Standard models of N and R: An Alice/Bob approach

Question

This is a question about a comment in a recent publication by Roman Kossak. Kossak wrote:

"Nonstandardness in set theory has a different nature. In arithmetic, there is one intended object of study - the standard model. There is no intended model in set theory. Set-theorists talk about universes of sets, but what happens in those universes very much depends on the axioms that they satisfy, and there are many axiomatic systems to choose from. There is no standard model of set theory, but still there is a notion of nonstandardness."

Model Theory of Nonstandard Structures with Applications

The point seems to be as follows. First, the purported base model $\mathbb N$ seems to please everybody, and therefore deserves the name "the standard model". Meanwhile, the corresponding model of $\mathbb R$ given by Goedel's computable reals does not please everybody because $V=L$ leads to consequences that are considered too strong. The point is not so much models of ZFC as models of $\mathcal P(\mathbb N)$ namely $\mathbb R$, where there is already a dispute with regard to the "minimal model". For example, if $V=L$ then $\mathcal P(\mathbb N)$ will satisfy CH, but other models will not.

Alice could argue regarding $\mathbb N$ that it has the special property that, under any ZFC-like foundational system, $\mathbb N$ is the least model of the Peano axioms. Bob would respond that Goedel's constructible reals are the least model of $\mathbb R$ in a similar sense to $\mathbb N$ being the minimal model of PA.

Alice could argue that $\mathbb N$ is routinely identified with a rather subtle thing that we may call the physical model of PA (counting pebbles, etc.) and practical computation.

Bob would ask, as far as counting pebbles and practical computation are concerned, $\mathbb N$ is "routinely identified" by whom exactly? It is perhaps the metalanguage integers that can be related to pebble-counting etc. However, the metalanguage integers form a sorites-like subcollection that cannot be assumed to satisfy PA (of course it would be different for formal computation). The point was argued in detail in a 2017 publication in Real Analysis Exchange https://u.cs.biu.ac.il/~katzmik/infinitesimals.html#17f Furthermore, where would one look for a "physical model of PA" ?

Alice could argue that positing a minimal $\mathbb N$ appeals to the broader mathematical public.

Bob would retort that Platonist notions may be appealing to such a public, but are they are justified?

Alice could argue that, by Goedel, ZFC does not prove the consistency of any model of ZFC; thus to get a model of ZFC, we would need to argue in a suitable stronger theory.

Bob would counter by noting that Goedel incompleteness applies to models of PA, as well.

Alice could argue that identifying a model $M$ of ZFC with any assemblage of things bearing any form of physical existence is well beyond any human vision of the world of things.

Bob would retort that, while $M$ is beyond any form of physical existence, both $\mathbb N$ and $\mathbb R$ are embedded in such an $M$, so it may be hard to argue for a difference between $\mathbb N$ and $\mathbb R$ based on the nature of $M$. While it seems incontestable that such an $M$ can't be endowed with "any form of physical existence", couldn't one argue the same for $\mathbb N$ and $\mathbb R$?

Such an analysis tends to confirm the following conclusions reached by Rittberg:

(1) the metaphysical views of mathematicians can shape what counts as relevant research; (2) mathematical results can shape the metaphysical beliefs of mathematicians; (3) metaphysical thought and mathematical activity develop in tandem in mathematical practices.

Mathematical Practices Can Be Metaphysically Laden

In conclusion, one could formulate the following query concerning Kossak's view on the difference between standard models for $\mathbb N$ and $\mathbb R$. Some people also find the so-called standard $\mathbb N$ displeasing, because it does not reflect the historical record of mathematicians from Leibniz to Cauchy who worked with infinite (in technical modern terminology, unlimited) integers. From this point of view, Edward Nelson's $\mathbb N$ (incorporating a distinction between standard and nonstandard integer, the latter being unlimited) would be more satisfying.

Could one then argue that there is little difference between $\mathbb N$ and $\mathbb R$ on account of the question of the existence of a standard/intended model?

