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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
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.