# Is arithmetic with infinite numbers fictitious?

In 1933 Skolem constructed models for arithmetic containing infinite numbers. In a 1977 article Stillwell emphasized the constructive nature of Skolem's approach; see here. Is this at odds with Tennenbaum's theorem on nonrecursivity?

This question is related to a comment exchange at Does evaluating hyperreal $f(H)$ boil down to $f(±∞)$ in the standard theory of limits? where terms like "fictitious" are being applied to nonstandard models, as well as the following comment:

"You wrote that hyperreals 'have precise definitions' (plural), which they do not. No matter how 'precise definition' is interpreted (definable, constructive, Borel, ZF+DC), there is no such way to name an individual H. The role of ZFC in your arguments is indistinguishable from a single axiom 'an H exists'. When we assume a Zeus exists, draw conclusions that apply to any element of the set of Zeuses, and the argument works equally well with Zeus replaced by any Greek male over 180cm tall, then Zeus is used only as a metaphor, (...)."

Note. The point about a nonstandard model of arithmetic is that one can do a significant fragment of calculus just using the quotient field of such a model. Avigad did something similar in his article in 2005: Avigad, Jeremy. Weak theories of nonstandard arithmetic and analysis. Reverse mathematics 2001, 19–46, Lect. Notes Log., 21, Assoc. Symbol. Logic, La Jolla, CA, 2005. See here.

• @zyx, who do you think you are kidding? I am referring to your fictitious, metaphor, and Zeus comments in the exchange following this answer: math.stackexchange.com/questions/1595793/… – Mikhail Katz Jan 6 '16 at 7:50
• I'm saying that you misrepresented and distorted the comments, which were not about nonstandard models but the description of $f(H)$ for individual elements of the models, the ambiguity of the expression $f(H)$, and other matters. The issues around those don't really relate to the models per se. The Zeus thing was about neither the models nor the elements, but an explanation of ways in which an argument can use its assumptions as metaphors and not genuine existence assertions. I predict you will have a hard time finding material that matches the distortion you added to the question. – zyx Jan 6 '16 at 8:50
• That is not a quotation that contains what you claim I said. The stuff you just edited into the question doesn't contain it, either. Maybe we need a thread addressing your distortion of Connes and Bishop. The arxiv papers are pretty dishonest. – zyx Jan 6 '16 at 8:57
• That's a view you are projecting onto the discussion. I am addressing mainly the lack of distinction between standard and nonstandard (in the stated context of your question). I did not say anything about rejection of AC or proof by contradiction, and answered in the negative your incorrect speculation about the latter. If anything, the difficulties with NSA may be less in constructive mathematics, given that theories of constructive NSA exist. I am not familiar with them, knowing only the Robinson approach. – zyx Jan 6 '16 at 12:38
• Compared to the standard alternative and to the motivating historical material, Robinson's framework falls far on the other side of objective dividing lines within that big umbrella of classical mathematics. Unremovable use of uncountable AC, undefinability of the individual nonstandard objects, noncomputability of the model. The complete inability to explain to students what is a concrete example of a nonstandard $H$ and how, exactly, to determine $f(H)$ and its standard part. Those differences are not empty word games. – zyx Jan 6 '16 at 19:37

## 3 Answers

I don't really see the connection with the linked question, but to address your question in the first paragraph: No, the two results are not at odds. Skolem's construction is not effective - see the Ramsey-style argument going from the end of page 149 to the middle of page 150. All of this is non-computable. (Note also that it's not unique, either: there are many different ways to follow this construction, which will produce non-isomorphic models.)

The sense in which it is constructive is that it can be used to produce a definable nonstandard model (actually, several definable nonstandard models) of arithmetic. However, definable is much broader than computable. (Also, I believe most logicians would disagree with calling this "constructive," and I note that Stillwell does not use that word in his article.)

A further edit: Perhaps surprisingly, there are definable hyperreals. This was proved by Shelah and Kanovei - see http://arxiv.org/abs/math/0311165 published in Journal of Symbolic Logic; see http://www.ams.org/mathscinet-getitem?mr=2039354

(Also notice that "hyperreals" and "nonstandard model of arithmetic" are very different things!)

