal compactum. (Cf. Theorem 5.1.)

In set theory without the Axiom of Choice we prove that the assertion “For every metric space (X, d) with a Borel measure (mu ) such that the measure of every open ball is positive and finite, (X, d) is separable.’ is implied by the axiom of choice for countable collections of sets and implies the axiom of choice for countable collections of finite sets. We also show that neither implication is reversible in Zermelo–Fraenkel set theory weakend to permit the existence of atoms and that the second implication is not reversible in Zermelo–Fraenkel set theory. This gives an answer to a question of Dybowski and Górka (Arch Math Logic 62:735–749, 2023. https://doi.org/10.1007/s00153-023-00868-4).

In this paper we consider some fragments of (textsf{IOpen}) (Robinson arithmetic (mathsf Q) with induction for quantifier-free formulas) proposed by Harvey Friedman and answer some questions he asked about these theories. We prove that (mathsf {I(lit)}) is equivalent to (textsf{IOpen}) and is not finitely axiomatizable over (mathsf Q), establish some inclusion relations between (mathsf {I(=)}, mathsf {I(ne )}, mathsf {I(leqslant )}) and (textsf{I} (nleqslant )). We also prove that the set of diophantine equations solvable in models of (mathsf I (=)) is (algorithmically) decidable.

We address some phenomena about the interaction between lower semicontinuous submeasures on ({mathbb {N}}) and (F_{sigma }) ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological (F_{sigma }) ideals. We give a partial answers to the question of whether every nonpathological tall (F_{sigma }) ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological (F_{sigma }) ideals using sequences in Banach spaces.