[FOM] 729: Consistency of Mathematics/1

[Arnon Avron]

I'll try to briefly provide my personal answers to some of 
Friedman's questions.

1. Is mathematics consistent - I.e., free of contradiction?

   This question is too vague to be answered. It depends on
what one includes in "mathematics". 

There are parts of mathematics
that are certainly consistent. This includes, e.g., what
is needed in order even to understand the question
whether  something is consistent (that is, structures
defined by finitary inductive definitions, and 
reasoning about them, including corresponding induction principles).

On the other hand there are things that once were definitely
considered as part of mathematics, and are still frequently  taught
in mathematical courses, which we know to be inconsistent,
like naive set theory.

And I wonder if you include in "mathematics" all axioms
of strong infinity that have ever been suggested? What about
MK set theory? Or Category theory that goes well beyond ZF?

3. Is the continuum hypothesis true or false? 

 This question presumes platonic views about an absolute, unique
universe of "sets". To people like me, who are sure that we can
be sure about objects not accessible to us in *any* sense,
the question is meaningless, or at best has no absolute
answer, because the answer depends on the universe of sets
in which it is sought.  (Natural numbers and elements of HF are 
examples of objects which *are* accessible to me, "arbitrary" 
subsets of N or HF are examples of "objects" that are not).

0. What are the most monumental issues in the foundations of mathematics?

 This depends on one's views about what  are the issues 
in the foundations of mathematics that have the greatest general
intellectual interest (a very nice term of H. Friedman). For me,
the most important issue is to defend and restore the search for TRUTH
as the main goal of science, in a post-modern world in which even
talking about truth is considered as an intellectual sin. Accordingly,
for me the main issues in the foundations of mathematics
is to find out what parts of "mathematics" are absolutely true
and certain, what is the justification to see them as such,
and to what extent they suffice for the needs of science.


Arnon Avron