FOM: Re: wrap on universes

[Harvey Friedman]

Reply to McLarty 11:44AM 4/16/99.

Before getting started, let me reiterate that something productive has come
out of this argument with McLarty. Namely, my posting 3:18PM 4/16/99,
"grand conjectures." Especially the second conjecture and the discussion
preceding that.

Now I want to respond to McLarty again. He has recycled his real hidden
agenda - which is this egregiously absurd, absolutely wrong-headed, grossly
misleading, totally backward analogy between Friedman and Grothendieck that
is intellectually incoherent, contains completely false assertions about my
work, and completely misses the point. Furthermore this ridiculously
bizarre analogy was thoroughly refuted when after it first surfaced in an
earlier posting of McLarty, which I responded to after recovering (with
difficulty) from my severe shock upon reading.

This ludicrous analosy first appeared in McLarty McLarty 5:53PM 4/2/99,
and was refuted in Friedman 4/3/99, 2:07. Here, in 11:44AM 4/16/99, McLarty
repeats this refuted outrage:

>        Friedman has combinatorial theorems formally provable in ZFC plus
>some consistency assumptions on cardinals. Friedman regards those proofs as
>essential uses of the cardinals themselves since the consistency statements
>have no genuine mathematical meaning.
>
>        Grothendieck and his heirs have theorems of number theory formally
>provable in ZFC. Grothendieck regards those proofs as essential uses of
>universes since the proofs which eliminate universes have no genuine
>mathematical meaning.

First of all, I do not "regard those proofs as
>essential uses of the cardinals themselves since the consistency statements
>have no genuine mathematical meaning." I informed you of this before in
>the earlier postings. If you wish to recycle nonsensical statements about
>other people's thoughts which have already been directly contradicted,
>then you are creating a nuisance on the FOM.

As I said earlier, the reason that the cardinals are considered necessary
for proving these combinatorial statements is the following:

1. Every proposed system for foundations of mathematics (e.g., those
proposed by set theorists) which proves these combinatorial statements,
also proves the relevant large cardinals exist.

2. Every system that proves these combinatorial statements contains an
interpretation of ZFC + the relevant large cardinals.

This has absolutely nothing to do with "mathematical meaningfullness" of
consistency statements about large cardinals. The
mathematical/metamathematical distinction comes into play regarding the
combinatorial statements being presented and analyzed. The ultimate
significance of this work rests substantially on the mathematical character
of the examples. As far as the choice of examples for the point of this
work is concerned, both the relevant consistency statements and the large
cardinal statements are completely unsuitable.

I said this before, and you are (I think) not (too) stupid. Yet you ignored
me and repeated this convoluted montrosity of an analogy.

As far as the statement about Grothendieck: in my second "grand conjecture"
I conjectured that these "uses" of universes are fake in the sense that
they can be elliminated by restriction. Now elimination by restriction is
going to result in perfectly mathematically meaningful assertions. For
instance, eliminating

every field has an algebraic closure

in favor of

every countable fiield has a countable algebraic closure

is perfectly mathematically meaningful in any normal sense of the word.

>        Joe Shipman asked whether Grothendeick universes were ever used to
>prove a theorem of ordinary mathematics. Privately he confirmed that he was
>not asking whether they were formally indispensible to any such theorem, but
>whether they occur in the published proof. I answered that they do occur, in
>famous results, though it is known they are formally eliminable. The experts
>agree with this.

But so does "rings have maximal ideals" and "fields have algebraic
closures." These are not real uses - they are fake uses. They are not uses
in the sense of Joe Shipman.

>        I say that a proof "uses universes" if it not only quantifies over
>universe-sized sets, but also defines those sets by properties that quantify
>over universe sized sets.

I agree that universes must be used to prove a theorem about universes. But
so what?

>Every textbook on cohomological number theory does
>this. Hartshorne ALGEBRAIC GEOMETRY does extensively. Wiles's paper
>justifies a key step by reference to a proof that uses universes in this
>sense. Papers in MODULAR FORMS AND FERMAT'S LAST THEOREM routinely do this.
>They could avoid it. But the authors want people to understand the argument.

This is where you make a claim that I directly refuted by talking to 3
experts.

These 3 experts, in no order, have the following properties:

They are fancy, very fancy, and very very fancy. All of them are algebraic
number theorists. Two of them know the Wiles' proof of FLT intimately and
have been involved in various ways, publicly and privately, with it before,
after, and during its publication.

YET NOT ONE OF THESE THREE EXPERTS COULD RECALL WHAT A UNIVERSE OR A TOPOS
IS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

They all heard of them, knew about them vaguely, but virtually never think
about them!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! And don't want to think
about them!!!!!!!

WHY? Because they have mentally eliminated them by restriction long, long,
long, long, ago.

So McLarty must be full of something or another.

>        The real questions of how to formalize universes, and how to
>interpret their use in the literature, have been obscured by sweepingly
>false claims such as "The use of universes in FLT - or any serious number
>theory has never, even remotely, been any kind of issue". Can anyone name a
>single textbook on cohomological number theory that does not use universes,
>in the sense defined above?

How many textbooks on modern mathematics don't refer to books that refer to
books that refer to books ... that mention arbitrary sets, groups, rings,
or fields? They just about all do. So what? Do they "use" arbitrary sets?

*******From my discussion with these three experts in number theory,
universes are one of the very best kept secrets in all of
mathematics!!!!!!!

McLarty writes:

>        I said Grothendieck (and not all his heirs) believes that using
>these methods at all levels to eliminate all use of universes in practice,
>would yield mathematically meaningless proofs.

Can you back this up with specific quotes from Grothendieck?

>The general theorems, which
>explicitly refer to universe-sized sets, are indispensible to mathematical
>meaning.

I believe this to be totally absurd. Is

countable fields have algebraic closures

mathematically meaningless?

>I see that no one yet publishes proofs in cohomological number
>theory without these theorems, even though it is well known how you could in
>principle. So I incline to say Grothendieck is right.

Just like: noone publishes algebra without using

every field has an algebraic closure
every ring has a maximal ideal?

Is that what you mean?

 >        Is this "excessive generality"? In a proof theoretic sense, if you
>only want the theorems in number theory, it sure is. In a practical sense,
>no one would pursue a career in algebraic number theory without studying
>Hartshorne.

In a practical sense, no one would pursue a career in algebraic mathematics
without studying Lang's Algebra (or equivalent). So what? Does that mean
that everybody is "using"

every field has an algebraic closure
every ring has a maximal ideal?