[FOM] Lecture Abstracts

[Harvey Friedman]

VIRTUAL TO ACTUAL
UCLA Philosophy of Math Workshop
April 18, 2005

We provide a general framework for discussing contexts for Russell's
Paradox. We focus on the context of binary operations. Unexpectedly simple
principles of a most fundamental character form systems which are mutually
interpretable with countable set theory, ZF(C), Ramsey cardinals, and other
higher and lower levels of abstract set theory. We discuss the prospects for
a general calculus of (normally informal and vague) concepts, providing a
robust measure of their logical strength in terms of their equivalence with
levels of abstract mathematics (through abstract set theory).

UNPROVABLE THEOREMS
Cal Tech Math Colloq
April 19, 2005

We discuss the growing list of examples of simply stated concrete theorems
which are known to have no concrete proofs. It has been established that all
proofs must use methods that are far more abstract than the concepts in the
statement.

The most concrete examples discussed involve only finite objects (A,B
below). Most of them lie in discrete mathematics (A,B,C,D,H). The most
nonconcrete level of proof required goes well beyond the accepted axioms for
mathematics (ZFC), and uses "large cardinal" axioms (H).

A. In sufficiently long finite sequences from a finite set, certain blocks
are subsequences of certain later blocks.
B. In sufficiently tall finite trees, certain initial segments are embedded
in certain taller initial segments.
C. Multivariate functions on the integers have nonsurjective infinite
restrictions.
D. Any two countable sets of reals are pointwise continuously comparable.
E. Any permutation invariant Borel function from infinite sequences of reals
into itself maps some sequence into a subsequence.
F. For every Borel set in the plane, either it or its complement has a Borel
selection. 
G. Any Borel set in the plane that has a Borel selection on every compact
set has a Borel selection.
H. For any two multivariate functions on the natural numbers of expansive
linear growth, there are three infinite sets which bear a certain Boolean
relation with their images under the two functions. (Boolean Relation
Theory).

THE INEVITABILITY OF LOGICAL STRENGTH
UCLA Logic Seminar
April 22, 2005

An extreme form of logic skeptic claims that "the present formal systems
used for the foundations of mathematics are artificially strong, thereby
causing headaches such as the Godel incompleteness phenomena". The skeptic
continues by claiming that "logician's systems always contain overly general
assertions. and/or assertions about overly general notions, that are not
used in any significant way in normal mathematics. For example, induction
for all statements, or even all statements of certain restricted forms, is
far too general - mathematicians only use induction for natural statements
that actually arise. If logicians would tailor their formal systems to
conform to the naturalness of normal mathematics, then various logical
difficulties would disappear, and the story of the foundations of
mathematics would look radically different than it does today". This talk
presents some specific results that sharply refute aspects of this
viewpoint. 

BOOLEAN RELATION THEORY
ULCA Logic Colloquium
April 22, 2005

Boolean Relation Theory is the study of the Boolean relations between sets
and their forward images under multivariate functions. More precisely,
consider (V,K), where V is a natural set of multivariate functions and K is
a natural family of associated one dimensional sets. BRT studies statements
of the form "for all f1,...,fn in V, there exists A1,...,Am in K such that a
given Boolean relation holds between the A's and their forward images under
the f's". Experience shows that the analysis of such statements, even for
small n,m, and basic discrete (V,K), has diverse connections with various
mathematical topics, including large cardinals. We discuss the contents of
my book Boolean Relation Theory, nearing completion.

Harvey Friedman