• Thanks. I take it Skolem's procedure is a construction to the extent that it takes place in ZF (rather than ZFC)? – Mikhail Katz Jan 5 '16 at 13:15
• @user72694 It actually takes place in much less than ZFC - the theory called "$ACA_0^+$" should suffice, if I read it correctly, which is vastly weaker. – Noah Schweber Jan 5 '16 at 13:17
• You mean to say that it takes place in much less than ZF? – Mikhail Katz Jan 5 '16 at 13:24
• @user72694 Yes, that was a typo. – Noah Schweber Jan 5 '16 at 13:25
• "one can do a significant fragment of calculus just using the quotient field of such a model." In logic papers, yes. Is there any working version of this used to "do significant amounts" (or teach them) of infinitesimal calculus as an applicable math/science subject? If not, it seems like a pointless distraction to raise it in conversations about nonstandard analysis. To argue that deficits of Robinson's theory might be real, but some other system, that nobody ever implemented, in some theoretical sense is free from those deficits, is just changing the subject. – zyx Jan 6 '16 at 10:45

The exact content of the cited K - Shelah result is that there is a concrete formula $A(x)$ in the set theoretic language such that ZFC (not ZF!) proves that 1st there is unique $x$ satisfying $A(x)$ and 2nd every $x$ satisfying $A(x)$ is a ctbly saturated elementary extension of the reals. In brief, there is a definable ctbly saturated elementary extension of the reals, in ZFC. By some reasons known to those working in nst models, this is not true wrt ZF.

Also, it is not asserted that the mentioned extension necessarily contains a definable nonstandard real.

The key tool of the proof is to assemble all really relevant ultrafilters in a sort of superfilter containing all of them in some sense, by means of a product of Fubini type.

• ::choosing a nonstandard $H$ and calculating $f(H)$ and then taking standard part This is utterly meaningless. You cannot pretend to calculating $f(H)$ prior you calculate' $H$ (whatever that means anyway). – Vladimir Kanovei Jan 11 '16 at 9:46
• The utterly meaningless thing seems to be the same as the description (say by @user72694 here at math.stackexchange.com/questions/1582790, that started all the discussions ) of how to use NSA instead of the standard theory of limits at $+\infty$. I called it a metaphor, rather than meaningless, to talk (in this context, limits/asymptotics) about standard part st( ) and $f(H)$ as functions that exist independent of each other. Other than word substitutions like choosing H --> calculating H, metaphor ---> meaningless, I don't see any disagreement (yet) between my remarks and this answer. – zyx Jan 11 '16 at 10:40
• You know, methafor' is a term suitable for poetics, belles lettres, fantasy or something like that, and people who are versed in those things understand it perpectly. But if you use it in a scientifically oriented piece (let alone specifically mathematical) people start thinking what the hell is meant by e.g. calculate $f(H)$? Like what is $\sin(H)$ or the standard part of it? And sans any specifying which way a generic inf. large number is chosen? Does it make any sense at all in view of the de nihilo nihil criteria? – Vladimir Kanovei Jan 11 '16 at 11:18
• (in continuation as it eclipsed the allowed size) Just to make my argument clearer, let me ask, what is $\sin(H)$ where $H$ is an arbitrarily large real. Does it make any sense? – Vladimir Kanovei Jan 11 '16 at 11:19
• Is the 1st part, unique x satisfying A(x), done in ZF or ZFC? It would be surprising, and a very nice example, if one needs choice to define a unique object. Also, what is the cardinality of the countably saturated extension? The paper builds the extension in uncountably many iterations, and it was not obvious from looking at the paper if there is any increase of cardinality during that process. – zyx Jan 11 '16 at 18:44

Is arithmetic with infinite numbers fictitious?

It depends on your definition of "arithmetic with infinite numbers" and "fictitious". The meaning of fictitious that this question was written to oppose, was in reference to certain descriptions of how Robinson's nonstandard analysis is used for calculus. Those descriptions don't have any obvious equivalent for Skolem arithmetic, because Skolem arithmetic is not used as a tool for doing or teaching calculus, or for any other application outside of mathematical logic and model theory.

In 1933 Skolem constructed models for arithmetic containing infinite numbers. In a 1977 article Stillwell emphasized the constructive nature of Skolem's approach. [...]

