Archive for Mathematical Logic最新文献

In the framework of Stewart Shapiro, computations are performed directly on strings of symbols (numerals) whose abstract numerical interpretation is determined by a notation. Shapiro showed that a total unary function (unary relation) on natural numbers is computable in every injective notation if and only if it is almost constant or almost identity function (finite or co-finite set). We obtain a syntactic generalization of this theorem, in terms of quantifier-free definability, for functions and relations relatively intrinsically computable on certain types of equivalence structures. We also characterize the class of relations and partial functions of arbitrary finite arities which are computable in every notation (be it injective or not). We consider the same question for notations in which certain equivalence relations are assumed to be computable. Finally, we discuss connections with a theorem by Ash, Knight, Manasse and Slaman which allow us to deduce some (but not all) of our results, based on quantifier elimination.

We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of (textsf {ZFC}), then (textsf {DC}_{<kappa }) can be preserved in the symmetric extension of V in terms of symmetric system (langle {mathbb {P}},{mathcal {G}},{mathcal {F}}rangle ), if ({mathbb {P}}) is (kappa )-distributive and ({mathcal {F}}) is (kappa )-complete. Further we observe that if (delta <kappa ) and V is a model of (textsf {ZF}+textsf {DC}_{delta }), then (textsf {DC}_{delta }) can be preserved in the symmetric extension of V in terms of symmetric system (langle {mathbb {P}},{mathcal {G}},{mathcal {F}}rangle ), if ({mathbb {P}}) is ((delta +1))-strategically closed and ({mathcal {F}}) is (kappa )-complete.

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).