[FOM] A proof that ZFC has no any omega-models

joeshipman at aol.com joeshipman at aol.com
Tue Feb 26 11:30:02 EST 2013

Eskew makes a useful distinction. If we want to justify mathematical practice, there are two distinct types of critics to answer: those who question the ontological assumptions of modern math as formalized in ZFC, and those who question the concrete arithmetical consequences of those assumptions. Both types of critics have valid epistemological concerns.

I am sympathetic to the first kind of critic because, in my opinion, there is no way we can obtain true knowledge about very large infinite sets. Even if we knew somehow that the existence of a measurable cardinal Kappa is consistent with ZFC, we can't tell whether there "really is" such a large set because of the possibility of deflating the "universe" down to V_Kappa or even to a much smaller submodel. Kronecker famously said "God created the integers, all else is the work of Man", but that gets the epistemology backwards: God can know what the true situation is about the Higher Infinite but we can only be confident about concrete arithmetical statements. I don't think we will run into any trouble by assuming the ZFC axioms, but I won't pretend there isn't an act of faith involved in accepting them as true or even as meaningful.

The second kind of critic is making more scientifically based objections. I can agree with this critic about the meaningfuless of statements of arithmetic, and can argue that the arithmetical consequences of ZFC (or just ZF since Choice is dispensable for arithmetical statements) are actually "true", but ultimately I need to justify the axioms, and some of that justification is based on intution or pragmatic experience. 

If Eskew will allow me to take the axioms of Peano Arithmetic as "true", I can justify other arithmetical statements that I can't prove directly from PA by various other means. For example, in the discussion on physics I showed how we might get EVIDENCE for statements of logical type as high as Sigma^0_4 from performing physical experiments (though Quantum Electrodynamics turns out to be less useful for this than I'd hoped). I'd be interested in hearing how far Eskew and others are willing to go in accepting axioms as "true" with no qualification or interpretive subtleties: is there any doubt about the truth of a theorem of PA? Is there any doubt about the "truth" of the Robertson-Seymour Graph Minor theorem, which has an arithmetical form but cannot be proved in PA?

-- JS

-----Original Message-----
From: Monroe Eskew <meskew at math.uci.edu>
To: tchow <tchow at alum.mit.edu>; Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Mon, Feb 25, 2013 12:16 pm
Subject: Re: [FOM] A proof that ZFC has no any omega-models

On Feb 24, 2013, at 3:12 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

> O.K., so in your scenario nothing really changes.  Is that supposed to be a 
bad thing?
> I mean, the whole point of Hilbert's program, as I understood it, was to 
enable us to stay in Cantor's paradise while easing ontological concerns that 
the "government" might have.  Although we now know that Hilbert's program can't 
succeed in its original form, it seems to me that you would have challenged 
Hilbert at the outset, asking him what the point of his program was all about if 
it wasn't going to lead to any change in mathematical practice.  But surely it's 
clear that the whole point of the program was to find good reasons *not* to 
change mathematical practice.

My understanding is that Hilbert's concerns were not about metaphysics but about 
consistency and the epistemology thereof.  In response to the paradoxes, he 
hoped to prove consistency from the bottom-up.  If something like this were 
possible and were discovered, we might imagine that it would have a profound 
effect mathematical method.  Furthermore, the failure of Hilbert's program led 
to very important mathematical developments, new fields, and new practices.

On the other hand, purely ontological concerns don't seem to lead to anything 
interesting.  If we can just add simple prefixes and suffixes to mathematical 
work in a uniform way to satisfy the ontologically conservative, then it seems 
we've found a rather boring solution to their problems.  And of course this is 
something anyone can do on their own: Have a metaphysical concern about some 
piece of mathematics?  Then if it suits you, imagine starting with a statement 
of the formal assumptions, and then when you're done reading, close off the 
conditional proof and draw your conclusion as to what those assumptions prove.  
If the "government" had its way in my scenario, then it might look to most 
people that they weren't concerned with the meat of what the mathematicians were 
doing, just that they weren't doing it with good manners.

My overall point is that if the only concerns about foundations of math are 
ontological, then this is not very interesting.  But if we move beyond ontology 
to epistemology and methodology, there are many interesting questions, like why 
should we think it is reasonable to assume the consistency of ZFC + there exists 
a measurable cardinal?  We can't just say because no one has found a 
contradiction in many years, because that applies to too many things, like the 
Riemann Hypothesis and its negation.  More generally, whether we adorn them with 
ontologically deflationary ornaments or not, what kinds of statements should we 
count as legitimate starting points for proofs?  Standard large cardinals?  
Determinacy?  The omega conjecture?  Saturated ideals?  Unique branches 
hypothesis?  PCF conjecture?  Perhaps eschew set theory and work from a category 
theoretic framework?  This type of question seems to me like a serious inquiry 
about foundations of mathematics, whereas purely metaph!
 ysical debates just seem like quibbling.

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