Interpretations of the alternative set theory
type: Item
instance of: scholarly article
title: Interpretations of the alternative set theory
zbMATH Open document ID: 0792.03036
DOI: 10.1007/BF01270464
author: Antonín Sochor
published in: Archive for Mathematical Logic
publication date: 1994-07-26T00:00:00Z
review text: The paper is another one in a series of papers concerning the author's main interest: the metamathematics of alternative set theory (AST) and higher-order arithmetics (here, the third-order arithmetic and some modifications are considered). The author shows the strength of the full comprehension scheme (the quantification of class variables is allowed in formulas defining classes) and the axiom of prolongation (being a form of expressing \(\omega_ 1\)-saturation). Adding only extensionality and existence of sets \((\text{Set}(0)\&(\forall x,y)(\exists z)(z= x\cup\{y\}))\) we obtain a theory strong enough for interpreting the whole AST or third-order arithmetic. None of the four given axioms can be deleted. Another interesting result of the paper is that adding a suitable axiom (an adaptation of constructibility) of AST we obtain a theory in which the system of all classes is fully described by the system of all sets and the class FN (the class of all natural numbers having the property that every subclass is a set, by prolongation there are natural numbers without this property). The principal technical means used in the paper are an adaption of the constructible process (described formerly), the ultrapower construction and the fact that saturated elementary equivalent models are isomorphic. Note that in a previous paper of the author and J. Sgall the consistency of AST + there is a nonconstructible class is proved by an adaptation of the forcing method.
reviewed by: Karel Čuda
Mathematics Subject Classification ID: 03E70
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03F35
zbMATH DE Number: 492851
zbMATH Keywords: interpretation
zbMATH Keywords: metamathematics of alternative set theory
zbMATH Keywords: higher-order arithmetics
zbMATH Keywords: third-order arithmetic
zbMATH Keywords: full comprehension scheme
zbMATH Keywords: axiom of prolongation
zbMATH Keywords: ultrapower
MaRDI profile type: Publication
Wikidata QID: Q60016470
cites work: Q3955351
cites work: Constructibility in higher order arithmetics
cites work: Choices of Convenient Sets
cites work: Q3037425
cites work: Q3996751
full work available at URL: https://doi.org/10.1007/bf01270464
OpenAlex ID: W1988223587
description: scientific article; zbMATH DE number 492851