Nonstandard Models of Arithmetic

June 5, 2019

Posted by John Baez

$MathML-enabled post (click for more details).$

A nice quote:

There seems to be a murky abyss lurking at the bottom of mathematics. While in many ways we cannot hope to reach solid ground, mathematicians have built impressive ladders that let us explore the depths of this abyss and marvel at the limits and at the power of mathematical reasoning at the same time.

This is from Matthew Katz and Jan Reimann’s nice little book An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics. I’ve been been talking to my old friend Michael Weiss about nonstandard models of Peano arithmetic on his blog. We just got into a bit of Ramsey theory. But you might like the whole series of conversations.

$MathML-enabled post (click for more details).$

Part 1: I say I’m trying to understand “recursively saturated” models of Peano arithmetic, and Michael dumps a lot of information on me. The posts get easier to read after this one!
Part 2: I explain my dream: to show that the concept of “standard model” of Peano arithmetic is more nebulous than many seem to think. We agree to go through Ali Enayat’s paper Standard models of arithmetic.
Part 3: We talk about the concept of “standard model”, and the ideas of some ultrafinitists.
Part 4: Michael mentions “the theory of true arithmetic”, and I ask what that means. We decide that a short dive into the philosophy of mathematics may be required.
Part 5: Michael explains his philosophies (plural!) of mathematics, and how they affect his attitude toward the natural numbers and the universe of sets.
Part 6: After explaining my distaste for the Punch-and-Judy approach to the philosophy of mathematics (of which Michael is not guilty), I point out a strange fact: our views on the infinite cast shadows on our study of the natural numbers. For example: large cardinal axioms help us name larger finite numbers.
Part 7: We discuss Enayat’s concept of “a $T$ -standard model of $PA$ ”, where $T$ is some set of axioms extending $ZF$ . We conclude with a brief digression into Hermetic philosophy: “as above, so below”.
Part 8: We discuss the tight relation between $PA$ and $ZFC$ with the axiom of infinity replaced by its negation. We then chat about Ramsey theory as a warmup for the Paris–Harrington Theorem.
Part 9: Michael sketches the proof of the Paris–Harrington Theorem, which says that a certain rather simple theorem about combinatorics can be stated in PA, and proved in ZFC, but not proved in PA. The proof he sketches builds a nonstandard model in which this theorem does not hold!
Part 10: Michael and I talk about “ordinal analysis”: a way of assigning ordinals to theories of arithmetic, that measures how strong they are.
Part 11: We finally get serious about working through Ali Enayat’s paper Standard models of arithmetic. Michael introduces $PA^T$ , the set of all closed formulas in the language of Peano arithmetic that hold in all $T$ -standard models where $T$ is a consistent recursively enumerable extension of ZF. He explains how to recursively axiomatize $PA^T$ using “Craig’s trick”, and as a bonus explains “Rosser’s trick”.
Part 12: Some examples. $PA^ZF$ is strictly stronger than $PA$ . $PA^{ZFC+GCH} = PA^ZF$ where $GCH$ is the generalized continuum hypothesis. $PA^{ZFL} = PA^{ZF}$ where $ZFL$ is $ZF$ plus an axiom saying all sets are constructible in Gödel’s sense. But $PA^{ZFI}$ is strictly stronger than $PA^{ZF}$ , where $ZFI$ is $ZF$ plus the existence of an inaccessible cardinal!
Part 13: Enayat’s “natural” axiomatization of $PA^T$ , and his proof that this works. A digression into Tarski’s theorem on the undefinability of truth, and how to work around it. For example, while truth is not definable, we can define truth for statements with at most a fixed number of quantifiers.

Posted at June 5, 2019 6:41 PM UTC

TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3116

41 Comments & 0 Trackbacks

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

A joke:

If we add up all of a nonstandard set of natural numbers, do we still get $-\frac{1}{12}$ ?

Posted by: Blake Stacey on June 6, 2019 12:24 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Taking the joke seriously, the answer has to be yes, right? Suppose we take one of the various theorems making rigorous the statement that $\sum_{n \in \mathbb{N}} n = -1/12$ . It’s still a theorem for any nonstandard $\mathbb{N}$ , isn’t it?

I admit, I don’t know the mathematical definition of “nonstandard model of the natural numbers”. (Or of “standard model”.) I skimmed through almost all of John and Michael’s conversations, rather fast, and I understood this to be actually one of the central points. Is that right?

Posted by: Tom Leinster on June 6, 2019 10:21 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

I’m pretty sure “It’s still a theorem for nonstandard $\mathbb{N}$ ” is a sentence that would make most logicians squirm… though it’s truer than is false. I feel the pedantic need to say it better. Pardon me if you know all this:

Theorems are things you prove starting from a theory—a bunch of axioms. Peano arithmetic is a theory. You can ask if some sentence is a theorem of Peano arithmetic. It either is or isn’t. Nothing about models here—not directly, anyway.

