[FOM] repairing a bridge between mainstream mathematics and f.o.m.
joeshipman at aol.com
Tue Jul 21 14:43:00 EDT 2009
From: Vaughan Pratt <pratt at cs.stanford.edu>
This seems far more relevant than worries over the
consequences of the obvious fact that aleph-1 cannot be split evenly in
half, which I would take as a good argument for not reading too much
into the relevance of uncountable ordinals to analysis.
Concluding from that reasoning that aleph-2 is a better ordinal for
analysis is like jumping out of the 300 degree frying pan into the 600
You need the continuum to be larger than aleph-omega for this. Dunion,
following Freiling, wants Fubini-type theorems to be true for
non-measurable functions as well, but to make this work for
n-dimensional integrals you need n uncountable cardinals (plus some
technical conditions on measure-0 sets). It works even better if you
have a real-valued measurable cardinal (which means weakly inaccessibly
many cardinals less than the continuum). These were the main results of
my thesis (see the October 1990Transactios of the AMS).
That doesn't mean uncountable ordinals are relevant to analysis;
they're not. All it means is that there is no problem, and some
benefit, from assuming stroger regularity properties for measures on
product spaces than ZFC gives us. The "disconnect" arise when
explaining why exactly one is not supposed to make these assumptions
when doing analysis.
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