[FOM] The nature of set theory and why V \not= L
meskew at math.uci.edu
Wed Feb 4 20:26:39 EST 2009
There is a prevalent idea that the business of set theory is very
profound: To provide for and strengthen the foundations of
mathematics, to solve the unsolvable by discovering new axioms. This
of course leads many to worry: By what standards do we judge this
enterprise? How do we know these new axioms track the truth? What is
mathematical truth anyway?
But is this conception too grandiose? I've only been studying set
theory for a few years, but it seems to me that an accurate
description of the discipline is simply this: Set theory is the study
of infinity. Now this study necessarily touches upon foundations, but
we need not take foundations as the purpose or focus of set theory.
Most practitioners don't spend much time trying to decide on new
axioms, wringing their hands over whether they are right or true.
They just study different kinds of infinities and their
With this in mind, examine V=L. V=L implies that many very
interesting kinds of infinities are impossible. Thus the set theorist
does not take V=L as an axiom because it inhibits her study of
infinities. This is simple means-ends reasoning without resort to
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