axiomatic set theory


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axiomatic set theory

(theory)
One of several approaches to set theory, consisting of a formal language for talking about sets and a collection of axioms describing how they behave.

There are many different axiomatisations for set theory. Each takes a slightly different approach to the problem of finding a theory that captures as much as possible of the intuitive idea of what a set is, while avoiding the paradoxes that result from accepting all of it, the most famous being Russell's paradox.

The main source of trouble in naive set theory is the idea that you can specify a set by saying whether each object in the universe is in the "set" or not. Accordingly, the most important differences between different axiomatisations of set theory concern the restrictions they place on this idea (known as "comprehension").

Zermelo Fr?nkel set theory, the most commonly used axiomatisation, gets round it by (in effect) saying that you can only use this principle to define subsets of existing sets.

NBG (von Neumann-Bernays-Goedel) set theory sort of allows comprehension for all formulae without restriction, but distinguishes between two kinds of set, so that the sets produced by applying comprehension are only second-class sets. NBG is exactly as powerful as ZF, in the sense that any statement that can be formalised in both theories is a theorem of ZF if and only if it is a theorem of ZFC.

MK (Morse-Kelley) set theory is a strengthened version of NBG, with a simpler axiom system. It is strictly stronger than NBG, and it is possible that NBG might be consistent but MK inconsistent.

NF ("New Foundations"), a theory developed by Willard Van Orman Quine, places a very different restriction on comprehension: it only works when the formula describing the membership condition for your putative set is "stratified", which means that it could be made to make sense if you worked in a system where every set had a level attached to it, so that a level-n set could only be a member of sets of level n+1. (This doesn't mean that there are actually levels attached to sets in NF). NF is very different from ZF; for instance, in NF the universe is a set (which it isn't in ZF, because the whole point of ZF is that it forbids sets that are "too large"), and it can be proved that the Axiom of Choice is false in NF!

ML ("Modern Logic") is to NF as NBG is to ZF. (Its name derives from the title of the book in which Quine introduced an early, defective, form of it). It is stronger than ZF (it can prove things that ZF can't), but if NF is consistent then ML is too.
References in periodicals archive ?
The book then explores the general question of isomorphism between mathematics and ontology before introducing axiomatic set theory alongside Badiou's ontological interpretation.
This is what permits him to think of axiomatic set theory as the basis of a materialist ontology: "Zermelo's axiom is therefore materialist in that it breaks with the figure of idealinguistery--whose price is the paradox of excess--in which the existential presentation of the multiple is directly inferred from a well-constructed language.
It is for this reason, among other reasons, that in his later work he abandons axiomatic set theory and introduces a new mathematical instrument, topos theory, that allows a plurality of "local" set theories and a plurality of logics.
Ludwig has proposed in the years between 1970 and 1978 Cermelo--Fraenkel axiomatic set theory.
We can reconstruct the theories of physics and technology by means of axiomatic set theory.
True, they had decided to use the context of Axiomatic Set Theory, ZF, as developed by Zermelo and Fraenkel for their work.
109], that "Cantor's set theory is so copious as to admit absolutely non-denumerable sets while axiomatic set theory [in particular, ZFC] is so limited [Skolem's paradox] that every non-denumerable set becomes denumerable in a higher system or in an absolute sense".
It is feasible that a symbiosis of the proposed theory and Vdovin set theory [1, 2] will permit to formulate a (presumably) non-contradictory axiomatic set theory which will represent the core of Cantor set theory in a maximally full manner as to the essence and the contents.
An untyped formalism based on axiomatic set theory, the standard way of formalizing everyday mathematics, can provide a simple, powerful foundation for writing formal specifications.
In axiomatic set theory (such as ZF), paradoxes are avoided by preventing the creation of sets that are too big.
Hintikka's proposal involves a rejection, inter alia, of the views that ordinary first-order logic ("standard logic") is the basic elementary logic and that axiomatic set theory is a natural framework for theorizing about mathematics.
One of the principal stakes of Alain Badiou's Being and Event is the articulation of a theory of the subject against the backdrop of the thesis that ontology, the science of being qua being, is none other than axiomatic set theory (specifically, the Zermelo-Fraenkel axiomatization ZF).