FOM: Subsets: Reply to Podnieks on V=L

Fri Apr 24 08:10:48 EDT 1998

Whether the principle "all sets can be built by using the axioms of ZF"
justifies not only Regularity but also V=L depends on whether you consider a
set "built" if you already have it as an element of another set.  If you do,
then you don't need to place an additional restriction on elements of the Power
Set of N by requiring constructibility.  If you require that each set be
*individually* built then V=L is reasonable if you accept all the ordinals as
primordial "stages" implicit in the word "built".  If you want to also restrict
the collection stages so each set is constructed in a completely specifiable way
you get the "strongly constructible sets", which form Cohen's "minimal model", a
countable model of ZFC defined in section III.6 of his book, in which every set
has a name.  The practical distinction is that regularity only asks us to ignore
non-well-founded sets, which we don't need for math, while V=L asks us to
ignore certain real numbers, or rather asks us to believe that all the reals we
got in another way (as elements of a powerset) are "already there" in L.  Those
with no strong notion of "arbitrary subset of N" may prefer your proposal. - JS

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