The words like constructed and construction have no particular meaning here beyond "formal existence proof". Stillwell did not use the word constructive whose precise interpretations do not apply to Skolem's proof.

Is this at odds with Tennenbaum's theorem on nonrecursivity?

There are computable number systems that extend integer arithmetic with additional objects that can be interpreted as infinitely large, and operations extending the familiar ones to the larger system. Polynomials with integer coefficients and computable ordinal notations are two examples. Tennenbaum's theorem shows that Skolem arithmetic cannot be presented in that way, with discrete computable data and operations on them.

This question is related to a comment exchange at Does evaluating hyperreal $f(H)$ boil down to $f(±∞)$ in the standard theory of limits? where terms like "fictitious" are being applied to nonstandard models,

"Fictitious" was applied to descriptions of what is done with nonstandard analysis, not the models themselves. The idea that nonstandard models constructed using the Axiom of Choice have a lesser form of existence than constructs that do not, is certainly an objection that arises in discussions of NSA, just not in the one that you linked to.

The metaphors and fictions relating to NSA occur not (as far as I was asserting) so much in the existence of the objects, but in the descriptions of how the theory is used, such as the idea that there is an ability to take the standard part of bounded $f(H)$ (going beyond the standard rubric of taking limits as $H \to \infty$ when they exist) when this ability never materializes except as the standard thing.

To the extent there is a problem on the existence front, it is that taking individual elements of the nonstandard models is more elusive than just constructing the models, so that the description of "choosing a nonstandard $H$ and calculating $f(H)$ and then taking standard part" can only mean a procedure that is independent of $H$, which is standard analysis dressed in very marginally different words. It doesn't matter whether one considers the individual $H$ to really exist or not, there just is no way to do things like compute standard part of $\sin(H)$ or other functions that depend nontrivially on infinite $H$.

Note 2. The point about a nonstandard model of arithmetic is that one can do a significant fragment of calculus just using the quotient field of such a model.

Only in logic papers. This is not a real "use" of nonstandard arithmetic to do calculus as something taught to and utilized by nonlogicians.

• Many words with little if any sense. Arithmetic with infinite numbers (if you mean NSA) is the field (or extended field) structure of any (such-and-such) non-Archimedean extension of R. As a part of mathematics it is not less fictitious than 2+2=4 or any other application of your favourite Turing machine. – Vladimir Kanovei Jan 11 '16 at 15:52
• Yes, I gave an example of "arithmetic with infinite numbers" in your generalized sense of the term, and it is universally assumed that NSA and nonstandard and nonArchimedean extensions of R are well defined theories in the same sense as any other theory in mathematics. There can be descriptions (or sales advertisements) of the use of a non-fictitious theory that are more or less fictitious, because (eg.) they do not accurately describe the way in which the theory is used. – zyx Jan 11 '16 at 18:32
• ::nonstandard and nonArchimedean extensions of R are well defined theories::: They are not theories, not methaphors, not sales advertisements, but mathematical structures as legitimate in principle as the number 7. A theory (in logic) is a list of axioms which describe some category of structures. There is no such a property of a theory as being fictitious, let alone `more or less' such, the closest thing is a theory which has no models at all, which is inapplicable to NSA. – Vladimir Kanovei Jan 11 '16 at 22:14
• For example, it is said that NSA gives a way to obtain the derivative as a ratio of infinitesimal increments dy/dx , which then justifies algebraic manipulation of differentials. A look at books like Keisler and how they handle separation of variables will show that this is fictitious advertising. The justification is not through nonstandard infinitesimals, but by making the equation $dy = f'(x) dx$ a definition(!), and using some other limit (sorry, "standard part") to make sense of $f'(x)$. The situation with st($f(H)$) is similar. – zyx Jan 12 '16 at 5:52
• what is your problem ... what offends you that much? /// I did not have a "problem" or take offense at anything. I am giving examples and definitions of how descriptions of (the use of) a theory can be metaphorical or fictitious either in absolute terms, such as the description of Leibniz dy/dx ratio being rescued by NSA, or relative to more accurate descriptions (such as st(f(H)) being a metaphor for standard calculations). All those things that are correctly or fictitiously described are well-defined as syntactic objects, there is no claim that the theory is ill-defined or fictional. – zyx Jan 14 '16 at 4:37