Athanassios Tzouvaras

Let ⊢ be the ordinary deduction relation of classical first-order logic. We provide an "analytic" subrelation ⊢a of ⊢ which for propositional logic is defined by the usual "containment" criterion Γ ⊢a φ iff Γ⊢φ and Atom ⊆ Atom, whereas for predicate logic, ⊢a is defined by the extended criterion Γ⊢aφ iff Γ⊢aφ and Atom ⊆' Atom, where Atom ⊆' Atom means that every atomic formula occurring in φ "essentially occurs" also in Γ. If Γ, φ are quantifier-free, then the notions "occurs" and "essentially occurs" for atoms between Γ and φ coincide. If ⊢ is formalized by Gentzen's calculus of sequents, then we show that ⊢a is axiomatizable by a proper fragment of analytic inference rules. This is mainly due to cut elimination. By "analytic inference rule " we understand here a rule r such that, if the sequent over the line is analytic, then so is the sequent under the line. We also discuss the notion of semantic relevance as contrasted to the previous syntactic one. We show that when introducing semantic sequents as axioms, i.e. when extending the pure logical axioms and rules by mathematical ones, the property of syntactic relevance is lost, since cut elimination no longer holds. We conclude that no purely syntactic notion of analytic deduction can ever replace successfully the complex semantico-syntactic deduction we already possess.

In the context of a distributive lattice we specify the sort of mappings that could be generally called ''negations'' and study their behavior under iteration. We show that there are periodic and nonperiodic ones. Natural periodic negations exist with periods 2, 3, and 4 and pace 2, as well as natural nonperiodic ones, arising from the interaction of interior and quasi interior mappings with the pseudocomplement. For any n and any even, negations of period n and pace s can also be constructed, but in a rather ad hoc and trivial way.

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 _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 on a set _A_ of atoms (or urelements) can be naturally represented by a pre-order relation \(\preccurlyeq \) of _A_ with the extra condition that it contains no minimal elements. Then, working in a mild strengthening of the theory \(\textrm{ZFA}\), 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 \(\langle A,\preccurlyeq \rangle \). The non-minimality condition for \(\preccurlyeq \) implies that all sets of \(LO(A,\preccurlyeq )\) are infinite and none of them is \(\subseteq \) -minimal. Next the pre-ordering \(\preccurlyeq \) is shifted (by a kind of simulation) to a pre-ordering \(\preccurlyeq ^+\) on \({{\mathcal {P}}}(A)\), 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 \(M_\alpha (A)\), along all ordinals \(\alpha \ge 1\), the“magmatic hierarchy”, such that \(M_1(A)=LO(A,\preccurlyeq )\), \(M_{\alpha +1}(A)=LO(M_\alpha (A),\subseteq )\), and \(M_\alpha (A)=\bigcup _{\beta, for a limit ordinal \(\alpha \). For every \(\alpha \ge 1\), \(M_\alpha (A)\subseteq V_\alpha (A)\), where \(V_\alpha (A)\) are the levels of the universe _V_(_A_) of \(\textrm{ZFA}\). The class \(M(A)=\bigcup _{\alpha \ge 1}M_\alpha (A)\) is the “magmatic universe above _A_.” The axioms of Powerset and Union (the latter in a restricted version) turn out to be true in \(\langle M(A),\in \rangle \). Besides it is shown that three of the five principles about magmas that Castoriadis proposed in his writings (mildly modified and adapted to the needs of formalization), \(M2^*\), \(M3^*\) and \(M5^*\), are true of _M_(_A_). A selection of excerpts from these writings, in which the concept of magma was first introduced and elaborated, is presented in the Introduction.

We adapt C. Freiling's axioms of symmetry 190–200) to models of set theory with classes by identifying small classes with sets getting thus a sequence of principles An, for n2, of increasing strength. Several equivalents of A2 are given. A2 is incompatible both with the foundation axiom and the antifoundation axioms AFA considered in Aczel . A hierarchy of symmetry degrees of preorderings is introduced and compared with An. Models are presented in which this hierarchy is strict. The main result of the paper is that a class X satisfies ¬An iff it has symmetry degree n−2

We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types , TST4. The result says that if is a standard transitive and rich model of TST4, then satisfies the 0,0,n-property, for all n≥2. This property has arisen in the context of the consistency problem of the theory New Foundations . The result is a weak form of the combinatorial condition that was shown in Tzouvaras [5] to be equivalent to the consistency of NF. Such weak versions were introduced in Tzouvaras [6] in order to relax the intractability of the original condition. The result strengthens one of the main theorems of Tzouvaras [5, Theorem 3.6] which is just equivalent to the 0,0,2-property

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, then there exist typical elements and only non‐typical ones. In particular this is true for the standard model 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 ⊆) 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).

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(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 LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ0-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 ${\Pi_1^1}$ -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