[FOM] A question about isomorphic structures

Sat Mar 26 16:31:12 EDT 2016

Re: your second paragraph, the answer is no - isomorphism is *not*
absolute. For example, let A be the linear order of rationals, and B be the
linear order of constructible (in the sense of Godel) reals. Then A and B
are isomorphic if e.g. there is a measurable cardinal, but not if e.g. V=L.
(This is a very coarse analysis of the situation.)

This is explored in more detail in a number of places, e.g.
http://arxiv.org/abs/math/9301208.

The "absolute version" of isomorphism is *potential isomorphism*, or
equivalently elementary equivalence in the infinitary logic $L_{\infty
\omega}$. It's a theorem essentially of Karp that A and B are isomorphic in
some forcing extension iff A and B are $L_{\infty\omega}$-equivalent.

 - Noah Schweber

On Sat, Mar 26, 2016 at 10:01 AM, Arnon Avron <aa at tau.ac.il> wrote:

> I cannot quite understand the claim that
> in mathematics "isomorphic" means  "equal". Showing that two
> *different* structures are actually isomorphic is sometimes
> a non-trivial mathematical theorem. It would not make much sense
> to prove such theorems if the two structures are completely identified.
> It is also certainly not true that mathematicians always identify two
> structures whenever they are isomorphic. indeed, Bruno Bentzen
> was careful to write: "mathematicians OFTEN identify two structures
> whenever they are isomorphic", adding the fuzzy word "often" here. So
> I wonder: does HoTT make this "often" precise and does not the fact
> that mathematicians do not always make this identification
> "clearly in conflict with foundational theories such as" HoTT?
>
> But the main question I would like to pose here is the following.
> The only definition I know of "isomorphic" is that there *exists
> a function* which is an isomorphism. Is there any theorem, or just
> a reason to believe, that this is an absolute notion, that is: that
> the answer to the question whether two given structures A and B are
> isomorphic or not never depends on the universe we are working in (and so,
> in my understanding, the set-theory we are working in), and what
> functions between  the carrier of A and the carrier of B are available
> in that universe?
>
> By the way,  is "the carrier of a structure A" not
> a *set*? If not - what is it? And if yes - does not it means that
> we still need some set theory for the alleged "alternative foundations"?
>
> I guess that I'll get here answers of the type "you are trying
> here to understand things from the set-theoretical point of view,
> using the set-theoretical language, instead of trying to understand
> them from the internal HoTT point of view, using its own language".
> This might well be true. But then I see no point in arguing with a
> camp with which I do not share even my basic  language and concepts.
>
> This brings me back to a point I have made here in the past: I often
> feel that some discussions on FOM  are pointless because people
> sometimes speak completely different  languages, although
> they seem to use the same words. An example I have given here several
> times is that my main trouble with devoted constructivists
> (as, if I am not totally mistaken, most HoTTists are)
> is that the "not" they use in doing mathematics (though *not*
> in ordinary life or even in describing their own work in
> the metalanguage) is *not* the "not" all people normally use, and they
> even pretend *not* to understand the classical "not" that all
> people (including them) normally use. (All the "*not*"s above
> are classical, of course).
>
> A clarification: after writing "HoTTist" (in the style
> of "constructivist") completely innocently,
> I discovered that it might *sound* (especially in the
> context of what I was writing about) as having an hidden second
> meaning. I did not and do not mean it. Obviously, when two groups
> do not understand each other, the problem is the same for both.
>
> Arnon Avron
>
> On Fri, Mar 25, 2016 at 03:13:39AM +0000, Bruno Bentzen wrote:
> > Harvey Friedman:
> > > A FOUNDATION FOR MATHEMATICS
> > > There is a standard classical meaning for this. Namely, to provide a
> > > precise "workable" rule book for mathematical "activity", that
> > > provides suitable criteria for "correctness" or "legitimacy".
> >
> > If I am interpreting you correctly, a key aspect of a foundational
> theory's "philosophical coherence" would be the question - perhaps
> motivated by Wittgenstein's famous slogan "meaning is use" - of whether or
> not it is compatible with the way mathematics is done in practice. In this
> sense, I think that the set-theoretical approach has some drawbacks.
> >
> > 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.
> >
> > Harvey Friedman:
> > > I favor the approach of carefully adding a lot of sugar to ZFC, rather
> > > than bringing in foreign elements or replacing aspects of ZFC.
> >
> > I believe this issue is not a matter of adding more "sugar" to ZFC or
> other set theory, it is a limitation of the very set-theoretical framework,
> since equality, as given by the axiom of extensionality, is a much stronger
> requirement than isomorphism.
> >
> > In contrast, this idea can be embodied within type theory. It is
> (roughly) the univalence axiom: two structures are equal if they are
> isomorphic.This is made possible by the fact that type theory, like
> category theory, has the nice property that every definable predicate of in
> the system is isomorphic-invariant. Therefore, HoTT admits a structuralist
> (and constuctivist) foundations of mathematics.
> >
> > Harvey Friedman:
> > >>> Among the obvious merits of ZFC as a foundation for mathematics,
> there
> > >>> is one that I often see not sufficiently emphasized. That is, its
> > >>> PHILOSOPHICAL COHERENCE.
> > >>>
> > >>> ../..
> > >>>
> > >>> But my impression is that the more radical proposals, particularly
> > >>> HOTT, philosophical coherence is proudly thrown away.
> > >
> > > That is my understanding, which may or may not be correct. (However,
> > > see below for a pointer to an internet manuscript ).This certainly
> > > prima facie precludes HOTT as any kind of reasonable "foundation for
> > > mathematics" in any standard sense.
> >
> > In this respect, I cannot easily understand why HoTT would lack of
> "philosophical coherence" while ZFC would not.
> >
> > I am learning a lot with this great discussion, however, I think we
> cannot objectively move forward with it since a better clarification of the
> word "philosophical coherence" is obviously required.
> >
> > Bruno Bentzen
>
> > _______________________________________________
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> > FOM at cs.nyu.edu
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>
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