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This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the abstract approaches to NSA (e.g., via the compactness theorem), as compared to the more concrete approach using ultrapowers? One can name generic reasons such as naturality, functoriality, categoricity, etc., but I am hoping for a concrete illustration of why a more abstract approach may be advantageous for understanding NSA concepts and/or proving theorems.

Note 1. One of the existing answers provided a bit of information about advantages of the more abstract approach in terms of saturation. I would appreciate an elaboration of this if possible, in terms of a concrete application of saturation.

Note 2. These issues are explored in more detail in this 2017 publication in Real Analysis Exchange.

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    $\begingroup$ Don't most logicians view the approaches as fundamentally the same? They move easily from one approach to the other. The ultrapower approach simply provides a detailed reason why the compactness approach is correct, just as one can prove the compactness theorem generally via ultrapowers. But sometimes those details are not important, and one can appeal to general principles. $\endgroup$ – Joel David Hamkins Dec 15 '15 at 15:29
  • $\begingroup$ Strictly speaking the approach via compactness is actually weaker since you can't axiomatize semantic truth (this remark was made by Martin Davis in a private message). Be that as it may, while the logicians move easily from one to the other, the general mathematical public doesn't, and may well be interested in motivation for the more abstract approach. I should mention that by "concrete" I don't mean "constructive" for obvious reasons, but rather "more congenial" to a broader public that finds quotient constructions more accessible than creation ad nihilo via compactness. $\endgroup$ – Mikhail Katz Dec 15 '15 at 15:46
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    $\begingroup$ @JoelDavidHamkins, that's getting pretty close to an answer provided one spells out the meaning of "smooth" :-) $\endgroup$ – Mikhail Katz Dec 15 '15 at 15:51
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    $\begingroup$ OK, I'll write up something by tomorrow fleshing out that perspective (I'm busy writing an exam just now). But meanwhile, I encourage NSA experts to post their views... $\endgroup$ – Joel David Hamkins Dec 15 '15 at 15:57
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    $\begingroup$ From what I've gathered from conversations with Udi Hrushovski, it can be convenient to work with models far larger than ultrapowers in order to gain access to stronger saturation properties, more automorphisms, a better theory of types and indiscernibles, etc.. So in some ways one can view "model theory" as being more advantageous than either of the two approaches to nonstandard analysis that you listed. though for the majority of mathematicians not trained in model theory, "ordinary" nonstandard analysis (in either approach) usually suffices. $\endgroup$ – Terry Tao Dec 16 '15 at 17:21
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To my way of thinking, there are at least three distinct perspectives one can naturally take on when undertaking work in nonstandard analysis. In addition, each of these perspectives can be varied on two other dimensions, independently. Those dimensions are, first, the order of nonstandardness (whether one wants nonstandardness only for objects, or also for functions and predicates, or also for sets of functions and sets of those and so on), and second, how many levels of standardness and nonstandardness one desires. Let me describe the three views I have in mind, and give them names, and then discuss how they relate to one another.

Classical model-construction perspective. In this approach, one thinks of the nonstandard universe as the result of an explicit construction, such as an ultrapower construction. In the most basic instance, one has the standard real field structure $\newcommand\R{\mathbb{R}}\newcommand\Z{\mathbb{Z}}\langle\R,+,\cdot,0,1,\Z\rangle$, and you perform the ultrapower construction with respect to an fixed ultrafilter on the natural numbers (or on some other set, if this was desirable). In time, one is led to want more structure in the pre-ultrapower model, so as to be able to express more ideas, which will each have nonstandard counterparts. And so very soon one will have constants for every real, a predicate for the integers $\Z$, or indeed for every subset of $\mathbb{R}$ and a function symbol for every function on the reals, and so on. Before long, one wants nonstandard analogues of the power set $P(\R)$ and higher iterates. In the end, what one realizes is that one might as well take the ultrapower of the entire set-theoretic universe $V\to V^{\omega}/U$, which amounts to doing nonstandard analysis with second-order logic, third-order, $\alpha$-order for every ordinal $\alpha$. One then has the copy of the standard universe $V$ inside the nonstandard realm $V^*$, which one analyzes and understands by means of the ultrapower construction itself.

Some applications of nonstandard analysis have required one to take not just a single ultrapower, but an iterated ultrapower construction along a linear order. Such an ultrapower construction gives rise to many levels of nonstandardness, and this is sometimes useful. Ultimately, as one adds additional construction methods, this amounts as Terry Tao mentioned to just adopting all of model theory as one toolkit. One will want to employ advanced saturation properties or embeddings or the standard system and so on. There is a very well-developed theory of models of arithmetic that uses quite advanced methods.

