How effective indeed is present-day mathematics?

We argue that E. Wigner’s well-known claim that mathematics is unreasonably effective in physics (and not in the natural sciences in general, as the title of his article suggests) is only one side of the hill. The other side is the surprising insufficiency of present-day mathematics to capture the uniformities that arise in science outside physics. We describe roughly what the situation is in the areas of (a) everyday reasoning, (b) theory of meaning and (c) vagueness. We make also the point that mathematics, as we know it today, founded on the concept of set, need not be a conceptually final and closed system, but only a stage in a developing subject

Propositional superposition logic

We extend classical Propositional Logic (PL) by adding a new primitive binary connective $\varphi|\psi$, intended to represent the "superposition" of sentences $\varphi$ and $\psi$, an operation motivated by the corresponding notion of quantum mechanics, but not intended to capture all aspects of the latter as they appear in physics. To interpret the new connective, we extend the classical Boolean semantics by employing models of the form $\langle M,f\rangle$, where $M$ is an ordinary two-valued assignment for the sentences of PL and $f$ is a choice function for all pairs of classical sentences. In the new semantics $\varphi|\psi$ is strictly interpolated between $\varphi\wedge\psi$ and $\varphi\vee\psi$. By imposing several constraints on the choice functions we obtain corresponding notions of logical consequence relations and corresponding systems of tautologies, with respect to which $|$ satisfies some natural algebraic properties such as associativity, closedness under logical equivalence and distributivity over its dual connective. Thus various systems of Propositional Superposition Logic (PLS) arise as extensions of PL. Axiomatizations for these systems of tautologies are presented and soundness is shown for all of them. Completeness is proved for the weakest of these systems. For the other systems completeness holds if and only if every consistent set of sentences is extendible to a consistent and complete one, a condition whose truth is closely related to the validity of the deduction theorem.Comment: 55 page

Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by $Loc({\rm ZFC})$, says that every set belongs to a transitive model of ZFC. LZFC consists of $Loc({\rm ZFC})$ plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All $\Pi_2$ consequences of ZFC are provable in ${\rm LZFC}$. LZFC strongly extends Kripke-Platek (KP) set theory minus $\Delta_0$-Collection and minus $\in$-induction scheme. ZFC+``there is an inaccessible cardinal'' proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define $\alpha$-Mahlo models and $\Pi_1^1$-indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form $Loc({\rm ZFC}+\phi)$ are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.Comment: 38 page

Consequences of Vop\v{e}nka's Principle over weak set theories

It is shown that Vop\v{e}nka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product, $\Delta_0$-Separation and Induction along $\omega$, then ${\rm EST+VP}$ proves the axioms of Infinity, Replacement (thus also Separation) and Powerset. The result was motivated by previous results in \cite{Tz14}, as well as by H. Friedman's \cite{Fr05}, where a distinction is made among various forms of VP. As a corollary, ${\rm EST}+$Foundation$+{\rm VP}$=${\rm ZF+VP}$, and ${\rm EST}+$Foundation$+{\rm AC+VP}={\rm ZFC+VP}$. Also it is shown that the Foundation axiom is independent from ZF--\{Foundation\}+${\rm VP}$. It is open whether the Axiom of Choice is independent from ${\rm ZF+VP}$. A very weak form of choice follows from VP and some similar other forms of choice are introduced.Comment: 22 page

Asymptotic typicality degrees of properties over finite structures

In previous work we defined and studied a notion of typicality, originated with B. Russell, for properties and objects in the context of general infinite first-order structures. In this paper we consider this notion in the context of finite structures. In particular we define the typicality degree of a property $\phi(x)$ over finite $L$-structures, for a language $L$, as the limit of the probability of $\phi(x)$ to be typical in an arbitrary $L$-structure ${\cal M}$ of cardinality $n$, when $n$ goes to infinity. This poses the question whether the 0-1 law holds for typicality degrees for certain kinds of languages. One of the results of the paper is that, in contrast to the classical well-known fact that the 0-1 law holds for the sentences of every relational language, the 0-1 law fails for degrees of properties of relational languages containing unary predicates. On the other hand it is shown that the 0-1 law holds for degrees of some basic properties of graphs, and this gives rise to the conjecture that the 0-1 law holds for relational languages without unary predicates. Another theme is the ``neutrality'' degree of a property $\phi(x)$ ( i.e., the fraction of $L$-structures in which neither $\phi$ nor $\neg \phi$ is typical), and in particular the ``regular'' properties (i.e., those with limit neutrality degree $0$). All properties we dealt with, either of a relational or a functional language, are shown to be regular, but the question whether {\em every} such property is regular is open.Comment: 32 pages, 3 figure

Large transitive models in local {\rm ZFC}

This paper is a sequel to \cite{Tz10}, where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and $\Pi_1^1$-indescribable models, were considered. By analogy we refer to such models as "large models", and the properties in question as "large model properties". Continuing here in the same spirit we consider further large model properties, that resemble stronger large cardinals, namely, "elementarily embeddable", "extendible" and "strongly extendible", "critical" and "strongly critical", "self-critical'' and "strongly self-critical", the definitions of which involve elementary embeddings. Each large model property $\phi$ gives rise to a localization axiom $Loc^{\phi}({\rm ZFC})$ saying that every set belongs to a transitive model of ZFC satisfying $\phi$. The theories ${\rm LZFC}^\phi={\rm LZFC}$+$Loc^{\phi}({\rm ZFC})$ are local analogues of the theories ZFC+"there is a proper class of large cardinals $\psi$", where $\psi$ is a large cardinal property. If $sext(x)$ is the property of strong extendibility, it is shown that ${\rm LZFC}^{sext}$ proves Powerset and $\Sigma_1$-Collection. In order to refute $V=L$ over LZFC, we combine the existence of strongly critical models with an axiom of different flavor, the Tall Model Axiom ($TMA$). $V=L$ can also be refuted by $TMA$ plus the axiom $GC$ saying that "there is a greatest cardinal", although it is not known if $TMA+GC$ is consistent over LZFC. Finally Vop\v{e}nka's Principle ($VP$) and its impact on LZFC are examined. It is shown that ${\rm LZFC}^{sext}+VP$ proves Powerset and Replacement, i.e., ZFC is fully recovered. The same is true for some weaker variants of ${\rm LZFC}^{sext}$. Moreover the theories LZFC$^{sext}$+$VP$ and ZFC+$VP$ are shown to be identical.Comment: 32 page

Semantics for first-order superposition logic

We investigate how the sentence choice semantics (SCS) for propositional superposition logic (PLS) developed in \cite{Tz17} could be extended so as to successfully apply to first-order superposition logic(FOLS). There are two options for such an extension. The apparently more natural one is the formula choice semantics (FCS) based on choice functions for pairs of arbitrary formulas of the basis language. It is proved however that the universal instantiation scheme of FOL, $(\forall v)\varphi(v)\rightarrow\varphi(t)$, is false, as a scheme of tautologies, with respect to FCS. This causes the total failure of FCS as a candidate semantics. Then we turn to the other option which is a variant of SCS, since it uses again choice functions for pairs of sentences only. This semantics however presupposes that the applicability of the connective $|$ is restricted to quantifier-free sentences, and thus the class of well-formed formulas and sentences of the language is restricted too. Granted these syntactic restrictions, the usual axiomatizations of FOLS turn out to be sound and conditionally complete with respect to this second semantics, just like the corresponding systems of PLS.Comment: 35 page