Models of a theory are structures in which the theory is “valid”. In other words, all the axioms, and all theorems following from them, are “satisfied”.

Gödel’s completeness theorem ties these concepts together. If a sentence is satisfied in every model of your theory, it must be a theorem!

So, you really meant to say something like:

Suppose we take one of the various theorems making rigorous the statement that $\sum_{n \in \mathbb{N}} n = -1/12$ . These sentences are still valid in nonstandard models of Peano arithmetic, because they are theorems in Peano arithmetic.

And that would be true if these really are theorems in Peano arithmetic. But I’ve never seen someone try to develop complex analysis using just Peano arithmetic. After all, Peano arithmetic is just about the natural numbers! But you can do a surprisingly large amount using surprisingly little, using various clever coding tricks. So it’s an interesting question: can you develop complex analysis to the point of studying the Riemann zeta function, using just Peano arithmetic?

Usually people, even logicians trying to develop math on bare-bones axioms, would use a bit more. An expert on reverse mathematics could tell us.

If you can’t even say $\zeta(-1) = -1/12$ in Peano arithmetic, then it doesn’t make sense to ask if it’s valid in some models and not others.

In short: now you see, Blake: when talking to logicians, no joke will go unpunished!

Posted by: John Baez on June 7, 2019 12:58 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Thanks, John, and I probably made you squirm too. I was indeed being careless.

However, I’d like to argue that what I wrote was less squirmworthy than it might appear. My argument is to do with the way in which categorical logic blends (a critic might say “muddles up”) syntax and semantics in a way that logicians of a more traditional mindset might find disturbing. I dimly remember some quote by Tarski or some such figure about Lawvere, along the lines of “that guy doesn’t know the difference between syntax and semantics”. It was, I believe, prompted by Lawvere theories.

More specifically: as I said, I never understood quite what was meant by “(non)standard model”. But I had in my head that it would be something like this.

We have “the” category of sets, which is really just a topos satisfying a couple of extra axioms. It has a unique-up-to-iso natural numbers object. No ambiguity there… except that there isn’t really such a thing as “the” category of sets. There are very many inequivalent categories satisfying the topos-plus-a-bit-more axioms, and they all have a uniquely-determined natural numbers object.

Perhaps we wave our hands a bit and distinguish one category satisfying these axioms as “the” category of sets — or as you might call it, the “standard” category of sets, $\mathbf{Set}$ — while calling the other ones “nonstandard”. This is certainly handwavy, but it looks to me to be in a similar spirit to what Michael was saying, and no vaguer.

It’s then natural to refer to the NNO of $\mathbf{Set}$ as the “standard $\mathbb{N}$ ” and the NNO of the nonstandard categories of sets as “nonstandard $\mathbb{N}$ s”.

The point is this. You’ve got all these different, inequivalent, toposes. There are statements that are true in some of them and not in others. In particular, there are statements about the NNO that are true in some but not others. So you really can talk about a statement being “true for any nonstandard $\mathbb{N}$ ”.

I shouldn’t have carelessly used the word “theorem” in a discussion about logic, but I was using it in the mathematicians’ rather than logicians’ sense, like if I was to ask you:

is the classification theorem for finite abelian groups true in an arbitrary topos?

Anyway, that’s how I was thinking. I realize your discussion with Michael wasn’t at all categorical. But when you think about logic categorically, the boundary between syntax and semantics is different and blurrier than in the classical treatment.

Posted by: Tom Leinster on June 7, 2019 10:21 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

By the way, I did eventually track down that half-remembered sort-of-quote of Tarski on Lawvere: see here.

Posted by: Tom Leinster on March 22, 2024 10:37 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Tom wrote:

More specifically: as I said, I never understood quite what was meant by “(non)standard model”. But I had in my head that it would be something like this.

We have “the” category of sets, which is really just a topos satisfying a couple of extra axioms. It has a unique-up-to-iso natural numbers object. No ambiguity there… except that there isn’t really such a thing as “the” category of sets. There are very many inequivalent categories satisfying the topos-plus-a-bit-more axioms, and they all have a uniquely-determined natural numbers object.

Perhaps we wave our hands a bit and distinguish one category satisfying these axioms as “the” category of sets — or as you might call it, the “standard” category of sets, $\mathbf{Set}$ — while calling the other ones “nonstandard”. This is certainly handwavy, but it looks to me to be in a similar spirit to what Michael was saying, and no vaguer. It’s then natural to refer to the NNO of $\mathbf{Set}$ as the “standard $\mathbb{N}$ ” and the NNO of the nonstandard categories of sets as “nonstandard $\mathbb{N}$ s”.

This is all very much worth thinking about, but this is not what classical logicians mean by “nonstandard models of the natural numbers”. So, when I’m talking to Michael Weiss, and I mention nonstandard models of the natural numbers, that’s not what I mean.