To give a sample consequence of saturation: every infinite graph, no matter how large, arises as an induced subgraph of a nonstandard-finite graph in any sufficiently saturated model of nonstandard analysis. This often allows you to undertake finitary constructions with infinite graphs, modulo the move to a nonstandard context.

Standard Axiomatic approach. Most applications of nonstandard analysis, however, do not rely on the details of the ultrapower or iterated ultrapower constructions, and so it is often thought worthwhile to isolate the general principles that make the nonstandard arguments succeed. Thus, one writes down the axioms of the situation. In the basic case, one has the standard structure $\R$ and so on, perhaps with constants for every real (and for all subsets and functions in the higher-order cases), with a map to the nonstandard structure $\R^*$, so that every real number $a$ has its nonstandard version $a^*$ and every function $f$ on the reals has its nonstandard version $f^*$. Typically, the main axioms would include the transfer principle, which asserts that any property expressible in the language of the original structure holds in the standard universe just in case it holds of the nonstandard analogues of those objects in the nonstandard realm. The transfer principle amounts precisely to the elementarity of the map $a\mapsto a^*$ from standard objects to their nonstandard analogues. One often also wants a saturation principle, expressing that any sufficiently realizable type is actually realized in the nonstandard model, and this just axiomatizes the saturation properties of the ultrapower. Sometimes one wants more saturation than one would get from an ultrapower on the natural numbers, but one can still achieve this by larger ultrapowers or other model-theoretic methods.

Essentially the same axiomatic approach works with the high-order approach, where one has nonstandard version of every set-theoretic object, and a map $V\to V^*$, with nonstandard structures of any order.

And similarly, one can axiomatize the features one wants to use in the iterated ultrapower case, with various levels of standardness.

As with most mathematical situations where one has a construction approach and an axiomatic approach, it is usually thought to be better to argue from the axioms, when possible, than to use details of the construction. And most applications of nonstandard analysis that I have seen can be undertaken using only the usual nonstandard axioms.

Nonstandard Axiomatic approach. This is a more radical perspective, which makes a foundational change in how one thinks about one's mathematical ontology. Namely, rather than thinking of the standard structures as having analogues inside a nonstandard world, one essentially thinks of the nonstandard world as the real world, with "standardness" structures picking out parts of it. So one has the real numbers including both infinite and infinitesimal reals, and one can say when two finite real numbers have the same standard part and so on. With this perspective, we think of the real real numbers as what on the other perspective would be the nonstandard reals, and then we have a predicate on that, which amounts to the range of the star map in the other approach. So some real numbers are standard, and some functions are standard and so on.

One sometimes sees this kind of perspective used in arguments of finite combinatorics, where one casually considers the case of an infinite integer or an infinitesimal rational. (I have a colleague who sometimes talks this way.) That kind of talk may seem alien for someone not used to the perspective, but for those that adopt the perspective it is very useful. In a sense, one goes whole-hog into the nonstandard realm.

More extreme versions of this idea adopt many levels of standardness and nonstandardness, extended to all orders. Karel Hrbáček has a very well-developed theory like this for nonstandard set theory, with an infinitely deep hierarchy of levels of standardness. He spoke on this last year at the CUNY set theory seminar, and I refer you to his articles on this topic. In Karel's system, one doesn't start with a standard universe and go up to the nonstandard universe, but rather, one starts with the full universe (which is fundamentally nonstandard) and goes down to deeper and deeper levels of standardness. Every model of ZFC, he proved, is the standard universe inside another model of the nonstandard set theories he considers.

Ultimately, my view is that the choice between the perspectives I mentioned is a matter of taste, and that in principle any achievement of nonstandard analysis that can be undertaken with one of the perspectives has natural analogues that can be expressed with the other perspectives.

