[FOM] three upcoming talks
Harvey Friedman
friedman at math.ohio-state.edu
Mon Feb 27 09:13:49 EST 2012
MAXIMALITY AND INCOMPLETENESS
Harvard University
March 1, 2012
Harvey M. Friedman
Distinguished University Professor of Mathematics, Philosophy,
Computer Science
Ohio State University
Prepared February 1, 2012
We show how the usual axioms for mathematics (ZFC) are insufficient
even in transparent countable and finite contexts. We begin with the
familiar "every countable binary relation contains a maximal square -
(an A x A)". The proof is entirely constructive. We formulate "every
'nice' binary relation contains a 'nice' maximal square", using
ambient spaces with modest structure. I.e., "every 'invariant' binary
relation on rational [0,16]^32 contains an 'invariant' maximal
square". This statement can be analyzed for purely order theoretic
notions of invariance that treat 1,...,16 as distinguished. We discuss
cases that can only be proved by going well beyond the usual ZFC axioms.
BOOLEAN RELATION THEORY AND INCOMPLETENESS
MIT Mathematics Department
March 2, 2012
by
Harvey M. Friedman
Distinguished University Professor of Mathematics, Philosophy,
Computer Science
Ohio State University
Prepared February 1, 2012
Boolean Relation Theory provides a general framework for formulating
diverse elementary mathematical problems. We start with the Thin Set
Theorem: "for all f:N^k into N there exists infinite A contained in N
such that f[A^k] is not N", and the Complementation Theorem: "for all
strictly dominating f:N^k into N there exists (unique) infinite A
contained in N such that f[A^k] = N\A". These Theorems assert that
"for any multivariate function of a certain kind there exists a one
dimensional set of a certain kind such that a given Boolean relation
holds between the set and its (Cartesian power) image under the
function". In the general theory, we fix a basic collection V of
multivariate functions and a basic collection K of one dimensional
sets, and consider "for any f_1,...,f_k in V there exists A_1,...,A_n
in K such that a given Boolean relation holds between A_1,...,A_n and
the (Cartesian power) images of A_1,...,A_n under the f_1,...,f_k". We
show how the usual axioms of mathematics (ZFC) are insufficient
already with k = 2, n = 3, V = the multivariate functions from N into
N of expansive linear growth, and K = the infinite subsets of N.
GOEDEL'S SECOND THEOREM: ITS MEANING AND USE
AMS Special Session
Washington, D.C.
March 17, 2012
by
Harvey M. Friedman
Distinguished University Professor of Mathematics, Philosophy,
Computer Science
Ohio State University
Goedel's Second Incompleteness Theorem is a spectacular finding of the
greatest general intellectual interest. The Theorem was established in
the early 1930's, and we discuss some transparent rigorous
formulations that have come much later. A weak form of the Theorem has
a particularly transparent proof that provides a certain kind of
information, raising the question of whether the full theorem can be
treated analogously. The Theorem is used in an essential way for
Concrete Mathematical Incompleteness. The Theorem also has finite
forms, which raise a number of open issues. We use Strict Reverse
Mathematics to address the consistency of Peano Arithmetic. We close
by comparing the inconsistency of Peano Arithmetic to such
developments as spontaneous disintegration of the sun, annihilation of
human life by black holes, gamma ray bursts, or comets, practical
finite P = NP, perpetual motion machines, time travel, fast neutrinos,
cold fusion, Jurassic Park, and million year life spans.
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