Unformatted text preview: 1 INTERPRETING SET THEORY IN ORDINARY
Harvey M. Friedman*
Nineteenth Annual Tarski Lectures
Interpretations of Set Theory in Discrete Mathematics and
Delivered, April 13, 2007
Expanded, May 24, 2007
7. Interpretation hierarchy (lecture #1).
The general nature of concept calculus.
Better than, much better than.
Single varying quantity.
Single varying bit.
Persistently varying bit.
Some models. 1. INTERPRETATION HIERARCHY (Lecture #1).
The first of these Tarski lectures was devoted to a
discussion of Tarski interpretability. This bears
STARTLING OBSERVATION. Any two natural theories
to interpret EFA, are known (with small numbers
exceptions) to have: S is interpretable in T or
interpretable in S. The exceptions are believed
have comparability. S,T, known
to also Exceptions to linearity are known, which have clearly
identifiable elements of artificiality. It would be
interesting to explore just how natural one can
(artificially!) make the incomparability.
Because of this observation, there has emerged a rather
large linearly ordered table of “interpretation powers”
represented by natural formal systems. Generally, several
natural formal systems may occupy the same position.
We call this growing table, the Interpretation Hierarchy.
2. THE GENERAL NATURE OF CONCEPT CALCULUS. 2
Concept Calculus is a new mathematical/philosophical
program of wide scope. The development of Concept Calculus
began in Summer, 2006. There is a report in .
Concept Calculus promises to connect mathematics,
philosophy, and commonsense thinking in a radically new
Advances in Concept Calculus are made through rigorous
mathematical findings, and promise to be of immediate and
growing interest to philosophers.
Developments in Concept Calculus generally consist of the
a. An identification of a few related concepts from
informal thinking. In the various developments, the choice
of these concepts will vary greatly. In fact, all concepts
from ordinary language are prime targets.
b. Formulation of a variety of fundamental principles
involving these concepts. These various principles may have
various degrees of plausibility, and may even be
incompatible with each other. There may be no agreement
among philosophers as to just which principles to accept.
Concept Calculus is concerned only with logical structure.
c. Formulation of a variety of systems of such fundamental
principles in b. These systems generally combine several
such fundamental principles in some attractive way.
d. An identification of the location of the theory in the
e. In particular, for each of the resulting systems, a
determination of whether they interpret mathematics – as
formalized by ZFC.
A system T having interpretation power at least that of
mathematics (ZFC) has special significance. This means that
if T is without contradiction then mathematics (ZFC) is
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