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    $\begingroup$ In the context of your idea of multiverse and switches I wonder whether it is accurate to describe Nelson's approach as "thinking of the nonstandard world as the real world". This would seem to presuppose that what is described as the "intended model" is the appropriate primary point of reference. The said intended model assumes a certain homogeneity about numbers that could be contested. Nelson would argue that there is a vast difference between the ordinary counting numbers on the one hand and an integer that's so big that it cannot be expressed by a computer the size of the universe... $\endgroup$ – Mikhail Katz Dec 20 '15 at 8:37
  • $\begingroup$ ...even in the total time allotted to our civilisation. In other words, inhomogeneity as captured in Leibniz's distinction between assignable and inassignable numbers should be a starting point rather than the second story. To summarize, this way of describing Nelson's framework seems based on philosophical assumptions that go counter to the ideas of mutliverse and switches. $\endgroup$ – Mikhail Katz Dec 20 '15 at 8:38
  • $\begingroup$ I find nonstandard analysis to be fundamentally about the interaction between two different worlds -- the standard world and the nonstandard counterpart -- and therefore fundamentally it involves multiverse issues. Nevertheless, when I used the phrase, "thinking of the nonstandard world as the real world," I had meant to refer to something a bit milder. Namely, which world has the real numbers? On the third perspective, what others might call the nonstandard reals are effectively the real reals. One can consider this difference merely semantics or terminology rather than philosophy. $\endgroup$ – Joel David Hamkins Dec 20 '15 at 10:48
  • $\begingroup$ Certainly Nelson's take on the story is that the real world has the real numbers, definitely. Also, this real number line contains elements that test negative for the standardness predicate. The standard points does not form a set so one can't speak of a pair of worlds one being embedded inside the other. Your expression "the real numbers" involves a rather confident use of the definite article that seems to be at variance with the philosophy of the multiverse, but I am certainly not an expert. $\endgroup$ – Mikhail Katz Dec 20 '15 at 11:36
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I feel like this is an instance of a larger question:

When might it be nice to work with an axiomatic description of a theory rather than an explicit construction?

This comes up all the time, e.g. in topology, cohomology, algebra (e.g. abstract groups rather than permutation groups), and more recently with homotopy type theory.

Some possible answers to this question:

  • Working with axioms provides the right level of abstraction: proofs often become much easier since you're left with only the essential facts, rather than the forest of theorems being obscured by the trees of the particular construction.

    In NSA, you're trying to justify the algebraic manipulations done by Newton or Euler on paper: but these are naturally stated in the abstract language of calculus, and involving ultrafilters just complicates matters. Of course you can just re-derive all the required inference rules, but the point is that the logical rules themselves are a useful conceptual framework.

  • Having an axiomatic framework opens the door to unexpected realizations of a concept. This is of course easy to see for a concept with many different realizations like groups, where things as disparate as field automorphisms, toplogical braids and integers all have a group structure, but can be more surprising for things like, e.g. cohomology or real closed fields, where there are only few immediately apparent models. It's probably useful to note that the concept of group itself has turned out to be much more general than envisioned originally. Once you have an axiom system, it becomes very easy to try to find "non-standard" models by various techniques (including ultrafilters, of course), or even dropping axioms, as in the famous "extra-ordinary" cohomology theories. One might even suggest that NSA itself is the result of finding non-standard models of an axiomatic theory of real numbers. It's not surprising therefore that people working with NSA might be more amenable to the advantages of the axiomatic approach.

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    $\begingroup$ Cody, thanks for this. I am not sure this is really about axiomatics though. There is an axiomatic approach to Robinson's framework, developed by Hrbacek, Nelson, and others. Here you find infinitesimals within the real number system itself. But Robinson constructs an extension of $\mathbb{R}$ by compactness theorem and transfer principle is a consequence of the fact that it is an elementary extension, whereas for ultrapowers the transfer principle is a consequence of Los' theorem. So in this sense you end up with transfer in either case. The key seems to be in saturation that's... $\endgroup$ – Mikhail Katz Dec 17 '15 at 18:22
  • $\begingroup$ ... apparently handled more successfully in the more abstract approaches. Hopefully this can be clarified. $\endgroup$ – Mikhail Katz Dec 17 '15 at 18:22
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For most purposes, I think the premise is wrong: in many situations ultraproducts simply are the preferable approach, so I'll try to discuss some of the purposes which are exceptions. @cody has already mentioned the issue of axiomatics, which is a reason to avoid ultraproducts.

Another reason to avoid ultraproducts is that nonstandard analysis actually handles iterated infinitesimals more easily. It's common in nonstandard arguments to take a nonstandard number H, and then take a second number which is not only nonstandard, but nonstandard (i.e. sufficiently big) relative to H. I've never seen an argument like this done in the ultraproduct version. Though of course there's no formal obstacle, for arguments like these the concreteness of the ultraproduct becomes a disadvantage, and getting it right would require either taking multiple ultraproducts or some fiddling with the underlying construction to choose the right elements. The abstract approach, on the other hand, is very natural: having gotten used to working in NSA, doing it repeatedly adds no new complications.*

This is really a special case of the issue of saturation. The point of nonstandard analysis/ultraproducts is that certain infinite intersections of sets are guaranteed to be non-empty. There's a lot of room to detail exactly which infinite intersections this applies to. Ultraproducts are useful in part because they give a definitive answer which is the right one for many purposes.

