[FOM] Hamkins's multiverse and ultrafinitism

[Dennis Kavlakoglu]

Tim (if I may):

It might help to look at the paper “Satisfaction is not absolute” co-authored by Hamkins and Yang: https://arxiv.org/abs/1312.0670. This paper expands on the claims in Hamkins’s “The Set-Theoretic Multiverse” (2012; the published version) paper, where he claims that the apparent “absoluteness” of arithmetic ultimately depends on our “murkier understanding of which subsets of the natural numbers exist” (p. 428). He’s generally skeptical of second-order categoricity proofs since he holds they assume a background concept of set, and one of the morals he takes from independence phenomena is that we don’t have any univocal, absolute, background concept of set. 

In “Satisfaction is not absolute”, Hamkins and Yang show that two models of set-theory with the same natural numbers and the same standard model of arithmetic, yet different theories of arithmetic truth. In line with John Baldwin’s comment, they also show that every countable model of set theory has elementary extensions with the same natural numbers and successor structure, as well as agree on addition and order, but disagree regarding the multiplication structure on the natural numbers (Corollary 7, p. 13).

Regarding: "In particular, a ‘generic' choice of viable candidate would 
*not* be the minimal model.” 

Minimal in what sense? How much structure must be captured? Which subsets of the set of numbers correspond to “genuine” properties of the natural numbers? In the appendix to “The Set-Theoretic Multiverse”, Hamkins briefly discusses “set-theoretic geology”. Instead of constructing outer models via forcing over a ground model, “set-theoretic geology” starts with the universe V and explores “grounds”, transitive class models of ZFC from which V can be obtained via forcing. The “mantle” is the intersection of all grounds, what results if you “strip away” all the extra structure introduced in the forcing construction of V. However, a theorem of Fuchs, Hamkins, and Reitz shows that every model of ZFC is the mantle of another model of ZFC (Theorem 10.7, p. 444). This suggests that there is no univocal forcing-minimal model, but instead what results from “stripping away” the forcing constructions in V is just an arbitrary model of ZFC which is, itself, a mantle for another model of ZFC, and so on.

I think that similar worries would apply to the property structure of the natural numbers. If we’re using full second-order semantics, that’s just the full powerset. But powerset isn’t absolute so what reason do we have to think the powerset of the set of natural numbers is? It’s all comes back to what we mean by “subset of the natural numbers”. The arithmetic sets? But then theorem 10 from “Satisfaction…” (p. 18) shows that countable models of set theory have elementary extensions that agree on the structure of their standard natural numbers, yet disagree over whether a particular subset of N is first-order definable.

The key point for the skeptic, I think, is that (they claim) we don’t have a clear picture of the property structure of the natural numbers that doesn’t seem to suffer the same independence-related skeptical worries afflicting our conception of the property structure of the set-theoretic universe.

Hope some of this helps to clarify how Hamkins is thinking of things (and that I haven’t garbled too much of it).

- Dennis



On November 29, 2017 at 4:24:33 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  
_______________________________________________  
FOM mailing list  
FOM at cs.nyu.edu  
https://cs.nyu.edu/mailman/listinfo/fom  
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20171129/942503e4/attachment-0001.html>