What do the traditional logicians mean? Fix your favorite version of the category of sets. Call it $Set$ . There are many nonisomorphic structures in $Set$ obeying the axioms in first-order classical logic called PA. There are lots of different countable ones, lots of different uncountable ones, etc. These are called “models of PA” or more loosely “models of the natural numbers” (which covers situations where we add or remove some axioms from PA to get some other closely related theory in first-order classical logic).

There’s a whole category of models of PA in our chosen category $Set$ . The initial objects in this category are all uniquely isomorphic to each other. People call any one of them “the standard model”. The rest are called “nonstandard models”. Each one has a copy of the standard model sitting inside it.

I’m also very interested in varying the category we call $Set$ , and Enayat’s paper Standard models of arithmetic does a bit of that too.

But it’s good to remember that while the second-order Peano axioms have a unique model up to isomorphism, the first-order system called PA does not. It can’t, even if we add finitely many more axioms expressed in first-order logic, or even a recursively enumerable infinite list of such axioms, thanks to Gödel’s incompleteness theorem.

The point is that in the first-order version, we only have mathematical induction for predicates that can be expressed in the language of arithmetic, while in the second-order version we have mathematical induction for all predicates — which is only possible because in second-order logic we can quantify over predicates.

Topos theory is a form of higher-order logic, and the definition of “natural numbers object” accomplishes something equivalent to mathematical induction for all predicates.

Posted by: John Baez on June 8, 2019 12:32 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Thanks; that’s a much-needed reminder. I used to know this stuff! I did the relevant course as an undergraduate and knew it like the back of my hand. But I’ve never needed this stuff since then, so it’s faded away in the way things do.

The statement:

There’s a whole category of models of PA in our chosen category $Set$ . The initial objects in this category are all uniquely isomorphic to each other. People call any one of them “the standard model”. The rest are called “nonstandard models”.

is very crisp and clear. I hadn’t picked that up from your conversation with Michael.

Posted by: Tom Leinster on June 8, 2019 9:01 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Tom wrote:

The statement:

There’s a whole category of models of PA in our chosen category $Set$ . The initial objects in this category are all uniquely isomorphic to each other. People call any one of them “the standard model”. The rest are called “nonstandard models”.

is very crisp and clear.

Thanks! Traditional model theorists never say this, perhaps because they’re not comfortable enough with category theory to say “initial object”. But they know the relevant facts:

The standard model $\mathbb{N}$ is initial. It maps via a monomorphism to every model $N$ . The elements in the range of this inclusion are called “standard” elements of $N$ , while the rest are called “nonstandard”.

You can define a linear ordering on any model, and the standard elements always form an initial segment. The rest, $N - \mathbb{N}$ , is isomorphic as an ordered set to a bunch of copies of $\mathbb{Z}$ . There are infinitely many of these copies, and between any two copies there are infinitely many more.

This is sort of easy to see: for example, if $N$ is nonstandard, there’s a nonstandard number $\lfloor N/2 \rfloor$ with

$0\; < \; \lfloor N/2 \rfloor \; < \; N$

and the difference between $N$ and $\lfloor N/2 \rfloor$ is too big to be standard, so $N$ and $\lfloor N/2 \rfloor$ live in different copies of $\mathbb{Z}$ .

By the Löwenheim–Skolem theorem, the cardinality of $N$ can be any infinite cardinal. So, the number of these copies of $\mathbb{Z}$ in $N$ can be any infinite cardinal.

All countable nonstandard models of PA have the same order type. However, they are not isomorphic in other ways! Indeed there are uncountably many nonisomorphic countable models of PA. To cook them up, just create a set $S$ of infinitely many sentences that are independent of PA, and each other. For any subset $X \subseteq S$ , there’s a countable model of PA where the sentences in $X$ are valid and those in $S - X$ are not.

I hadn’t picked that up from your conversation with Michael.

Sorry: we never said any of the above stuff, since we’re just chatting and we both know this stuff.

I am also starting a conversation with him on category-theoretic approaches to the foundations of first-order logic and arithmetic, and I’ll announce it here after it’s gone on for a while. You just raised a fun question: does every nonstandard model of Peano arithmetic in $Set$ become a natural numbers object in some suitable topos? There’s some wiggle room in making this question precise.

Posted by: John Baez on June 8, 2019 7:29 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Analytic continuation – i.e.,

“For any analytic $f : \mathbb{C} \to \mathbb{C}$ on some open region $U \subseteq \mathbb{C}$ , there is a unique analytic continuation $f^{+}: \mathbb{C} \to \mathbb{C}$ ”

will be a theorem of second-order arithmetic $Z_2$ using coding tricks. I suspect it’s a theorem of $\mathsf{ACA}_0$ , since it doesn’t seem to be impredicative, or to involve any fancy recursion. I looked this up the Reverse Mathematics literature, but couldn’t find that exact theorem.

Letting $f(s)$ be the usual sum $\sum_{n=1}^{\infty} n^{-s}$ , then this function converges when $Re(s)$ > $1$ . The 1+2+3…-theorem required is then “ $f^{+}$ exists and is unique and $f^{+}(-1) = -1/12$ ”.