Note. In response to a comment below the question to the effect that "standard means that the model is well-founded", note that Nelson's $\mathbb N$ (incorporating the standard/nonstandard distinction, as mentioned above) is well-founded in precisely the same technical sense as in ZFC. Another comment claimed that Kossak "is just making a sociological observation. People agree that the minimal model of first-order PA deserves to be called the standard model of the natural numbers". My question concerns the meaning the definite article before "standard model", as per above.

Answer 1

At the request of Mikhail, I am turning a comment of mine into a partial answer, even though what I am going to write down are well-known, presumably by Mikhail as well. While I am not going to directly argue for that there is little difference between $\mathbb{N}$ and $\mathbb{R}$ in terms of the plausibility of the existence of a standard/intended model, I shall argue that one cannot argue against this claim, at least not without strong (philosophical) assumptions regarding the underlying axiomatic system.

Since the metatheory-theory distinction matters in what follows, let me name a few things that I shall talk about. I shall work in ZFC whose language is augmented with a constant symbol $0$ for the empty set and a unary function symbol $S$ for the successor function. Let me introduce four different worlds.

• $\text{(R)}$ is the real physical world in which I write symbols on a paper and build my axiomatic system. The terms $0$, $S0$, $SS0$,... of my axiomatic system shall be referred to as numerals. Let me say that ZFC is physically-consistent if I can never derive a contradiction in my axiomatic system in the physical world we live in even when we are provided with unlimited physical power, space and time. Observe that this is not a mathematical statement in my axiomatic system, but rather is a statement about a physical phenomenon.

• $\text{(V)}$ is the set-theoretic world. In this world, the set $\mathbb{N}$ of natural numbers is defined to be the unique object $N$ for which the statement $$\varphi(N): 0 \in N \wedge \forall w (w \in N \rightarrow Sw \in N) \wedge (\forall x ((0 \in x \wedge \forall w (w \in x \rightarrow Sw \in x))\rightarrow (\forall w (w\in N \rightarrow w \in x)))$$ holds, i.e. it is the smallest inductive set. Observe that when I talk about a natural number $n \in \mathbb{N}$ in my axiomatic system, I only write statements that involve the variable symbol $n$. Therefore, when I talk about natural numbers, I do not necessarily talk about numerals; although, I can prove one by one that each numeral belongs to $\mathbb{N}$. Let me write $\text{Con(ZFC)}$ for the statement in my axiomatic system that formalizes the idea of physical-consistency that I had in mind. So $\text{Con(ZFC)}$ is a statement in my axiomatic system that is of the form $\forall n \in \mathbb{N} \dots$

• $\text{(M)}$ is a model of (various extensions) of $\text{ZFC}$. In this model, there is an object $\mathbb{N}^{\text{M}}$ which this model believes satisfies $\varphi$. This model has its own understanding of the statement $\text{Con(ZFC)}$ that is determined by $\mathbb{N}^{\text{M}}$.

• $\text{(M)}'$ is an object inside $\text{M}$ which $\text{M}$ thinks is a model of $\text{ZFC}$.

Here is my claim.

Philosophical claim. There cannot be any methods to argue that the collection of numerals $0$, $S0$, $SS0$,... coincide with $\mathbb{N}$. In other words, we cannot argue that the numerals of $\text{R}$ and the natural numbers of $\text{V}$ are the same.

How can we argue for this claim? Since some of the objects to be discussed are not mathematical objects but rather physical strings of symbols in real life, surely we cannot argue for this claim mathematically as a theorem in our axiomatic system.

