[FOM] the "best way to do set theory"
Martin Davis
martin at eipye.com
Thu Jul 7 18:38:43 EDT 2016
In a recent FOM post, Harvey contrasted an "absolute truth" stance on set
theory with what he sees as the more appropriate "good ways" and even a
"best way" to do set theory when going beyond ZFC. To my mind putting the
question in these terms is somewhat question begging.
I have been a great fan of Harvey's work on Pi-0-1 statements that are
provable only when the existence of a large cardinal is assumed. In many
cases the statement in question is equivalent to the consistency of ZFC +
"the existence of some large cardinal". To someone who is not a crude
formalist or an ultra finitist, the truth or falsity of a Pi-0-1 statement
is an objective fact (whether we will ever know which). Why would Harvey
even devote himself to finding nice combinatorial forms for such
propositions, if he didn't believe they were likely mathematical truths of
interest to mathematicians? Now there is Hilbert's famous dictum, that in
mathematics for an entity to exist means no more or less than that the
assumption of its existence is consistent.
Of course we know that we cannot hope for a mathematical proof of the
consistency of large cardinal axioms. In fact given the increasing
consistency strength as one marches up the large cardinal hierarchy, the
ice on which one is marching gets ever thinner as one ascends. Kanamori's
famous chart emphasizes this by placing "0=1" at the top! Certainly the
illusion of certainty is gone, but we are no worse off than empirical
scientists.
I think it also worth noting that because Pi-0-1 propositions can be
expressed as a polynomial equation having no natural number solutions, a
counter-example to a false Pi-0-1 proposition can be verified by a finite
number of additions and multiplications of integers.
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