This (I suspect) is a theorem of $\mathsf{ACA}_0$ .

Posted by: Jeff Ketland on June 8, 2019 9:01 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Jeff wrote:

Analytic continuation – i.e.,

“For any analytic $f : \mathbb{C} \to \mathbb{C}$ on some open region $U \subseteq \mathbb{C}$ , there is a unique analytic continuation $f^{+}: \mathbb{C} \to \mathbb{C}$ ”

will be a theorem of second-order arithmetic $Z_2$ using coding tricks. I suspect it’s a theorem of $\mathsf{ACA}_0$ , since it doesn’t seem to be impredicative, or to involve any fancy recursion.

Nice! Of course the Riemann zeta function is not analytic on $\mathbb{C}$ , but we can probably work around that since we know exactly the domain on which it is analytic.

I’m actually hoping we need even less, though. To analytically continue the Riemann zeta function, we don’t need a general theorem on analytic continuation. I think we can write down a few explicit power series that prove we can analytically continue this function from the region $\mathrm{Re}(z) > 1$ to $latex z = -1$ . We should be able to show by explicit computations that these power series agree on the regions where they’re both defined. I haven’t done this, but I think I could if required. All the arguments should be on the same order of logical complexity as proving that

$\sum_{n = 0}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$

Trig identities, convergence of series where you have explicit formulas for how fast they converge, etc. I don’t think you really need anything as abstract as, say, the intermediate value theorem… which is essential when you’re trying to prove something about all continuous functions.

This is why I suggested that PA may be enough.

Posted by: John Baez on June 8, 2019 9:25 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

John, I’m not sure your response hits at the crux of what Jeff was saying. Sure, there are any number of ways of showing there exists an analytic continuation of $\zeta(s) = \sum_{n \geq 0} \frac1{n^s}$ to larger regions, and one of those might be codable in PA. The question though is how you know there’s only one – and that, it seems to me, is the crucial thing one wants to know to justify the summing of the divergent series.

For what it’s worth, the nicest proof I know of that $\zeta(s)$ , or more precisely $\Psi(s) = \zeta(s) - \frac1{s-1}$ , analytically continues the region $Re(s) \gt 0$ uses some integrals. Here is what I wrote into the nLab some while ago:

Let us bound the summands in the definition of $\Psi(s)$ :

$\array{ \left| \int_n^{n+1} \frac1{n^s} - \frac1{x^s}\; d x \right| & \leq & \underset{n \leq x \leq n+1}{\max}\; \left|\frac1{n^s} - \frac1{x^s} \right|\\ & \leq & \underset{n \leq x \leq n+1}{\max}\; \underset{n \leq t \leq x}{\max}\; \left| \frac{s}{t^{s+1}} \cdot (x-n)\right| \\ & \leq & \underset{n \leq t \leq n+1}{\max}\; \left|\frac{s}{t^{s+1}}\cdot 1\right| \\ & \leq & \frac{{|s|}}{n^{Re(s) + 1}} }$

where the second inequality follows from the mean value theorem. Since

$\sum_{n=1}^\infty \frac1{n^{Re(s) + 1}}$

converges over the region $Re(s) \gt 0$ , it follows that the series for $\Phi(s)$ converges absolutely, and $\Psi(s)$ is a holomorphic function, in that region.

Over the region $Re(s) \gt 1$ , we have

$\array{ \Psi(s) & = & \sum_{n=1}^\infty \frac1{n^s} - \int_n^{n+1} \frac1{x^s}\; d x \\ & = & \sum_{n=1}^\infty \frac1{n^s} - \sum_{n=1}^\infty \int_n^{n+1} \frac1{x^s}\; d x \\ & = & \zeta(s) - \int_1^\infty \frac1{x^s}\; d x \\ & = & \zeta(s) - \frac1{s-1} }$

as asserted.

Granting the uniqueness clause of analytic continuation, this would allow us to extend the equation

$\zeta(s) = \frac{\eta(s)}{1 - 2^{-s}}$

into the region $Re(s) \gt 0$ (where $\eta(s)$ is the eta function), and then, provided with the knowledge that one can analytically continue still further to the left (integral representations, etc.), evaluate the formally divergent series $\eta(-1)$ by any of a number of methods such as series acceleration (mentioned in the eta function article), or generalized Cesàro summations, etc. But I’m still puzzled about how you would address the uniqueness aspect.

Posted by: Todd Trimble on June 9, 2019 3:28 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Sorry for this additional comment. The nLab page I meant to link to was this, and the proof I quoted may look a little garbled because I should have added a bit more context (namely, how the proposition was stated). Let me fix that now.

Proposition: The function $\Psi$ defined by the expression

$\Psi(s) = \sum_{n=1}^\infty \int_n^{n+1} \frac1{n^s} - \frac1{x^s}\; d x$

converges absolutely for $Re(s) \gt 0$ , giving a holomorphic function in that region, and

$\Psi(s) = \zeta(s) - \frac1{s-1}$

over the region $Re(s) \gt 1$ .