On the other hand, we can prove theorems that will translate into this once appropriately interpreted. By Gödel's theorem, we know that $$\text{Con(ZFC+Con(ZFC))} \rightarrow \text{Con(ZFC+Con(ZFC)}+\neg\text{Con(ZFC+Con(ZFC)))}$$ Hence, assuming that there is a model of $\text{ZFC+Con(ZFC)}$ at all, we can obtain a model $\text{M}$ of $$\text{ZFC+Con(ZFC)}+\neg\text{Con(ZFC+Con(ZFC))}$$ Inside this model, the formal statement $\text{Con(ZFC)}$ holds and so there are objects $\text{M}' \in \text{M}$ which $\text{M}$ thinks are models of $\text{ZFC}$. However, $\text{M}$ also believes $\neg\text{Con(ZFC+Con(ZFC))}$ and so believes that $\text{ZFC} \vdash \neg \text{Con(ZFC)}$. Therefore, the objects $\mathbb{N}^{\text{M}'}$ can never be equal to $\mathbb{N}^{\text{M}}$ from $\text{M}$'s perpective. That is, we can have a model of set theory that thinks that all of the set models of $\text{ZFC}$ inside it are $\omega$-nonstandard from its perspective.

Now let us pull this argument back layer by layer. $\text{M}'$ is to $\text{M}$ as $\text{M}$ is to $\text{V}$. Consequently, assuming that ZFC is physically-consistent and $\text{Con(ZFC)}$, we cannot prove that $\mathbb{N}$ and $\mathbb{N}^{\text{M}}$ are the same for an arbitrary set model $\text{M}$ of $\text{ZFC}$, not without additional assumptions, because all models of $\text{ZFC}$ may be $\omega$-nonstandard as simulated above.

In the same fashion, $\mathbb{N}^{\text{M}}$ is to $\mathbb{N}$ as $\mathbb{N}$ is to the collection of numerals. Therefore, we cannot hope to even argue that the collection of numerals and $\mathbb{N}$ are the same since any argument would have to break down somewhere once it is pushed back through these formal layers.

The idea that there is a standard/intended model of arithmetic presupposes that the formalization of the collection of numerals as the set $\mathbb{N}$ correctly captures the essence of the informal concept of numerals. Why should we believe in this?

Answer 2

Like Burak, I am responding to the OP's request to promote my comments to an answer, with the caveat that I want to avoid wading too deeply into philosophical debates that I think are beyond the scope of MO. I will try to focus on the question of what Kossak meant in the quoted paragraph. (The best authority on that topic is of course Kossak himself, but I think something can be said without asking him directly.)

The focus of the paragraph is the term "intended model of set theory" or "standard model of set theory." (Side remark: as François G. Dorais commented, "standard" in set theory typically means "well-founded," but I think that in this paragraph, "standard model of set theory" is being used as a synonym for "intended model of set theory." I will use the latter term to avoid confusion.) I maintain that "There is no intended model of set theory" is primarily intended here as a sociological claim, about mathematical practice. In particular, Kossak is not claiming that "There is no intended model of set theory" is a mathematical theorem. One can cite theorems that are relevant to the discussion, but the claim itself is not a formal mathematical claim.

Evidence for my claim can be found by looking around at how other people use the term "intended model of set theory." For example, Virginia Klenk's paper, Intended models and the Löwenheim-Skolem theorem, makes it clear that she regards debates about the existence of an intended model of set theory as philosophical debates. In the (unpublished?) note, Remarks on Intended Models of Mathematical Theories, Jerzy Pogonowski says, "We do not have a single intended model of set theory ZF," but again in a way that makes it clear that it is not so much a mathematical theorem as a comment about the practice of set theory (e.g., the search for additional axioms for set theory).

To quote my own comment, people agree that the minimal model of first-order PA deserves to be called "the standard (or intended) model of the natural numbers." People don't agree that the minimal (transitive) model of ZFC, or any other specific model of ZFC, deserves to be called "the intended model of set theory." I don't think there's anything deeper than that going on in Kossak's paragraph. Now, whether people are justified in having different attitudes in the two cases is another matter, but as I said, that is a philosophical debate that I don't want to wade into here.

Answer 3

I very much like Burak's and Timothy Chow's answers, and I hope that this answer complements theirs. Your imaginary conversation between Alice and Bob got me thinking a little differently about your question: "Could one then argue that there is little diffence between N and R on account of the question of the existence of a standard/intended model?" Particularly, I like how your question has used the word "intended". To that end, please view this answer as an invitation to another Alice-and-Bob-like conversation.

