[FOM] Clarification on Higher Set Theory
Harvey Friedman
friedman at math.ohio-state.edu
Thu Feb 13 01:13:37 EST 2003
>One of the topics of recurring interest on this list (and in the FOM
>community more generally) is the potential for 'justifying belief
>in' (despite the problems with this idiom I know no better) higher
>set theory by way of potential 'applications' to core mathematical
>theories outside set theory and, perhaps, to the mathematical
>theories central to physics.
>
>I take it that the goal of such research would be finding proofs of
>statements in such a mathematical or physical theory supplemented
>with higher set-theoretic assumptions that are
>
>(a) provably not provable or refutable in the (non-supplemented)
>mathematical or physical theory in question, and
>
>(b) have a clear sense when formulated in that theory, and which
>perhaps 'strike one as true' (in the sense that the (extensionally
>formulated) axiom of choice does this) given the concepts of that
>theory; or, alternatively and perhaps better, which have clear and
>important consequences for 'going problems' in the rest of that
>theory.
>
>My understanding from the list is that such results are in fact
>emerging from the work of Professor Friedman and others, but it
>would be nice to have some specific examples to chew on: a
>proposition formulable within intuitive number theory, geometry,
>group theory, etc. which one could present to the moderately well
>mathematically educated and say: "this is why you should believe in
>higher set theory."
>
You may be referring to certain propositions in discrete mathematics
which I prove using large cardinals, and show cannot be proved in ZFC.
However, I do not claim that they "strike one as true". Also, there
are not formulated within "intuitive number theory, geometry, group
theory".
When such discrete consequences of large cardinals, not provable
without them, reach a certain level of "beauty, depth, and breadth",
the mathematical community will come to accept them as an important
tool for obtaining mathematical results. The case could be bolstered
by results of a more philosophical nature that I am polishing now.
See "Equational Boolean Relation Theory" under my name at the
preprint server http://www.mathpreprints.com/math/Preprint/show/
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