arXiv subjects

Athanassios Tzouvaras

Publications and source records attributed to Athanassios Tzouvaras.

12 recordsLinked to original sources

arXiv2026

The magmatic universe revisited: we define ordered pairs, relations, numbers and a special form of Separation

This is a companion article to \cite{Tz24}. We address the following two questions: 1) Can we define in the magmatic universe $M$ of \cite{Tz24} counterparts, or just analogues, of some very basic set-theoretic objects which are missing from $M$, specifically ordered pairs, binary relations, especially functions, as well as natural and ordinal numbers? 2) Are there restricted forms of the Separation and, perhaps, Replacement schemes that hold in $M$? We show the following: 1) Magmatic analogues of ordered pairs can indeed be defined by means of certain magmas called ``magmatic pairs''. However when we use them to generate relations and especially functions, some unsurmountable problems come up. These problems are due to the peculiarity of the elements of magmas to be distinguished into ``intended'' and ``collateral'' ones, a distinction due to their inherent relation of dependence. So magmatic functions are defined under very special conditions. 2) A certain class of formulas, called ``magmatic formulas'' is isolated, and the scheme of Separation restricted to these formulas, called ``Magmatic Separation Scheme'' (MSS), is proven to hold in $M$. On the other hand Replacement fails badly, and this is due to its functional form.

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arXiv2026

An observer-based approach to the sorites paradox and the logic derived from that

We approach the sorites paradox (SP) through an observer-based and time-dependent approach to truth of vague assertions. Formally the approach gives rise to a semantics, called fluxing-object semantics (FOS), because it involves models that contain ``fluxing objects'', that is, entities changing with time and observer. The models are equipped with agents (observers) and a linear and discrete time axis for time. The changing entities are represented by partial functions of time and agent, and this partiality causes truth-value gaps. If we interpret a truth-value gap as a third truth value, then FOS becomes a three-valued logic, which, quite interestingly, is proved identical to strong Kleene three-valued logic. The sorites phenomena can be represented in a structure of FOS as special objects that change imperceptibly with respect to an observer and with respect to a particular attribute. In this account the key point that eliminates the paradoxical character of sorites is the partiality of functions. When the observer is fixed the partiality corresponds to interrupted watching on his part. The interruption creates watching gaps during which the attributed property of the object as understood by the observer may change, without violating the imperceptibility condition. Interrupted watching has, according to experts on visual attention and focusing capabilities of humans, a firm physiological justification. The relationship of watching gaps with horizon crossing is also discussed.

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arXiv2026

Sets with dependent elements: A formalization of Castoriadis' notion of magma

We present a formalization of collections that Cornelius Castoriadis calls ``magmas'', especially the property which mainly characterizes them and distinguishes them from the usual cantorian sets. It is the property of their elements to {\em depend} on other elements, either in a one-way or a two-way manner, so that one cannot occur in a collection without the occurrence of those dependent on it. Such a dependence relation can be represented by a pre-order relation $\preccurlyeq$ Then, working in a mild strengthening of the theory ${\rm ZFA}$, where $A$ is an infinite set of atoms equipped with a primitive pre-ordering $\preccurlyeq$, the class of magmas over $A$ is represented by the class $LO(A,\preccurlyeq)$ of nonempty open subsets of $A$ with respect to the lower topology of $\langle A,\preccurlyeq\rangle$. Next the pre-ordering $\preccurlyeq$ is shifted (by a kind of simulation) to a pre-ordering $\preccurlyeq^+$ on ${\cal P}(A)$, which turns out to satisfy the same non-minimality condition as well, and which, happily, when restricted to $LO(A,\preccurlyeq)$ coincides with $\subseteq$. This allows us to define a hierarchy $M_α(A)$, along all ordinals $α\geq 1$, the``magmatic hierarchy'', such that $M_1(A)=LO(A,\preccurlyeq)$, $M_{α+1}(A)=LO(M_α(A),\subseteq)$, and $M_α(A)=\bigcup_{β<α}M_β(A)$, for a limit ordinal $α$. For every $α\geq 1$, $M_α(A)\subseteq V_α(A)$, where $V_α(A)$ are the levels of the universe $V(A)$ of ${\rm ZFA}$. The class $M(A)=\bigcup_{α\geq 1}M_α(A)$ is the ``magmatic universe above $A$.''

math.LO↗

arXiv2023

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 $ϕ(x)$ over finite $L$-structures, for a language $L$, as the limit of the probability of $ϕ(x)$ 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 $ϕ(x)$ ( i.e., the fraction of $L$-structures in which neither $ϕ$ nor $\neg ϕ$ 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.