Proof: Let us bound the summands in the definition of $\Psi(s)$ :

$\array{ \left| \int_n^{n+1} \frac1{n^s} - \frac1{x^s}\; d x \right| & \leq & \underset{n \leq x \leq n+1}{\max}\; \left|\frac1{n^s} - \frac1{x^s} \right|\\ & \leq & \underset{n \leq x \leq n+1}{\max}\; \underset{n \leq t \leq x}{\max}\; \left| \frac{s}{t^{s+1}} \cdot (x-n)\right| \\ & \leq & \underset{n \leq t \leq n+1}{\max}\; \left|\frac{s}{t^{s+1}}\cdot 1\right| \\ & \leq & \frac{{|s|}}{n^{Re(s) + 1}} }$

where the second inequality follows from the mean value theorem. Since

$\sum_{n=1}^\infty \frac1{n^{Re(s) + 1}}$

converges over the region $Re(s) \gt 0$ , it follows that the series for $\Phi(s)$ converges absolutely, and $\Psi(s)$ is a holomorphic function, in that region.

Over the region $Re(s) \gt 1$ , we have

$\array{ \Psi(s) & = & \sum_{n=1}^\infty \frac1{n^s} - \int_n^{n+1} \frac1{x^s}\; d x \\ & = & \sum_{n=1}^\infty \frac1{n^s} - \sum_{n=1}^\infty \int_n^{n+1} \frac1{x^s}\; d x \\ & = & \zeta(s) - \int_1^\infty \frac1{x^s}\; d x \\ & = & \zeta(s) - \frac1{s-1} }$

as asserted.

Posted by: Todd Trimble on June 9, 2019 4:08 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Todd wrote:

Sure, there are any number of ways of showing there exists an analytic continuation of $\zeta(s) = \sum_{n \geq 0} \frac1{n^s}$ to larger regions, and one of those might be codable in PA. The question though is how you know there’s only one – and that, it seems to me, is the crucial thing one wants to know to justify the summing of the divergent series.

Oh, I wasn’t thinking of proving any result in PA about uniqueness of analytic continuations. I was just thinking about writing down some explicit formula for the analytic continuation of $\zeta(s)$ to the region around $s = -1$ , and showing that it equals $-1/12$ at $s = -1$ . What you want to do sounds like it would require more general results — and thus, perhaps, a stronger theory.

Posted by: John Baez on June 16, 2019 1:20 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Come to think of it: if one is trying to ‘justify’ $1 + 2 + 3 + \cdots = -1/12$ , this method looks easier to carry out in PA:

This is from Dillon Berger on Twitter, based on a blog article by Terry Tao. It’s an example of a rather general method of extracting the ‘finite part’ of a divergent sum.

The proof that this method matches what we get from analytic continuation may require more sophisticated logic than the proof that this method, applied to the case at hand, gives $-1/12$ .

Posted by: John Baez on June 16, 2019 11:52 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

I dunno. I’m being a real killjoy, I know, but ultimately that tweet by itself is no more impressive to me than the infamous Numberphile video, which is like a magic trick where someone palms an infinity. Terry’s blog post looks interesting at a glance (glad to see mention of Euler-MacLaurin), but I don’t see immediately how this alternative would carry any advantage for implementation in PA (which I have to admit doesn’t interest me greatly in the first place, so maybe I should have kept my mouth shut).

But since I didn’t, maybe I can ask: is there a nice explanation somewhere of what the rules of the game are supposed to be? To belabor an obvious point: a similar method gives

$1 + 1 + 1 + \ldots = \zeta(0) = -\frac1{2}$

but then we are not allowed to do

$1 + 2 + 3 + \ldots = 1 + (1 + 1) + (1 + 1 + 1) + \ldots = 1 + 1 + 1 + \ldots$

Terry’s blog mentions some other examples which are not so dumb as this, but anyway, I’d like it if there were agreed upon rules for the formalism which forced everyone to agree that $-\frac1{12}$ is the only sensible finite answer. After all, this is not just messing around: physicists put this “calculation” to use, as you explained in your 24 talk.

Posted by: Todd Trimble on June 17, 2019 3:34 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Isn’t the answer in Tao’s post? You’re looking at the smoothed asymptotic of these sums. In particular, you can’t do the manipulation you mention because the two sides differ as a function of the smoothing parameter – the $N$ in the $\eta(n/N)$ in the post. Even with a not-so-smooth smoothing function like the number of terms in the sum, you can see that the asymptotic as a function of the number of terms is quadratic for the left side and linear for the right side.

I worked out how you can make sense of the “magic trick” manipulations to get $\zeta(-1) = -\frac{1}{12}$ in this math overflow answer.

Posted by: Aaron on June 17, 2019 5:31 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Aaron, I am sort of aware that my example was a dumb example, and it wasn’t meant to be a main focus. But maybe you can respond to my actual question: has someone written down formal rules of the game for assigning finite results to divergent series, which would be such that everyone would agree, for example, that $-\frac1{12}$ is the only sensible finite result for the particular sum? (I know a little bit about various methods of summing divergent series, but I don’t have a good feeling for this question.)

