[FOM] The Gold Standard

[Harvey Friedman]

On 2/21/06 12:26 AM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> I have said that impredicative mathematics has no clear
> philosophical basis, whereas predicative mathematics has
> a clear philosophical basis.  Here's a sharper formulation:
> 
> Impredicative systems like ZFC and Z_2 lack canonical models.
> Predicative systems like ACA_0 and PA have canonical models.

Set theorist: ZFC has a canonical model. It is called (V,epsilon).

Real line realist: Z_2 has a conical model. It is built out of the acutal
real line.

Finitist: PA has no model at all.
 
> Is there any reason to expect that ZFC, or even Z_2, has a
> natural model *unless one is a platonist*?

Finitist: Is there any reason to expect aht PA has a natural model, or any
model, *unless one is an infinitist* who believes in such nonsense as
"completed infinite totalities"?
> 
> If one is a platonist and believes in the objective existence
> of a well-defined universe of sets then one simply has to argue
> that ZFC holds in that universe.

Set theorist: This is evident by reflecting on the picture of the
hereditarily finite sets.

>But if one is not a platonist
> it seems that any legitimate model must be in some (possibly
> loose) sense constructed, and then you have a basic difficulty
> in capturing impredicativity.

Set theorist: I don't know how to convert predicativists out of their silly
position to my evident position.

Predicativists: I don't know how to convert set theorists out of their silly
position to my evident position.
 
> However, that
> doesn't address the question of there being natural models.
> (It also seems sketchy on supporting the truth of arithmetical
> theorems provable in the system.)

Set theorist working with medium large cardinals: But we have natural
models. These are the so called inner models in inner model theory. And we
are trying to find inner models for the large large cardinals. We have to
study them hard in order to find them. Once we find them, all will be clear,
we will have our natural models.
 
>> ZFC. GOLD STANDARD (rightly or wrongly). Justification:
>> extrapolation from finite set theory.
> 
> Just not a very convincing justification, in my opinion.
> As Arnon Avron points out, lots of properties of finite sets
> fail disastrously for infinite sets.

I just replied to Avron. Let's get to work!
> 
>> Z_2. Justification: realist view of the real number system.
> 
> Fine, if one is a platonist.

The mathematical community likes real numbers in this sense, don't they?
> 
> So I ask: is there any non-platonist justification of the
> assertion that Z_2 has a natural model?

Platonist: Is that your definition of Platonism? Then tautologically, the
answer may well be no. But so what? Why aren't you a Platonist?
> 
Finitist: We know that PA does not have a model. Is there any non-platonist
justification of the assertion that PA is consistent?

Weaver, 2/21/06, 12:34AM:

> I flatly "cannot"
> clearly conceive of possible worlds occupied by structures of type
> <N-union-P(N),epsilon>, but rather that I know of no plausible way
> to conceive of such a world, and think I have good reasons for
> believing this can't be done.

Easy to "conceive". Let A be a FINITE set of absolutely enormous size. Just
ponder P(A). Then pretend that A is so big, you just can't list it. This is
in fact, reality - you can't actually list it.

Weaver, 2/21/06, 12:44AM

> Yes, but Weyl didn't suggest any philosophical reason for
> stopping at arithmetic definability.  If I remember right
> he was very clear about the possibility of going further
> but felt that for his purposes (developing 19th century
> real analysis in a predicatively acceptable way) there
> was no need to do this.

Actually, the amount of core math or normal math you pick up by doing higher
but stopping at various proposals for the limits of predicativity is, by
your standards, rather minimal even today.

>In other words his criterion for
> stopping at ACA_0 was esthetic.

I would surmise that Weyl was largely concerned, at least at some point,
with mathematical practice, and not philosophy. So it's not esthetic.

> Friedman claimed that it is "child's play" to come up with
> a coherent foundational stance corresponding to ACA_0.  I'd
> still like to see one.

It appeared in my original Gold Standard posting
http://www.cs.nyu.edu/pipermail/fom/2006-February/009882.html

Harvey Friedman