[FOM] three upcoming talks

[Harvey Friedman]

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.