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 Fränkel 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.
(2003-09-21)
Nearby terms:
AXIOM* « Axiomatic Architecture Description Language
« axiomatic semantics « axiomatic set theory
» Axiom of Choice » Axiom of Comprehension » AXLE
|