When I think about what our formal systems of numbers are intended to model, I'd answer "the usual counting/arithmetic we do every day." We have this picture in our heads of an idealized notion of number, which we (pre-mathematically) assume has a Platonically idealized existence in the world of concepts. With the discovery that there are multiple models of the usual axioms that we (previously) thought should specify the numbers, and indeed with any computable list of axioms for said numbers, this throws a wrench in idea that we have a real picture of the Platonic ideal. Then, as a saving grace, we learn that inside all of these models is a smallest one. And of course, the picture in our head doesn't have any extraneous baggage thrown on top. When we write $0,1,2,3,\cdots$, we don't think of those dots hiding the unnecessary, just the necessary. This seems to save the day.

And yet, if one digs a bit deeper, problems seem to crop up again. We discover that this intended model might not model "the usual counting/arithmetic we do every day" as intended, so we start having to make caveats. We discover historical conversations about the difference between completed infinities and potential infinities. Is a completed infinity the true model, or just a potentialist version? The nearly universal acceptance of $\mathbb{N}$ seems to be more a matter of historical utility than of serious self-reflection.

I'd argue that all of that is somewhat irrelevant. Any idea, when drilled down on, needs caveats, explanations, and our understanding has to expand and grow. We went from believing the earth is flat, to being a sphere, to being an elongated ellipsoid. None of those is perfectly accurate, but they are getting us further understanding at each stage. Our understanding of the idealized numbers similarly continues to go through growth stages, which will necessitate a deeper understanding of what $\mathbb{N}$ is, but does not (for most of us) eliminate our hope that there is such an idealized notion.

The story is fundamentally different for ZFC. Sets were originally intended to model all possible idealized collections. The early paradoxes revealed that this idealized concept was inconsistent. There is no entire collection. There cannot be a "true" $V$. There cannot even be a completed collection of all ordinals.

The intended model of the numbers is based on a minimality property; the smallest collection doing something. The (original and naïve) intended model of sets is based on a maximality property; having the most freedom to talk about arbitrary collections. This rules out wanting $V=L$ as an axiom, as that limits the types of collections available.

Answer 4

This is way too long for a remark, therefore I post it as an answer.

Although I agree with the previous answers that there is probably no clear “intended” model of set theory, there are perhaps two principles for any “standard” model of set theory which many mathematicians could agree upon, though I am not sure how to formalize them. However, in contrast to the claim in the original question, I do not think that this is some general sort of “maximality” principle, but partially quite the opposite. To formulate the principles at least intuitively, let me first point out that set theory knows (at least) about two set hierarchies:

1. The hierarchy of ordinals.
2. The hierarchy of power sets.

Since there seems to be an agreement that the “standard“ model of $\mathbb N$ should be guided by some “minimality” principle, one should probably assume for a standard model of set theory as well that the set-theoretic analogue $\omega_0$ should correspond to such a “standard” model of $\mathbb N$. More general, it seems to me that many mathematicians could agree that the whole hierarchy of ordinals which reflect “infinite counting“ should obey such a form of “minimality” principle for a “standard” model. I think that this means, roughly speaking, that the considered models of set theory should not “skip” any ordinals which might exist in “smaller” models. (As mentioned, I am not sure whether or how this could be formalized; at least 1st order logic does not seem to be sufficient for this.)

For the hierarchy of power sets, with $\mathbb R$ and its power set being the “simplest“ examples in the infinite case, however, many mathematicians could perhaps agree indeed on some sort of “maximal comprehension” principle in the sense that the model should not have an artificial restriction on the “admissible” subsets: Anything which forms a “collection“ of elements of a set $M$ in any intuitive sense should be a set of the model. In particular, also non-constructible sets (opposing V=L) and “exotic“ sets (like non-measurable or non-Baire subsets of $\mathbb R$ or non-principal ultrafilters on $\mathbb N$) should be sets of the model, probably even many sets which (in contrast to the known constructions of “exotic” sets based on AC) cannot be obtained with AC alone. Such a “maximal comprehension” principle, of course, is even harder to formalize (if at all possible) than the minimality principle for ordinals (and very likely not possible at all to formalize in 1st order logic).

