[FOM] set theory, V = L, CH
Harvey Friedman
hmflogic at gmail.com
Sun Aug 24 00:33:45 EDT 2014
Joe Shipman wrote
I disagree that ordinary mathematicians would regard V=L as a good
axiom, because it cannot even be STATED without bringing in much more
set theory than most mathematicians are comfortable with.
Can you provide an alternative to V=L, consistent with it, that has
similar consequences at the level of statements of descriptive set
theory, but which is recognizably "mathematical"?
*****************
In NBG, formulate V = L as follows. If (A,epsilon) satisfies ZF, where A is
a transitive proper class, then V = A. For mathematicians sensitive to the
foundational issues, they are going to be comfortable with ZF, and so this
will work fine for them. Any alternative is far worse from their point of
view. Of course, as perfect Pi01 develops, this becomes a totally new
situation.
**********
Monroe Eskew has acted productively on my suggestion that he present some
examples of interaction between higher set theory and other areas of
mathematics. He came up with some items from functional analysis. This area
is put into a better category than general topology by the mathematical
community. I will do a little investigation on Eskew's letter and hopefully
get back to the FOM about it soon. One thing is already obvious -
mathematicians in many areas of math are going to have zero interest in
such things, and the issue is this: do they have zero interest in such
things simply because it is not their area, or because they are aware of a
heavy set theoretic component, or because they instinctively feel a heavy
set theoretic component?
Harvey Friedman
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