Are you claiming that Tao’s post answers that question? I was thinking maybe the question is too broad or too vague, but I’m predicting I won’t be satisfied by an answer that specifies a particular method in advance. Optimally, the rules would guarantee that you’ll get the same finite result no matter which method you apply, as long as one plays within the rules.

Posted by: Todd Trimble on June 17, 2019 10:13 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

The short answer is that I don’t know, especially beyond what’s in Hardy’s book. There certainly won’t be something that guarantees a unique solution for every divergent series — there are examples in Hardy’s book. But I think that Tao’s post gives a nice conceptual understanding of what’s ‘going on’ with a few different techniques. It seems to encompass both all levels of Cesaro summation and zeta function regularization (at least for the simple series in the post). I’d like to see a more general theory of the uniqueness of the constant term as the smoothing function varies. I’d also love to see how to incorporate more general zeta function regularizations. I suspect that the technique also encompasses Abel summation, but I haven’t proven it.

I was also happy that one could use this technique to make sense of the Eulerian manipulations in that old video. Most importantly, I appreciate it because it elucidates why the -1/12 is the relevant quantity for the physics (which presumably generalizes to other zeta functions), because adding a cutoff and taking the constant term is what we do.

Hope that makes some sense.

Posted by: Aaron on June 19, 2019 2:08 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Aaron, that’s a very good response – thanks for that. In particular, for mentioning Hardy’s book, which I don’t know. I’ve just opened up a pdf file of his book.

But I think that Tao’s post gives a nice conceptual understanding of what’s ‘going on’ with a few different techniques. It seems to encompass both all levels of Cesaro summation and zeta function regularization

Already I find that prospect exciting. Possibly somewhat related: I had recently been wondering whether the “value” of a formal series obtained by analytically continuing a (say) Dirichlet series must agree with its value in the case where it is generalized Cesaro summable.

I’d like to see a more general theory of the uniqueness of the constant term as the smoothing function varies. I’d also love to see how to incorporate more general zeta function regularizations. I suspect that the technique also encompasses Abel summation, but I haven’t proven it.

Good questions. To be honest, regularization and renormalization techniques, despite being well-studied for decades, is not something I’ve studied well myself. So maybe I should bow out and just say thanks again for providing me your references and thoughts.

Posted by: Todd Trimble on June 19, 2019 3:49 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

I haven’t had much to contribute to this conversation, but just yesterday I happened to be looking back at this interview with Yuri Manin from 10 years ago, which I like to quote because of his prevision of HoTT:

… after Cantor and Bourbaki … set theoretic mathematics resides in our brains. … we start with the discrete sets of Cantor, upon which we impose something more in the style of Bourbaki. But fundamental psychological changes also occur. … the place of old forms and structures … is taken by some geometric, right-brain objects. … I am pretty strongly convinced that there is an ongoing reversal in the collective consciousness of mathematicians: the right hemispherical and homotopical picture of the world becomes the basic intuition, and if you want to get a discrete set, then you pass to the set of connected components…. Cantor’s problems of the infinite recede to the background: from the very start, our images are so infinite that if you want to make something finite out of them, you must divide them by another infinity.

and I noticed that he ends it with the following intriguing passage about summing divergent series:

This is parallel to the way we envisage a Feynman integral. At first it is just a hieroglyph charged with an interpretational challenge. The first two, three, four steps of interpretation are all ad hoc, appealing to various analogies with other cases where the mathematics is clean (“toy models”). At a certain stage you may get a formal series that doesn’t just diverge, but consists of terms that are themselves divergent (although finite-dimensional) integrals. Then you artificially regularize each term, making it finite. But the series, in general, still diverges. So you invent an interpretation of the series. And finally, having forced your way through a crowd of infinities, you obtain a finite answer. As a bonus, you get a series of marvelous mathematical theorems. I see in this an analogy with a rebuilding of pragmatic foundations in terms of category theory and homotopic topology.

It took me a while to figure out what he might have meant by this analogy, but eventually I came up with one possibility:

Unjustified manipulations of divergent series are analogous to nonrigorous arguments about $\infty$ -categories, such as the precursor to HTT. (If I may dare to say so, I feel that one still finds this sort of argument unfortunately often in the literature on $\infty$ -categories.)
Tricks of modern analysis like smoothed asymptotics are analogous to HTT-style $\infty$ -categories, which are rigorous but highly technical. By a fortunate coincidence we nowadays also refer to this as “analytic” higher category theory.
Todd’s request for “rules of the game” resonates with me as a request for a formal system, analogous to what we now call synthetic higher category theory, e.g. homotopy type theory, where we can reason directly with the objects we originally wanted to think about and be rigorous without having to delve into details like simplicial sets or smoothed asymptotics.