But the abstract approach is more flexible: it can consider both less saturated versions (structures which are nonstandard but don't have "$\aleph_1$-saturation"), and also more saturated versions (like the double saturation implied by the item above, or by demanding $\kappa$-saturation for larger cardinals---that is, considering intersections of more than just countable sets).

*: In fact, I'm not certain one can do the analogous arguments using only a single ultraproduct---one might have to take the ultraproduct twice to get the right result.

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  • $\begingroup$ The iterated infinitesimals are the main reason people sometimes use iterated ultrapowers, as I mention in my answer. It is often useful to iterate the ultrapowers along a saturated linear order, and not just a well-ordered order, so that one gets a corresponding range of "levels" of standardness. $\endgroup$ – Joel David Hamkins Dec 19 '15 at 23:12
  • $\begingroup$ @JoelDavidHamkins: That makes sense. I'm curious: I've only seen them used in set theory and models of arithmetic. Is anyone using iterated ultrapowers as an alternative to NSA in applications outside of logic? $\endgroup$ – Henry Towsner Dec 19 '15 at 23:46
  • $\begingroup$ Karel Hrbáček's approach to nonstandard set theory is based fundamentally on iterated ultrapowers, or at least, since he proceeds axiomatically, on the features one gets in such kinds of iterated ultrapowers. But what he is doing is, as you mention, nonstandard set theory rather than NSA as such. $\endgroup$ – Joel David Hamkins Dec 20 '15 at 0:40
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    $\begingroup$ Some time ago, I attempted to rewrite Szemeredi's original proof of his theorem on k-term arithmetic progressions in NSA to see if it became any cleaner. I found out that it became natural to ask for (if I recall correctly) k-1 different levels of standardness, necessitating a k-2-fold iterated ultrapower construction or something comparable. At that point I decided that the proof was not, in fact, going to get any cleaner, and I never actually wrote it up. Maybe I'll get back to it one day, though. $\endgroup$ – Terry Tao Dec 28 '15 at 18:12
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You can quote me on this if you like.

This is such an old issue that I am surprised it is still up for discussion. I know you like ultra-powers, etc, but I have always thought they were wrong-headed; it was Luxemburg who made them popular for NSA originally.

The point is that you actually really NEED the transfer theorems; so you essentially need the logical apparatus; in most cases the compactness theorem and some form of saturation. Occasionally a type omitting argument could be used but that is rare.

The reason is that you are postulating that "all of mathematics" carries over to the non-standard model and in fact, that is the underlying intuition in the applications.

I also think that the "more concreteness" of the ultraproduct construction is just an illusion. You can not answer if the integer produced by (1,2,3,4,5, ...) is even or odd since it depends on the ultrafilter completion of e.g. any non-principal filter.

Moreover, the basic tools in almost all the proofs is dealing with the seam between "standard" and "non-standard" elements; which really depends on a feel for expressibility in the language being used. Nelson's approach was to try to help people manage without the "feel". That's a matter of taste.

Furthermore, I would differ with your point that NSA is to formalize classical mathematicians arguments. That is "cool" and Robinson greatly enjoyed it, but really he also was sure that it is a great way to prove new theorems. Since people sort of like to hear about the theorems consequence in "classical" settings; then one often had to find equivalence theorems (this was clear in the work on brownian motion etc) and in my work on inverse limits of finite groups (by the uniqueness of inverse limits, the non-standard finite groups are essentially equivalent to inverse limits of systems of finite groups and a lot of the very early work of Lubotzky on profinite groups can be easily carried out in this setting) but actually a more extreme position can be seen when you just take the position that the "standard" results are just one specific implementation and one could develop mathematics happily without them. The argument that the non-standard models are not "unique" is just a habit and not important.

Robinson himself put this viewpoint forward very nicely in his Brouwer medal address "Standard and NonStandard Number Systems"

Best regards Larry Manevitz manevitz@cs.haifa.ac.il

Regarding the use of limit ultrapowers etc; one needs to be careful there. I once proved there was no measureable cardinal by taking such a large ultrapower; but actually all the standard sets dont grow in that context :).

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IST easily maintains many levels of standardness (sort of). Take an inf. large integer $H_0$. The family $F_0$ of all values $f(H_0)$, $f:\mathbb N\to \mathbb N$ being standard, is bounded in (the nonstandard) $\mathbb N$, so there is an inf. large integer $H_1$ bigger than any number in $F_0$. Replace $H_0$ by $H_1$ in this argument, getting $H_2$, and so on. Then each $H_{k+1}$ behaves in many ways like an inf. large integer wrt $H_k$. My book with Reeken considers relative standardness in detail.

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