Department Colloquium
Department of Mathematics
Princeton University
October 25th, 2000
Does normal mathematics need new axioms
According to conventional wisdom (CW), normal mathematics
steers clear of foundational issues.
Only a minimal fragment of the currently
accepted axioms and rules for mathematics (ZFC)
are used (in any remotely essential way)
in current normal mathematics. The known
set theoretic independence results
from ZFC do not upset CW because they are known to involve
abnormal subsets of uncountable sets.
The known unprovability of consistency
does not upset this conventional wisdom since normal
mathematics is not concerned with properties of formal systems
for mathematical reasoning. The study of Diophantine equations is highly
normal, but the known impossibility of an algorithm does not upset CW since it does not lead to any need to reconsider the status of ZFC.
This CW has been attacked inconclusively at the margins:
every Borel subset of $R^2$ that is symmetric about y=x
contains or is disjoint from the graph
of a Borel function. It is necessary and
sufficient to use uncountably many
uncountable cardinalities to prove this Theorem.
Standards are very high for the genuine overthrow of CW. The new
Boolean relation theory (BRT) and its reduced forms, disjoint cover theory
(DCT) and formal partition theory (FPT),
promise to refute CW and ignite renewed interest in foundational issues.
Initial indications are that in
virtually any mathematical context (discrete or
continuous), these thematic investigations are deep, open ended, varied,
and explainable at the undergraduate level.
BRT grew out of two examples, which indicate its flavor.
The thinness theorem asserts that for F:N^k into N, there exists
an infinite subset A of N such that F[A^k] is not N.
The complementation theorem asserts that for any strictly
dominating F:N^k into N, there exists a (unique) infinite
subset A of N such that F[A^k]=N\A. We present statements of this
kind involving two functions and three sets provable using large cardinal axioms but not ZFC. Restricting to rather concrete functions does not change
matters.
We conjecture that the general theory of such statements
can be carried out with large cardinal axioms.
Partial results have been obtained.