I don’t know whether this is what Manin had in mind, but it makes a bit of sense to me. He was making the analogy in one direction, but now that we have synthetic homotopy theory, we can turn the question around: Is there a “synthetic theory of divergent series”?

Posted by: Mike Shulman on June 20, 2019 6:06 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Brilliant! And thanks for that generous analogizing of my floundering groping-in-the-dark questions. I feel slightly less ridiculous as a result. :-)

Posted by: Todd Trimble on June 20, 2019 8:53 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Reverse mathematics uses systems of second-order arithmetic, so not PA in the usual first-order sense, and even then, all sorts of weird restriction on induction are made to get the systems intermediate between the base system ( $RCA_0$ ) and full second-order arithmetic ( $Z_2$ ), the latter of which corresponds to a good old NNO in a topos.

Posted by: David Roberts on June 7, 2019 1:44 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

I was just thinking: RCA₀, which is the least powerful of the ‘big five’ systems of second-order arithmetic, is already able to state and prove very general theorems like the intermediate value theorem or the Baire category theorem for separable complete metric spaces. (I’m getting this from Wikipedia; I keep rereading this article and forgetting the details.) But an equation like $\zeta(-1) = -1/12$ is very specific, basically just the result of a calculation: we’re not quantifying over all continuous functions $f \colon [0,1] \to \mathbb{R}$ , or all separable metric spaces. To state this equation we don’t really need to talk about real numbers in general, just certain computable ones. So it might be true that this equation, or some other easier identities, can be encoded into Peano arithmetic somehow, and proved there.

Posted by: John Baez on June 7, 2019 2:03 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Well, I guess since there’s an elementary statement equivalent to RH (pdf), maybe there’s an elementary statement equivalent to the evaluation of the analytic continuation of $12\sum_{n=1}^\infty \frac{1}{n^s}$ at $s=-1$ being $-1$ , via some kind of estimates etc.

Posted by: David Roberts on June 7, 2019 12:27 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Clearly, I need to make jokes more often.

Posted by: Blake Stacey on June 14, 2019 12:51 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

It seems that two slightly different versions of this post were accidentally posted; see also.

Posted by: L Spice on June 6, 2019 6:10 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Yeah, John posted it a nonstandard number of times.

Posted by: Tom Leinster on June 6, 2019 10:14 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Whoops! I don’t know how that happened. Fixed.

Posted by: John Baez on June 7, 2019 12:38 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

More episodes:

Part 10: Michael and I talk about “ordinal analysis”: a way of assigning ordinals to theories of arithmetic, that measures how strong they are.
Part 11: Michael begins explaining Enayat’s paper Standard models of arithmetic. I pull him into explaining “Craig’s trick” and “Rosser’s trick”, two famous tricks in mathematical logic.

Posted by: John Baez on June 23, 2019 2:04 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Typo in part 10: there’s an “&epsilon” that doesn’t render as $\epsilon$ .

Posted by: Blake Stacey on June 27, 2019 6:17 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Finally fixed!

Posted by: John Baez on September 16, 2019 3:03 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

After a long summer break, a new part:

Part 12: Some examples. $PA^ZF$ is strictly stronger than $PA$ . $PA^{ZFC+GCH} = PA^ZF$ where $GCH$ is the generalized continuum hypothesis. $PA^{ZFL} = PA^{ZF}$ where $ZFL$ is $ZF$ plus an axiom saying all sets are constructible in Gödel’s sense. But $PA^{ZFI}$ is strictly stronger than $PA^{ZF}$ , where $ZFI$ is $ZF$ plus the existence of an inaccessible cardinal!

Posted by: John Baez on September 16, 2019 3:03 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Part 13: Enayat’s “natural” axiomatization of $PA^T$ , and his proof that this works. A digression into Tarski’s theorem on the undefinability of truth, and how to work around it. For example, while truth is not definable, we can define truth for statements with at most a fixed number of quantifiers.

Posted by: John Baez on September 22, 2019 9:06 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Here’s another article on nonstandard models of arithmetic:

John Baez, Enayat on Nonstandard Numbers, Azimuth.

It’s a gentle introduction to the most recent of my dialogs with Michael Weiss, part 18. This is where we finally get around to discussing all of Ali Enayat’s main results. I see I’m behind on listing this series:

Non-standard Models of Arithmetic 14: The previous post showed that PA^T implies Φ_T, where Φ_T is Enayat’s “natural” axiomatization of PA^T. Here we show the converse. We also interpret Φ_T as saying, “Trust T”.

Non-standard Models of Arithmetic 15: We start to look at Truth (aka Satisfaction). Tarski gave a definition of Truth, and showed that Truth is undefinable. Less enigmatically put, there is no formula True(x) in the language of Peano arithmetic (L(PA)) that holds exactly for the Gödel numbers of true sentences of Peano arithmetic. On the other hand, Truth for Peano arithmetic can be formalized in the language of set theory (L(ZF)), and there are other work-arounds. We give an analogy with the Cantor diagonal argument.

