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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 Frankel 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.