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Definitions of axiomatic set theory

  1. One of several approaches to set theory, consistingof a formal language for talking about sets and a collectionof axioms describing how they behave.There are many different axiomatisations for set theory.Each takes a slightly different approach to the problem offinding a theory that captures as much as possible of theintuitive idea of what a set is, while avoiding theparadoxes that result from accepting all of it, the mostfamous being Russell's paradox.The main source of trouble in naive set theory is the ideathat you can specify a set by saying whether each object inthe universe is in the "set" or not. Accordingly, the mostimportant differences between different axiomatisations of settheory concern the restrictions they place on this idea (knownas "comprehension").Zermelo Fränkel set theory, the most commonly usedaxiomatisation, gets round it by (in effect) saying that you canonly use this principle to define subsets of existing sets.NBG (von Neumann-Bernays-Goedel) set theory sort of allowscomprehension for all formulae without restriction, butdistinguishes between two kinds of set, so that the setsproduced by applying comprehension are only second-class sets.NBG is exactly as powerful as ZF, in the sense that anystatement that can be formalised in both theories is a theoremof 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 thanNBG, and it is possible that NBG might be consistent but MKinconsistent.NF (http://math.boisestate.edu/~holmes/holmes/nf.html) ("NewFoundations"), a theory developed by Willard Van Orman Quine,places a very different restriction on comprehension: it onlyworks when the formula describing the membership condition foryour putative set is "stratified", which means that it couldbe made to make sense if you worked in a system where everyset had a level attached to it, so that a level-n set couldonly be a member of sets of level n+1. (This doesn't meanthat there are actually levels attached to sets in NF). NF isvery different from ZF; for instance, in NF the universe is aset (which it isn't in ZF, because the whole point of ZF isthat it forbids sets that are "too large"), and it can beproved that the Axiom of Choice is false in NF!ML ("Modern Logic") is to NF as NBG is to ZF. (Its namederives from the title of the book in which Quine introducedan early, defective, form of it). It is stronger than ZF (itcan prove things that ZF can't), but if NF is consistent thenML is too. foldoc_fs
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