Non-standard Models of Arithmetic 16: We look at the nitty-gritty of True_d(x), the formula in L(PA) that expresses truth in PA for formulas with parse-tree depth at most d. We see how the quantifiers “bleed through”, and why this prevents us from combining the whole series of True_d(x)’s into a single formula True(x). We also look at the variant Sat_Σ_n(x,y).

Non-standard Models of Arithmetic 17: More about how True_d evades Tarski’s undefinability theorem. The difference between True_d and Sat_Σn, and how it doesn’t matter for us. How True_d captures Truth for models of arithmetic: PA ⊢ True_d(⌜φ⌝) ↔ φ, for any φ of parse-tree depth at most d. Sketch of why this holds.

Non-standard Models of Arithmetic 18: The heart of Enayat’s paper: characterizing countable nonstandard T-standard models of PA (Prop. 6, Thm. 7, Cor. 8). Refresher on types. Meaning of ‘recursively saturated’. Standard meaning of ‘nonstandard’; standard and nonstandard meanings of ‘standard’.

Posted by: John Baez on September 22, 2020 6:08 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

I wonder if it is possible to axiomatically define the category of first order models of Peano arithmetic, and, if so, what the axioms would be. Because in such a category, I would imagine the standard model of arithmetic would simply be the initial object of the category, and nonstandard models of arithmetic be the other objects in the category. (Of course, the category of second order models of Peano arithmetic is simply the terminal category, because there only exists one second order model of Peano arithmetic up to isomorphism.)

Posted by: Madeleine Birchfield on September 24, 2020 2:52 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Yes, there’s probably a couple of interesting categories where the objects are models of (first-order) PA in a given universe U of sets, but the one I like has morphisms being functions $f: N \to N'$ that preserve everything you can express in the language of PA. These are called elementary embeddings — click for a more precise (and general) definition.

In this category, the standard model of PA is an initial object.

Enayat’s results fascinate me because they talk about what happens when you think of U as sitting inside some larger universe V.

Posted by: John Baez on September 24, 2020 5:20 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

I had a look at the link to Enayat’s results, and in the comments of that article you wrote

I think the concept of “standard” model of PA is circular in a subtle way.

Very very crudely, the standard natural numbers are 0, 1, 1+1, 1+1+1, … But what does the “…” mean? It means that we keep on going. But how long do we keep on going? What sort of expressions 1+…+1 count as natural numbers? Only those where there’s a standard natural number of plus signs!

But this is circular.

Is the axiom that a NNO exists in the category a required axiom for defining a category of models of first order PA in some universe of sets (and elementary embeddings as morphisms if you wish)? If not, then the categorical approach would eliminate the circularity of defining standard models of first order PA, because as you said one could simply define the standard model as the initial object of said category, after defining the category.

Posted by: Madeleine Birchfield on September 24, 2020 8:01 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Is the axiom that a NNO exists in the category a required axiom for defining a category of models of first order PA in some universe of sets?

I’m not sure which category is “the category” here. A category of sets that’s not big enough to have a NNO is probably not going to have models of PA in it.

Anyway, later it sounds like you’re asking: if we have some category of models of PA with elementary embeddings as morphisms, can we define a ‘standard’ model to be an initial object in there? And I’d say yes, I think that’s something category theorists would be happy to do. (It’s not something I’ve ever seen traditional logicians do.)

Posted by: John Baez on September 25, 2020 12:12 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

I’m not sure which category is “the category” here. A category of sets that’s not big enough to have a NNO is probably not going to have models of PA in it.

In which case the circularity is inherent regardless. But that is true of the foundations in general; one needs some external concept of set or class or type in order to define a set theory or class theory or type theory, so it would seem one would need some external concept of the natural numbers to define arithmetic theories of the natural numbers.

Posted by: Madeleine Birchfield on September 25, 2020 1:24 AM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Here’s another blog article where Michael and I examine Enayat’s results on nonstandard models of arithmetic:

Non-standard Models of Arithmetic 19: We marvel a bit over Enayat’s Prop. 6, and especially Cor. 8. The triple-decker sandwich, aka three-layer cake: ω^U⊂U⊂V. More about why the omegas of standard models of ZF are standard. Refresher on Φ_T. The smug confidence of a ZF-standard model.

Posted by: John Baez on September 24, 2020 11:59 PM | Permalink | Reply to this

Re: Nonstandard Models of Arithmetic

$MathML-enabled post (click for more details).$

Here’s another:

Non-standard Models of Arithmetic 20: We start to develop the proof of Enayat’s Prop. 6. We get as far as a related result: any nonstandard model of PA is recursively d-saturated. (‘Recursively d-saturated’ is our user-friendly version of the professional-grade concept: recursively Σ_n-saturated.)

This was so fun that I decided to summarize it in simple terms here:

Recursive saturation.

Posted by: John Baez on October 13, 2020 6:56 AM | Permalink | Reply to this

The n-Category Café

Skip to the Main Content

June 5, 2019