FOM: Set theory vs category theory
Soren Moller Riis
smriis at brics.dk
Tue Jan 27 05:58:20 EST 1998
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Set theory vs category theory:
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I have a question which concerns the debate on
Set theoretical foundation vs Categorical foundation.
Am I right that one of the most truly and amazing discovery in
logic/mathematics is the fact mathematics can be formalized?
The discovery began to take shape around the turn of the century.
Dedekin, Cantor, Frege, Zermelo, Russel and Whitehead prepared the ground
for this.
In the early 1920s the insight culminated with Frankel's completion of the
formalization of Cantors Set-theory. This formalization is called ZFC
(Zermelo-Frankel with the axiom of Choice). The discovery have an
{\it ontological} as well as an {\it epistemological} component.
The language of ZFC is as simple as one could possible have hoped for.
It is a first order language $L(\epsilon)$ which besides
the logical constants only contain one binary relation symbol $\epsilon$.
Despite this simplicity the language is so rich that it is fair to say that
it can express all mathematical facts (and including those theorems which
traditionally is taken to belong to Computer Science). No one have (without
"cheating" by using truth predicates or other "artificial" means) been
able to produce a single well-defined proposition of Mathematics or
Computer Science which cannot be translated into an equivalent proposition
in the language of ZFC. This was the Ontological part.
Equally remarkable all mathematical arguments (with a few exceptions)
can be achieved by repeated use of the principles expressed in the axioms
and axiom-schemes of ZFC. There are only 10 such axioms (one is an axiom-schema).
If we add one or two powerful large cardinal assumptions (or reflection
principles) we are left with a system of 11 or 12 basic principles of
reasoning. And that really seems to be just about the limit of the amount of
deductive strength which can be achieved by axiomatic methods. This was
the epistemological part.
This meta-mathematical discovery forces us to place all ordinary and traditional
mathematics world in which all facts and conjectures can be expressed in just
about the most simple language which one could have hoped for, and in a world where
any theorem which is ever going to be proved can be established by iterating a
handful of basic principles from logic.
This amazing discovery can be summarized: fom is possible!!!!
I suggest this highly non a priory fact should be part of a definition of fom.
fom is possible because of the "phenomenon" I just referred to. In the same sense
as physics or chemistry happens to make sense in the world in which we live.
It is not a priory clear topics such as physics or Chemistry should be possible
let alone have the tremendous success they have.
Now a question to the people who take the position that category theory can serve
as foundation for mathematics. Is this a NEW discovery??? What was the historical
events leading to this new insight? It is possible to express (like I did for
set theory) what this discovery was?? If there was no discovery or new insight taking
place I am puzzled how you can claim category theory can replace set theory at this
point. If there was a new insight taking place I would like to understand
the history (including the debate and the disputes) of what exactly took place.
Søren Riis
Footnote and another question:
I am very fascinated of some of the more concrete aspects of category theory
(though it is not my own research area). In computer science (especially in
semantics) I am convinced that category theory have been much more successful than
set theory (which is not even an alternative).
In many areas of mathematics there exists a central representation theorem.
For C^*-algebras we have the Gelfand-Neumann-Segel theorem showing every
abstract C^*-algebra have a concrete representation as functions of suitable space.
We have a representation theorem for Boolean Algebras. Abstract groups has
a representation as subgroups of permutation groups.
I am wondering: Why is there no representation theorem for category theory? There
are many different categories with many different properties. If the axioms of a
category are well stated there ought to be a representation theorem.
Or is this against the ideology?
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