[FOM] Hamkins's multiverse and ultrafinitism

[tchow]

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