[FOM] Hamkins's multiverse and ultrafinitism

[katzmik at macs.biu.ac.il]

It is a mistake to use the term "ultra-finitist flavor" in reference to
Hamkins' position.

If anything his position can be more appropriately described as
"ultra-infinitest". Namely, our usual concept of familiar integer ends up
(more or less inevitably, depending on to what extent one is committed to the
multiverse scheme) incorporating ideal entities that behave like infinite ones
from the viewpoint of a more "economical" universe, and so on ad infinitum.

This was discussed thoroughly in https://mathoverflow.net/q/284747 where Tim
himself was an active participant.

A more concrete interpretation of the idea that every model of the integers
necessarily includes infinite ones from the viewpoint of another model is
discussed in our publication https://arxiv.org/abs/1703.00425

where we show more specifically that the Hamkins-Gitman "baby model" turns out
to be a model of a variant of Hrbacek-Nelson type formulation of Robinson's
nonstandard analysis.

MK

On Fri, November 24, 2017 18:27, tchow wrote:
> I recently took a closer look at the set-theoretic multiverse a la Joel
> David Hamkins, in particular his thoughts about the concept of "finite."
>
> http://jdh.hamkins.org/the-set-theoretic-multiverse/
>
> As far as I can see, Hamkins is not a card-carrying ultrafinitist, but
> he does ask questions with an ultrafinitist flavor, e.g., "Are we
> correct in thinking we have an absolute concept of the finite?"  I'm
> unsure what Hamkins means by "absolute" here, given that he wants us to
> be unsure what "finite" means, but setting that aside, what I'm curious
> about is his attempt to draw an analogy between set theory and
> arithmetic.
>
> As I understand it, Hamkins is strongly influenced by the fact that in
> set theory, we have the tools to create all kinds of "set-theoretic
> universes" that satisfy all kinds of combinations of axioms and their
> negations thereof.  Hamkins argues that the effect is to create the
> impression that there are a multitude of mutually incomparable universes
> that all exist on an equal footing.  If we were to ask, "Will the real
> universe please stand up?" we would be at a loss to point to any
> specific mathematical entity, except by fiat or convention.
>
> I won't contest Hamkins's claim about set theory here, but what
> interests me is his suggestion that we might be led to, or at least
> tempted by, similar conclusions in the finitary realm if we were to
> develop powerful tools for independence phenomena in arithmetic.  This
> is something that I find rather implausible, and is what I want to
> discuss here, particularly as there are some FOM readers who are
> sympathetic to what we might call ultrafinitism.
>
> I think that Hamkins is inviting us to imagine that someday we might
> develop tools that will allow us to show that, say, the Riemann
> Hypothesis (RH) and Goldbach's Conjecture (GC) and the Twin Prime
> Conjecture (TPC) and a host of other familiar unproved statements in
> elementary number theory are all independent of our favorite axioms for
> mathematics, and moreover we will be able to construct, at will, models
> for any combination of these statements that is not obviously
> contradictory.  I think that Hamkins is suggesting that if this were to
> happen, then we would start to lose confidence that we have an "absolute
> concept of the finite."
>
> This strikes me as a rather peculiar claim.  Let us spell out the
> conventional response to this scenario.  If we were to prove that, say,
> GC were unprovable in some strong system, then because GC is Pi_1, we
> would conclude that GC is simply *true*.  Models of PA in which GC fails
> would have an upper bound on the even numbers that are the sum of two
> primes, but this upper bound would be a nonstandard integer.  The
> independence of TPC would not immediately allow us to conclude that it
> is true; we might be able to construct nonstandard models in which TPC
> fails and others in which TPC holds, but this would not immediately tell
> us anything about whether TPC holds for the "standard" integers.
>
> The point is that under the conventional view, none of these technical
> developments would lead us to doubt whether we had a clear understanding
> of what the "standard natural numbers" are.  The nonstandard models that
> we would construct would all be *more complicated* than the standard
> natural numbers, and would contain the standard natural numbers as an
> initial segment.  I cannot imagine anyone forming a clear picture of a
> nonstandard model of PA without first forming a clear picture of the
> standard model of PA.  This strikes me as a qualitative difference
> between arithmetic and set theory.  A countable model of ZFC, or even
> something like L, is arguably "simpler" than V itself, and one can
> imagine getting a clear picture of it without first getting a clear
> picture of all of V.  I don't see how one can make the same claim in the
> realm of arithmetic.
>
> But let's try to give Hamkins the benefit of the doubt.  Suppose that
> technical advances of the sort mentioned above were to lead us to doubt
> whether we have a clear picture of what "finite" means and what the
> "standard natural numbers" are.  What kind of "arithmetic multiverse"
> might we offer instead?  As I said above, I do not think that a
> multiverse of nonstandard models of arithmetic (as conventionally
> conceived) really fits the bill, because it's easy for the "real natural
> numbers to please stand up."  Borrowing ideas from ultrafinitism, it
> would seem that the arithmetic multiverse would comprise a multitude of
> "feasible number systems" that all start off the same way (1,2,3,4,...)
> but then fade off into obscurity in different ways.  GC could be
> "conventionally" false, but the smallest counterexample might be
> infeasibly large.  Indeed, maybe our technology advances to the point
> where we are able to get good upper and lower bounds on the smallest
> counterexample to GC, but the bounds are manifestly "infeasibly
> large"---at least according to some definitions of "infeasible."  We
> might even develop theories of different levels of infeasibility, and
> imagine denizens of different feasible universes having varying levels
> of power to gain access to large natural numbers.  This is the closest
> that I can see coming to painting a picture of an "arithmetic
> multiverse."
>
> Yet even if our technology were to reach this point, would it really
> tempt us to abandon the conventional belief in the clarity of the word
> "finite"?  I don't see it.  In the above scenario, it seems to me that
> we would accept that GC is simply *false*.  We might argue about how
> explicit our counterexample is or how explicit we might be able to make
> it in the future, but would the majority opinion really be that whether
> GC is true or false depends on what feasible universe we're talking
> about?  Would we abandon talk of truth or falsity of GC altogether in
> favor of conditional statements about its demonstrability under various
> assumptions of physical feasibility?  I suppose that it is possible, but
> it strikes me as unlikely.
>
> I am curious as to whether any other FOM readers can make a stronger
> case for the "arithmetic multiverse" than I have here.
>
> Tim
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