axiomatic set theory

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 (http://math.boisestate.edu/~holmes/holmes/nf.html) ("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)