Archive for Mathematical Logic最新文献

In this work, we study computable families of Turing degrees introduced and first studied by Arslanov and their numberings. We show that there exist finite families of Turing c.e. degrees both those with and without computable principal numberings and that every computable principal numbering of a family of Turing degrees is complete with respect to any element of the family. We also show that every computable family of Turing degrees has a complete with respect to each of its elements computable numbering even if it has no principal numberings. It follows from results by Mal’tsev and Ershov that complete numberings have nice programming tools and computational properties such as Kleene’s recursion theorems, Rice’s theorem, Visser’s ADN theorem, etc. Thus, every computable family of Turing degrees has a computable numbering with these properties. Finally, we prove that the Rogers semilattice of each such non-empty non-singleton family is infinite and is not a lattice.

Answering a question of Mitchell (Trans Am Math Soc 329(2):507–530, 1992) we show that a limit of accumulation points can be singular in ({mathcal {K}}). Some additional constructions are presented.

We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence (alpha ) which extends a weak arithmetical theory (which we take to be ({{,mathrm{IDelta _{0}+exp },}})) such that for some formula (Theta ) and any arithmetical sentence (varphi ), (Theta (ulcorner varphi urcorner )equiv varphi ) is provable in (alpha ). We say that a sentence (beta ) is definable in a sentence (alpha ), if there exists an unrelativized translation from the language of (beta ) to the language of (alpha ) which is identity on the arithmetical symbols and such that the translation of (beta ) is provable in (alpha ). Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we generalize the result of Pakhomov and Visser showing that the set of (Gödel codes of) definitions of truth is not (Sigma _2)-definable in the standard model of arithmetic. We conclude by remarking that no (Sigma _2)-sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic.

In this paper we study essential hereditary undecidability. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below R. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation essential tolerance, or, in the converse direction, lax interpretability that interacts in a good way with essential hereditary undecidability. We introduce the class of (Sigma ^0_1)-friendly theories and show that (Sigma ^0_1)-friendliness is sufficient but not necessary for essential hereditary undecidability. Finally, we adapt an argument due to Pakhomov, Murwanashyaka and Visser to show that there is no interpretability minimal essentially hereditarily undecidable theory.

We show that there is a ( varvec{Sigma }_4^0) ideal such that it’s neither extendable to any ( varvec{Pi }_3^0) ideal nor above the ideal ( textrm{Fin}times textrm{Fin} ) in the sense of Katětov order, answering a question from M. Hrušák.

In this note, I will list instances where in the literature on subcomplete forcing and its forcing principles (mostly in articles of my own), the assumption of the continuum hypothesis, or that we are working above the continuum, was omitted. I state the correct statements and provide or point to correct proofs. There are also some new results, most of which revolve around showing the necessity of the extra assumption.

We study the theory of K-vector spaces with a predicate for the union X of an infinite family of independent subspaces. We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-fold sums of X with itself. If K is finite this is no longer true, but we still have that a natural completion is near-model-complete.

Let ({mathcal {R}}) be a polynomially bounded o-minimal expansion of the real field. Let f(z) be a transcendental entire function of finite order (rho ) and type (sigma in [0,infty ]). The main purpose of this paper is to show that if ((rho <1)) or ((rho =1) and (sigma =0)), the restriction of f(z) to the real axis is not definable in ({mathcal {R}}). Furthermore, we give a generalization of this result for any (rho in [0,infty )).

The consistency of a second-order version of Morley’s Theorem on the number of countable models was proved in [EHMT23] with the aid of large cardinals. We here dispense with them.

Suppose that (kappa ) is indestructibly supercompact and there is a measurable cardinal (lambda > kappa ). It then follows that (A_0 = {delta < kappa mid delta ) is a measurable cardinal and the Mitchell ordering of normal measures over (delta ) is nonlinear(}) is unbounded in (kappa ). If the Mitchell ordering of normal measures over (lambda ) is also linear, then by reflection (and without any use of indestructibility), (A_1= {delta < kappa mid delta ) is a measurable cardinal and the Mitchell ordering of normal measures over (delta ) is linear(}) is unbounded in (kappa ) as well. The large cardinal hypothesis on (lambda ) is necessary. We demonstrate this by constructing via forcing two models in which (kappa ) is supercompact and (kappa ) exhibits an indestructibility property slightly weaker than full indestructibility but sufficient to infer that (A_0) is unbounded in (kappa ) if (lambda > kappa ) is measurable. In one of these models, for every measurable cardinal (delta ), the Mitchell ordering of normal measures over (delta ) is linear. In the other of these models, for every measurable cardinal (delta ), the Mitchell ordering of normal measures over (delta ) is nonlinear.