[FOM] Hamkins's multiverse and ultrafinitism

[John Baldwin]

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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