[FOM] The Gold Standard

Nik Weaver nweaver at math.wustl.edu
Tue Feb 21 00:26:10 EST 2006

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.

Is there any reason to expect that ZFC, or even Z_2, has a
natural model *unless one is a platonist*?  The question is
not precise because I'm not saying what I mean by "natural".
Still, I think it's reasonably clear what I'm asking for.

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.  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.

Harvey Friedman wrote:

> Consider the following levels indicated roughly as follows:
> (there is nothing inclusive about them, and nothing unique
> about the justifications).
> ZFC + large large cardinals. Justification: "inconsistencies
> should be easy and not take long to find, like Kunen's for
> ZFC + j:V into V, and this hasn't happened yet over a 'long'
> period of time", and "go for it!"

I agree with "go for it!".  I'm not sure why inconsistencies
should be easy to find.  However, there may be plausible
reasons for thinking these systems are consistent.  For
example: "if you work with ZFC + some particular large
cardinal axiom for a while, you build up an intuition for
how the system works which leads you to believe it is

That could be a good reason for thinking the system is
consistent, and if the consequences of it being consistent
seem interesting, then indeed "go for it!"  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.)

> 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.

> Z_2. Justification: realist view of the real number system.

Fine, if one is a platonist.

So I ask: is there any non-platonist justification of the
assertion that Z_2 has a natural model?


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