[FOM] Hamkins's multiverse and ultrafinitism

Wed Nov 29 21:43:39 EST 2017

FOM:

We have a "clear idea" of plus and times from the natural model of PA, which "comes before" PA in an ontological sense.  Formal theories capture the nature of this model only approximately, as best as mathematical logicians can manage.

- Martin Dowd

-----Original Message-----
From: John Baldwin <jbaldwin at uic.edu>
To: tchow <tchow at alum.mit.edu>; Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Wed, Nov 29, 2017 5:01 pm
Subject: Re: [FOM] Hamkins's multiverse and ultrafinitism

I am indebted to Roman Kossak for the following insight.  It is a big leap from thinking we have a clear grasp of  the set of natural numbers with the successor operation to
thinking we have a clear grasp of the natural numbers with successor, addition and multiplication.  It seems to me that any illusion that we have such a clear grasp stems  (contra Tim's point 3) precisely
categoricity of 2nd order  Peano arithmetic.

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607

On Tue, Nov 28, 2017 at 9:36 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:

After reading a bit of the relevant sections of the paper by Fletcher et al. that Mikhail Katz cited, I have a few additional comments.

If I understand correctly, the frame of mind that we are being invited to adopt is that we don't have, and perhaps never have had, a particularly clear notion of what *the* natural numbers are.  Instead, because of independence phenomena (we are supposed to imagine that we have powerful tools available to construct nonstandard models of arithmetic almost at will, satisfying almost any properties that we want), we come to realize that "natural number system" is a vague term referring to any member of an infinite family of nonstandard models of arithmetic.  Unenlightened people of the past, when they thought they were talking about a specific member of this family, were really not singling out any particular member at all; rather, they were just narrowing down the field a little by picking a few properties that excluded some candidates but left many others in the running.  In particular, a "generic" choice of viable candidate would *not* be the minimal model.

I don't dispute that it's possible to adopt this frame of mind.  What I dispute is that the mathematical community at large will ever be tempted into adopting it, no matter what technical tools for constructing nonstandard models are developed.  There are a couple of reasons I say this.

1. We don't have to wait for those technical tools to materialize to imagine what it would be like to have them.  It would just mean that a book entitled "Models of Peano Arithmetic" would no longer be a specialist monograph; rather, it would be a standard graduate text like "Ring Theory" where some axioms are laid down and lots of examples and tools are studied.  On page 3, Example 1 would be the standard natural numbers. Everyone is still going to know exactly what Example 1 is and it will remain the the most important example.  For comparison, algebraic number theorists routinely study number fields and are interested in theorems that hold for arbitrary number fields as well as for various classes of number fields, but none of this causes anyone to think that "the rationals" is a vague term that might refer to any number field.  For another comparison, computational complexity theorists routinely construct oracles to separate or collapse various complexity classes, but this does not lead anybody to think that "P = NP" is a vague statement that doesn't have a privileged interpretation.  The empty oracle is privileged.

2. I said this before but it bears repeating: I cannot see any way of forming a clear grasp of a multiverse of nonstandard models of arithmetic without first forming a clear grasp of the standard model.  This is a key point where the analogy with set theory breaks down.  It is plausible to argue that we can form a clear concept of a countable model of ZFC without first forming a completely clear concept of the entirety of V.

3. I maintain that our confidence that we know what the standard natural numbers is does *not* stem from examining a set-theoretic proof that the second-order Peano axioms are categorical.  If it did, then I could see a devil's advocate trying to argue that the words in that set-theoretic proof could be interpreted as an argument in first-order set theory and hence not necessarily mean what we think it means.  A clear conception of the standard natural numbers must come first before we can even form a clear concept of, say, "well-formed formula."  No human being has ever arrived at a clear concept of "well-formed formula" at an earlier age than arriving at a clear concept of "natural number."  Theorems about the weak expressive power of first-order logic cannot lead us to confidently declare that "natural number" is vague, because if we take that conclusion seriously, then it automatically creates doubts about whether we really know what those theorems are asserting.  It doesn't make sense to saw off the branch one is sitting on.

4. Doubting that we have a clear concept of "natural number" can be done, but that takes us down the ultrafinitist route, not the multiverse route.

To summarize, I still see no reason to believe in the multiverse view at the level of the natural numbers.  It can of course be studied as a mathematical curiosity, but the analogy with set theory fails and I have trouble believing that it will ever catch on with the mathematical community at large.

Tim
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