Archive for Mathematical Logic最新文献

We have provided a model-theoretic proof for the decidability of the additive structure of integers together with the function f mapping x to (lfloor varphi xrfloor ) where (varphi ) is the golden ratio.

Given that (L(mathbb {R})models {text {ZF}}+ {text {AD}}+{text {DC}}), we present conditions under which one can generically add new elements to (L(mathbb {R})) and obtain a model of ({text {ZF}}+ {text {AD}}+{text {DC}}). This work is motivated by the desire to identify the smallest cardinal (kappa ) in (L(mathbb {R})) for which one can generically add a new subset (gsubseteq kappa ) to (L(mathbb {R})) such that (L(mathbb {R})(g)models {text {ZF}}+ {text {AD}}+{text {DC}}).

In Arai (An ordinal analysis of a single stable ordinal, submitted) it is shown that an ordinal (sup _{N<omega }psi _{varOmega _{1}}(varepsilon _{varOmega _{{mathbb {S}}+N}+1})) is an upper bound for the proof-theoretic ordinal of a set theory (mathsf {KP}ell ^{r}+(Mprec _{Sigma _{1}}V)). In this paper we show that a second order arithmetic (Sigma ^{1-}_{2}{mathrm {-CA}}+Pi ^{1}_{1}{mathrm {-CA}}_{0}) proves the wellfoundedness up to (psi _{varOmega _{1}}(varepsilon _{varOmega _{{mathbb {S}}+N+1}})) for each N. It is easy to interpret (Sigma ^{1-}_{2}{mathrm {-CA}}+Pi ^{1}_{1}{mathrm {-CA}}_{0}) in (mathsf {KP}ell ^{r}+(Mprec _{Sigma _{1}}V)).

An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point (({{mathbf {IUL}}^{fp}})). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic (({mathbf {IUL}}), posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd involutive FL(_e)-chains which have finitely many positive idempotent elements.

For a non-zero natural number n, we work with finitary (Sigma ^0_n)-formulas (psi (x)) without parameters. We consider computable structures ({mathcal {S}}) such that the domain of ({mathcal {S}}) has infinitely many (Sigma ^0_n)-definable subsets. Following Goncharov and Kogabaev, we say that an infinite list of (Sigma ^0_n)-formulas is a (Sigma ^0_n)-classification for ({mathcal {S}}) if the list enumerates all (Sigma ^0_n)-definable subsets of ({mathcal {S}}) without repetitions. We show that an arbitrary computable ({mathcal {S}}) always has a ({{mathbf {0}}}^{(n)})-computable (Sigma ^0_n)-classification. On the other hand, we prove that this bound is sharp: we build a computable structure with no ({{mathbf {0}}}^{(n-1)})-computable (Sigma ^0_n)-classifications.

The minimal ordinal-connection axiom (MOC) was introduced by the first author in R. Freire. (South Am. J. Log. 2:347–359, 2016). We observe that (MOC) is equivalent to a number of statements on the existence of certain hierarchies on the universe, and that under global choice, (MOC) is in fact equivalent to the ({{,mathrm{GCH},}}). Our main results then show that (MOC) corresponds to a weak version of global choice in models of the ({{,mathrm{GCH},}}): it can fail in models of the ({{,mathrm{GCH},}}) without global choice, but also global choice can fail in models of (MOC).

Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

Second-order arithmetic and class theory are second-order theories of mathematical subjects of foundational importance, namely, arithmetic and set theory. Despite the similarity in appearance, there turned out to be significant mathematical dissimilarities between them. The present paper studies various principles in class theory, from such a comparative perspective between second-order arithmetic and class theory, and presents a few new dissimilarities between them.