Side notes:

• In physics, apparently no “exotic” sets do exist. Therefore, if one uses set theory to “model“ in a physical sense some real-world problem, it might be more natural to consider a model without such exotic sets even at the cost of the full power of AC. That's why some applied mathematicians might consider the “maximal comprehension” principle above questionable.
• The axiomatic approaches to NSA (for instance of Nelson) mentioned in the original question violate both principles “formulated” above: Neither does ”their” $\omega_0$ obey some minimality principle, nor does the natural collection of standard elements of any (infinite) set $M$ of the model form a set in the sense of the model.

Answer 5

It's hard to pinpoint what question is being asked in this post, but here is my attempt at an answer.

Second-order number theory with full semantics pinpoints the intended model of the structure of natural numbers. First-order PA does not. Second-order induction, or the first-order approximation as an induction schema, attempt to pinpoint what "numbers" there aren't, i.e., to rule out nonstandard models.

ZF+$V=L$ quite effectively says what sets there aren't, in order to limit models of ZFC. It quite naturally implies AC, as well as the GCH. To me, it doesn't say that V is "skinny", rather, it says that On is "tall". There is no reason to reject V=L in favor of measurable cardinals and other "very large" cardinals, just like there is no reason to reject AC in favor of Reinhardt cardinals. Large cardinals are metamathematical objects, as are models of ZF in which AC is false.

Edit: I think the undue resistance to the axiom of constructibility stalemates any reasonable attempts at nailing down a unique category of sets. I don't believe "ultimate L" will ever gain popular traction, for reasons that should be apparent.

Regarding large cardinals: I have never seen them used in ring theory or number theory (my two main passions), and I see them fundamentally as metamathematical tools for gauging the consistency strengths of various theories. I would be probably ok with ruling out worldly cardinals and above, but I don't think of such objects enough to advocate banishing them or alternatively demanding that they must exist. I think the question of whether $V$ equals $L$ is far more important to settle.

My opinion about $V=L$ is in the minority, but it has had some important supporters.

Aside regarding Edward Nelson's $\mathbb{N}$ alluded to in the OP: I am a huge fan of that version of nonstandard analysis. Introducing a new predicate "standard" is an elegant way of doing infinitesimals in a conservative extension of PA. I actually prefer Nelson's elementary nonstandard analysis to standard calculus. Infinitesimals are often more easily grasped than $\epsilon$-$\delta$ limits are, e.g., by physicists. I hope that in 100 years we will be back to using infinitesimals in calculus.

Finally, regarding the question, is there a difference between $\mathbb{N}$ and $\mathbb{R}$ with regard to standard models, etc? My position is that there is a unique initial object (namely, $\mathbb{N}$) in the category of semirings with identity, and a unique terminal object (namely, $\mathbb{R}$) in the category of archimedean ordered fields, up to isomorphism, and these categories themselves are unique in the meta-category of categories, and that questions of standard models, etc., are metamathematical concerns having to do with limitations of first-order logic and formal systems. This is a "monist Platonist" stance. At the same time, I recognize the value of considering nonstandard models of arithmetic and analysis, or of sets where $V$ is not $L$, or where even AC and the axiom of foundation both fail. If $V = L$ is eventually accepted, as I hope it will be, then that will settle CH and GCH in the positive and thus afford us a more complete picture of the true $\mathbb{R}$.

Here are a few of many references I could provide.

https://link.springer.com/book/10.1007/BFb0070208 (Keith Devlin, The Axiom of Constructibility: A Guide for the Mathematician)

https://arxiv.org/abs/1210.6541 (Joel David Hamkins: A multiverse perspective on the axiom of constructiblity)

Why not adopt the constructibility axiom $V=L$? (MO question from 2019)