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arXiv2023

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 $Π_2$ consequences of ZFC are provable in ${\rm LZFC}$. LZFC strongly extends Kripke-Platek (KP) set theory minus $Δ_0$-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 $α$-Mahlo models and $Π_1^1$-indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form $Loc({\rm ZFC}+ϕ)$ 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.

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arXiv2023

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 $Π_1^1$-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 $ϕ$ gives rise to a localization axiom $Loc^ϕ({\rm ZFC})$ saying that every set belongs to a transitive model of ZFC satisfying $ϕ$. The theories ${\rm LZFC}^ϕ={\rm LZFC}$+$Loc^ϕ({\rm ZFC})$ are local analogues of the theories ZFC+"there is a proper class of large cardinals $ψ$", where $ψ$ is a large cardinal property. If $sext(x)$ is the property of strong extendibility, it is shown that ${\rm LZFC}^{sext}$ proves Powerset and $Σ_1$-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 Vopěnka'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.

math.LO↗

arXiv2023

Propositional superposition logic

We extend classical Propositional Logic (PL) by adding a new primitive binary connective $φ|ψ$, intended to represent the "superposition" of sentences $φ$ and $ψ$, 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 $φ|ψ$ is strictly interpolated between $φ\wedgeψ$ and $φ\veeψ$. 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.

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arXiv2023

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, $(\forall v)φ(v)\rightarrowφ(t)$, 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.

math.LO↗

arXiv2023

Consequences of Vopěnka's Principle over weak set theories

It is shown that Vopěnka'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, $Δ_0$-Separation and Induction along $ω$, 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.

math.LO↗

arXiv2023

Some structural similarities between uncountable sets, powersets and the universe

We establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras ${\cal P}(κ)$, for a regular cardinal $κ$, and ${\cal C}(V)$ (the class of subclasses of the universe $V$), endowed with some ideals, especially the ideal $[κ]^{<κ}$ for ${\cal P}(κ)$, and the ideal of sets $V$ for ${\cal C}(V)$.

math.LO↗

arXiv2023

Russell's typicality as another randomness notion

We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first-order structure. We argue that the notion parallels Martin-Löf (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of ``small'' cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero sets. It is shown that if the domain $M$ satisfies $cf(|M|)>\aleph_0$, then there exist $|M|$ typical elements and only $<|M|$ non-typical ones. In particular this is true for the standard model ${\cal R}$ of second-order arithmetic. By allowing parameters in the defining formulas, we are led to relative typicality, which satisfies most of van Lambalgen's axioms for relative randomness. However van Lambalgen's theorem is false for relative typicality. The class of typical reals is incomparable (with respect to $\subseteq$) with the classes of ML-random, Schnorr random and computably random reals. Also the class of typical reals is closed under Turing degrees and under the jump operation (both ways).

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arXiv2023

Typicality à la Russell in set theory

We adjust the notion of typicality originated with Russell, which was introduced and studied in a previous paper for general first-order structures, to make it expressible in the language of set theory. The adopted definition of the class ${\rm NT}$ of nontypical sets comes out as a natural strengthening of Russell's initial definition, which employs properties of small (minority) extensions, when the latter are restricted to the various levels $V_ζ$ of $V$. This strengthening leads to defining ${\rm NT}$ as the class of sets that belong to some countable ordinal definable set. It follows that ${\rm OD}\subseteq {\rm NT}$ and hence ${\rm HOD}\subseteq {\rm HNT}$. It is proved that the class ${\rm HNT}$ of hereditarily nontypical sets is an inner model of ${\rm ZF}$. Moreover the (relative) consistency of $V\neq {\rm NT}$ is established, by showing that in many forcing extensions $M[G]$ the generic set $G$ is a typical element of $M[G]$, a fact which is fully in accord with the intuitive meaning of typicality. In particular it is consistent that there exist continuum many typical reals. In addition it follows from a result of Kanovei and Lyubetsky that ${\rm HOD}\neq {\rm HNT}$ is also relatively consistent. In particular it is consistent that ${\cal P}(ω)\cap {\rm OD}\subsetneq{\cal P}(ω)\cap {\rm NT}$. However many questions remain open, among them the consistency of ${\rm HOD}\neq {\rm HNT}\neq V$, ${\rm HOD}={\rm HNT}\neq V$ and ${\rm HOD}\neq {\rm HNT}= V$.

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