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1 definition found
 for axiomatic set theory
From The Free On-line Dictionary of Computing (18 March 2015) :

  axiomatic set 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
     http://math.boisestate.edu/~holmes/holmes/nf.html)">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.

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