Is arithmetic with infinite numbers fictitious?

Question

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.

Answer 1

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.)

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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!)

Answer 2

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.

Answer 3

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.