MW: John, you wrote:
Roughly, my dream is to show that “the” standard model is a much more nebulous notion than many seem to believe.
and you gave a good elucidation in post 2 and post 4. But I’d like to defend my right to “true arithmetic” and “the standard model ℕ”.
Maybe being nebulous isn’t so bad! It doesn’t wipe out your ontological creds. Look at that cloud over there:
HAMLET: Do you see yonder cloud that’s almost in shape of a camel?
POLONIUS: By the mass, and ’tis like a camel, indeed.
HAMLET: Methinks it is like a weasel.
POLONIUS: It is backed like a weasel.
HAMLET: Or like a whale?
POLONIUS: Very like a whale.
No one’s saying there’s no cloud. Or that you can only talk about the cloud if you do so in the language of ZF set theory!
JB: I don’t mind nebulous philosophy. But if you’re going to say “true arithmetic” and “the standard model ℕ” in mathematical discourse, I think you need to define them. (That’s for the “hard core” of mathematical discourse, where one is stating theorems and conjectures, and proving them. It’s fine, indeed essential, to say a lot of vaguer stuff while doing mathematics.)
POLONIUS: Let G be a Lie group. If G is connected and simply connected, it’s determined up to isomorphism by its Lie algebra.
HAMLET: Sorry, could you remind me—what’s a “Lie group”?
POLONIUS: What?? Every mathematician worthy of the name has a clear intuitive concept of a Lie group! How can you possibly ask such a question?
MW: That reminds me of an incident from grad school. My friend Mark and I went to ask one of our favorite profs, Bert Walsh, something about Riemann surfaces. He began by saying, “Well, you take your Dolbeault complex…” At the time, neither Mark nor I knew a Dolbeault complex from a dollhouse. So Mark asked him, “What’s a Dolbeault complex?” To which Prof. Walsh replied, “Mark, it’s exactly what you think it is!”
Since this is a Philosophy-with-a-capital-P post, let me sling the jargon: do you mean that ℕ is nebulous epistemically, or ontologically?
Epistemology—what we know—sure that’s nebulous! There’s a heck of lot we don’t know about ℕ. Are there odd perfect numbers? An infinite number of prime pairs? The ABC conjecture! Even something like the Riemann ζ hypothesis can be “coded” into the language of arithmetic with a little work. (Or a lot of work, if you really mean “written” and not just “convince yourself it could be written”.)
But ontologically—are there lots of different ℕ’s, or just one? The very term “non-standard models of arithmetic” has a double-edged tinge to it. If there are non-standard models, then there must be a standard model!
You gave a sort-of justification for the phrase, “the standard model”, in post 4:
If I pick a set theory, say ZFC or whatever, then I can prove that there is an initial model of PA (one with a unique embedding in any other), and I’m happy to call that the “standard model of PA in ZFC”. Then I can talk about the theory consisting of all closed sentences in this model. All that’s fine with me.
But do we have to “pick a set theory, say ZFC or whatever”, to give the proof?
I said my philosophy of math was an incoherent mish-mash of intuitionism, formalism, and platonism. Let’s start with intuitionism. I think I read somewhere that the taproot of Brouwer’s philosophy (or maybe Poincaré’s) consists in this: mathematical intuition comes first, axioms later.
That’s all I’m taking from those guys. But it’s important. I’m betting you saw the proof that there is, up to isomorphism, only one model of Peano’s axioms, long before you learned about first-order theories and non-standard models and all that. (It’s basically in Dedekind’s famous essay, “Was sind und was sollen die Zahlen?”, paragraph 79.)
Of course, “Peano’s axioms” here isn’t the same as “PA”. The induction scheme is a single axiom quantifying over all subsets of ℕ, or as logicians like to say, it’s formulated in second-order arithmetic. (Or you can go all the way to ZFC, if you like.)
Here’s another way to put it: when you say, “pick a set theory”, I say, “OK, what if I pick the intuitive set theory of Dedekind and Cantor?”
I know what you’re thinking. “Has he totally forgotten about Russell’s paradox? Has he even studied axiomatic set theory? Does he really believe that has a single correct value, even if we may never know what it is?”
I’ll admit that oxygen can get a little scarce far up in V. Do I believe that Woodin cardinals really exist? When the air becomes hard to breathe, I might want to take refuge in Hilbert’s gambit, and just claim it’s all a game with symbols. (*)(That’s my formalism ingredient. Again, a very small part of Hilbert’s program, but important.)
(*) Added later: I should have mentioned that Hilbert didn’t actually say this, and it’s a (common) misinterpretation of his program to summarize it this way. “We note at once that there is no evidence in Hilbert’s writings of the kind of formalist view suggested by Brouwer when he called Hilbert’s approach ‘formalism’.” —Kreisel, “Hilbert’s Programme”
The Hilbert gambit may be philosophically unimpeachable, but you know, it just isn’t that much fun! G. H. Hardy famously refused to accept that Hilbert really believed it. When watching Game of Thrones, do you tell yourself the whole time, “Those aren’t real dragons, they’re just CGI.”?
Let’s take the “existence” of non-standard models of PA in the first place. From a strictly formalist standpoint, we’d have to say: “here’s a proof in ZFC that ∃N…”, where the ellipsis is a formalization of “N is a model of the PA axioms that is not isomorphic to ω”. Of course nobody does that. We carry out the proof that such beasties exist in an informal set theory, convinced (with good reason!) that it could (under duress) be turned into a ZFC proof.
And that’s the platonism part of my philosophy. The outermost layer will always be raw mathematical intuition. But when I’m swimming inside ZFC, it’s just easier, and more fun, to imagine that the ZF universe really exists. I pretend that the axioms are laws of this universe. The laws tell me there’s a unique ω, and (equipped with the usual paraphenalia), that’s what I mean by “the standard model”. Who knows, maybe it’s really true.
(Does The Matrix have any scenes of someone playing video games? Or Greg Egan’s Permutation City? Can’t remember.)
Now for the kicker. Say we go with the Hilbert gambit. You need some raw intuition about the natural numbers just to make sense of it! Anyone who will swallow “well-formed formula” and “proof tree” as sufficiently clear concepts, but balks at “natural number”, has a strange perspective, IMO. The Hilbert gambit buys you something, philosophically, when we’re talking about full-bore set theory. Not so much with arithmetic.
Put it this way. You wrote:
for any sentence that’s not decidable in Peano Arithmetic, there are models where this sentence is satisfied and models where it’s not. In which camp does the standard model lie?…[By the Overspill Lemma] if our supposed “standard” model were actually nonstandard, we still couldn’t pick out the nonstandard numbers. The aliens could be among us, and we’d never know!
The results you appeal to (the Completeness Theorem and the Overspill Lemma) are themselves theorems of either ZFC, or of informal set theory. But so is the uniqueness (up to isomorphism) of ℕ. Why regard these results differently?
On the other hand, if you meant “nebulous” epistemically, well, just about all math is nebulous in that sense. There’s a lot we don’t know about—fill in the blank!