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Unformatted text preview: uskal’s
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impredicativity cannot be used for proving normal
mathematical theorems that involve only finite objects.
Here is a refutation of this.
THEOREM 17. Let T be a sufficiently tall rooted finite tree
of bounded valence (splitting). There is an inf preserving
embedding of some truncation of T into a taller truncation of
T which sends the highest vertices of the former into the
highest vertices of the latter.
Results from math logic again show that there is no predicative proof of this finite version of Kruskal’s Theorem.
9. ZERMELO SET THEORY.
Zermelo set theory with the axiom of choice, ZC, is a very
powerful fragment of the usual axioms and rules of
mathematics (ZFC), and is far more than what is needed to
formalize nearly all of existing normal mathematics. ZC
consists of the axioms of exten-sionality, pairing, union,
separation (comprehension), infinity, power set, and choice.
We now give an example of a uniformization theorem from
normal real analysis that cannot be proved in ZC. It can,
however, be proved in ZFC, using the Replacement axiom.
THEOREM 18. (using D.A. Martin). Let E be a Borel measurable
subset of the ordinary unit square which is symmetric about
the diagonal. Then E contains or is disjoint from the graph
of a Borel measurable function from the unit interval into
10. ZFC AND BEYOND. 14
Are there examples of discrete or even finite normal mathematics which cannot be carried out within the usual axioms
and rules of mathematics as formalized by ZFC?
This question naturally arises since even ZC is overkill for
nearly all normal mathematical contexts.
There is ongoing work suggesting that not only are there such
examples, but that there is a new thematic subject which cuts
across nearly all mathematical contexts, readily digestible
at the undergraduate mathematics level, but which can be
properly carried out with and only with the use of certain
previously proposed new axioms for mathematics going under
the name of “large cardinal axioms.”
However, it would be premature for me to report on this work
with any specificity at this important gathering, and so I
will end this lecture at this time. Thank you very much....
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