[FOM] Fwd: FOMUS/alternative foundations

John Baldwin jbaldwin at uic.edu
Fri Mar 25 00:07:55 EDT 2016

Louis Garde wrote:

May be the real difference is one's intuitions of mathematical objects:
- with the classical foundations, mathematical objets are 'sets',
structures are constructions on sets.
- with category theory and HoTT, mathematical objets are 'structures', and
sets are just some specific structures.
But it's just my understanding of it as an amateur of logic, interested in
"foundation of mathematics".

end quote

Bruno  Bensen wrote

It turns out that in practice mathematicians often identify two structures
whenever they are isomorphic. This is clearly in conflict with
set-theoretic foundational theories such as ZFC, since, for example, if A
\cong B and \empty \in A then in general \empty \notin A.

end quote

>From the model theoretic point of view, the objects of mathematics are
structures (sets with a specified collection of relations) and in most
cases  they are identified up to isomorphism.

I suppose the conflict Bentzen sees is that when taking the quotient
by the equivalence relation of isomorphism, we are apparently working with
classes.  Of course, that is no difficulty in Godel-Bernays (which is
conservative over ZF); but even more  concretely we can restrict to
structures bounded by a sufficiently large cardinal.

(Strikingly, there are a few cases -- e.g. the `undecidability' of the
Whitehead problem' where the underlying set theory structure is a key tool
solving a problem stated strictly in the vocabulary of groups.  Is every
Whitehead group free?

Other examples where "isomorphism = identity" have nothing to do with set
theoretic independence. But only with the need to distinguish two
isomorphic structures where one's domain is a subset of the other.  And
this can easily be done by choosing the appropritate category; it's just
automatic in set theory.)

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
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