# Non-standard Models of Arithmetic 5

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 $\mathbb{N}$“.

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 $\mathbb{N}$” 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.)

Imagine this:

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 $\mathbb{N}$ is nebulous epistemically, or ontologically?

Epistemology—what we know—sure that’s nebulous! There’s a heck of lot we don’t know about $\mathbb{N}$. 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 $\mathbb{N}$‘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, I can prove in there that’s there’s 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 $\mathbb{N}$, 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 $2^{\aleph_0}$ 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 $\exists N(\ldots)$“, 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 $\mathbb{N}$. 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!

Filed under Conversations, Peano arithmetic

### 7 responses to “Non-standard Models of Arithmetic 5”

1. Vicente Sanchez-Leighton

I have been reading your interesting conversation and look forward to the next “episodes” 😉 While reading a question popped in my mind: how do you stand in relation to Voevodsky’s famous talk in 2010, What if the current foundations of mathematics are inconsistent?, https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2010_09_25_slides.pdf and in particular to his “workflow” proposal on slide 17 ? I feel Voevodsky’s stance could very well help taking care of the nebulousness you (JB) point, but with the kind of attitude you (MW) defend when you write “Maybe being nebulous isn’t so bad! It doesn’t wipe out your ontological creds.” Also the “incoherent mish-mash of intuitionism, formalism, and platonism” which one of you stands for (like probably most mathematicians) could also use such a flexible epistemological “workflow” to remain productive.

• First off, I should say that any philosophy at all can lead to fun math. Voevodsky’s piece seems (in part) to be an ad for constructive type theory. I don’t know enough about that to have an opinion on it. Maybe it’s really great!

Slides 17 and 18 (on constructing “reliable” proofs) made me think of George Kreisel’s critique of Hilbert’s program:

…the finitist reduction … consisted [for Hilbert] in clearing the fair name of mathematics which had been sullied by the paradoxes. [Hilbert also said] that the paradoxes simply have nothing to do with the theory of sets of numbers: it is hard to see why this remark, if true, has not cleared the name at least of analysis unless one believes that stained reputations can only be cleared with a great deal of ceremony.

We haven’t found an inconsistency in ZFC, even fortified with large cardinal axioms, for a good long while! If the issue is, “how can we ever trust set theory (or even arithmetic) again?” after Gödel’s theorem—well, like the advice columnists always say, you rebuild trust one day at a time.

IMO, Voevodsky is pulling a fast one on slide 5. The trick is the slipperiness of “know”. We “know” that PA is consistent only in the sense that (most of us) believe it pretty strongly, just like we believe we’re not really lobsters dreaming we’re humans. Voevodsky gives the main reason for this belief (the consistency of PA, not the thing about lobsters) in the first bullet on the next slide: the “formulas as subsets” interpretation of the induction schema. Well of course that does lead to a proof of consistency of sorts, namely the usual proof of Con(PA) in ZF. Claiming that this sort of “knowledge” means that “second incompleteness theorem is false as stated” is a non-sequitur.

The third bullet on slide 5 likewise contains a non-sequitur. So I guess I’d plump for the second bullet, except that I’d amend “‘transcendental’, provably unprovable knowledge” to “I’m really strongly convinced of this, but if an inconsistency in PA turns up tomorrow, I won’t kill myself.”

• Vicente Sanchez-Leighton

As I understand Voevodsky’s argument in this talk, he believed in the third bullet in slide 5 and so his ‘workflow’ in slide 17 applies to a situation in which we have inconsistencies, which, I believe, are related to the ‘nebulousness’ John Baez points out, or even his inconsistent mash of philosophies of mathematics. This appears similar to building in places with earthquakes: the houses you build look very much like the houses in places without earthquakes… but they need a more robust structure in order to last. And perfect earthquake robustness (Hilbert’s program) doesn’t exist (Gödel, Turing etc…).

• I am hoping to read “Univalent Foundations of Mathematics and Paraconsistency”, by Vladimir Vasyukov (in New Directions in Paraconsistent Logic) One Of These Days. Voevodsky (in his modified grant application) summarized the benefits of the Univalent Foundations Project this way:

1. 1. Univalent foundations naturally include ”axiomatization” of the categorical and higher categorical thinking.
2. 2. Univalent foundations can be conveniently formalized using the class of languages called dependent type systems.
3. 3. Univalent foundations are based on direct axiomatization of the ”world” of homotopy types instead of the world of sets.
4. 4. Univalent foundations can be used both for constructive and for non-constructive mathematics

All these motivations strike me as more appealing than as a defense against some as yet undiscovered inconsistency.

As for the third bullet on slide 5: “Admit that the sensation of knowing in this case is an illusion and that the first order arithmetic is
inconsistent.” I’m sort-of OK with the first half (depending on what you mean by “illusion”), but the inconsistency of PA just doesn’t follow at all.

• Vicente Sanchez-Leighton

I really encourage you to to watch the video of that conference https://video.ias.edu/voevodsky-80th.
And to get back to your dialogue with JB, my point in recalling Voevodsky’s conference was the ‘nebulousness’ you were discussing with JB could very well resonate with Voevodsky’s position in the talk and his proposal to overcome the shakiness of the foundations.

• Ok, I’ve watched it.

To anyone else, I recommend starting at the 25 minute mark; that’s where Voevodsky starts discussing his own ideas. (Before that it’s just a review of very basic formal logic.)

The Q&A period begins at the 45 minute mark. It’s worth listening to the first question and Voevodsky’s reply. The questioner posed another possibility besides the three choices on slide 5: “We know we can’t prove the consistency of mathematics, but we can always hope.”

To reiterate, although I have a bemused view of Voevodsky’s philosophical claims (in this lecture), I don’t want to cast shade on the Univalent Foundations Project. Motivations are one thing, math another!

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