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according to compactness theorem in logic, there are models for the Reals of all infinite cardinalities, and these are elementary-equivalent to those of the "Standard" Reals ( Reals with uncountably-infinite cardinality). Questions: 1) Are any of these non-standard Reals metrizable? Without having an a priori topology, I cannot apply metrization theorems. I assume we can also try to tell if metrizability is a 1st-order property. If so, what is the (a) metric, up to topological equivalence? I assume since there is a nonstandard analysis that the answer is yes, since analysis , AFAIK, necessarily makes use of a metric. And this metric would restrict to the usual one on Standard pairs of Reals. What would $d(x,y)$ be if either of $x,y$ is a non-standard Real?

2) Once the axioms for the Standard Reals are given, the Real line is a model for these axioms. Are there "nice", maybe geometric models for nonstandard Reals? Is the Real line ( together with a choice of metric) a model for the nonstandard Reals?

Thank you.

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  • $\begingroup$ There is a natural topology, the order topology. But you need to ask yourself a question as well, what good is a metric if it doesn't give you a "reasonable valuation" like $|a|=d(a,0)$? And in case of a non-standard field, if $p$ is any non-standard point, what is $d(p,0)$? $\endgroup$ – Asaf Karagila May 31 '15 at 18:56
  • $\begingroup$ (Also, "Standard reals" are standard, almost all the models you'd get have uncountable cardinality (if only because only "a few" countable fields which are elementarily equivalent to $\Bbb R$ exist, up to isomorphism). The standard reals are standard, there's no need to explain what they are. $\endgroup$ – Asaf Karagila May 31 '15 at 18:57
  • $\begingroup$ You mean there is no valuation for nonstandard Reals so that, for the metric $d$, $|a|=d(a,0)$? This is part of my question, how do we define $d(x,y)$ when at least one of $x,y$ is nonstandard? $\endgroup$ – Gary. May 31 '15 at 18:59
  • $\begingroup$ Yes, because a metric, by definition, can only attain real values. If $a$ is transfinite, what would that value be? $\endgroup$ – Asaf Karagila May 31 '15 at 19:00
  • $\begingroup$ But how can we then do Analysis without a metric? I though analysis was necessarily tied to a metric. And, is it then the case that metrizability is not a first-order property? Because otherwise, I believe, by elementary equivalence, the non-standard Reals would be metrizable. $\endgroup$ – Gary. May 31 '15 at 19:01
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The key to your questions, I think, is to understand how to look at these things from an internal perspective. Recall that the transfer principle says that the standard and nonstandard models look identical when viewed internally: the technical term is "elementarily equivalent". That is, any statement you can even write down (and even those you can't) in a suitably bounded fragment of set theory is either true for both models or false for both models.

Since analysis is done in a suitably bounded fragment of set theory, this says that analysis in the standard model behaves identically to analysis done internally to the nonstandard model.

e.g. an internal metric space is an internal set $X$ together with an internal function $d : X \to {}^\star \mathbb{R}$ satisfying

  • $d(x,y) \geq 0$
  • $d(x,y) \geq d(x,z) + d(z,y)$
  • $d(x,y) = 0 \implies x = y$

the nonstandard reals are an internal metric space. And they even have internal completeness properties, such as the fact that any internal subset that is bounded and nonempty has a least upper bound.

Now, an internal metric space is usually not an external metric space -- i.e. a metric space in the "ordinary" sense -- because the distance function is nonstandard-real-valued, not real-valued.

Worse, things that behave nicely internally are usually very poorly behaved externally; e.g. while the nonstandard reals do have a topology (given by the order topology, or even by the internal metric), it has horrible external properties like being a totally disconnected space.

For the most part, you don't want to study arbitrary things in the nonstandard model from the external perspective, since they simply don't work well. Instead, they should be understood from the internal perspective.

Once you've grokked that, (what I know of) nonstandard analysis is all about studying standard objects by transferring them to the nonstandard model and looking for external properties that simplify their study.

e.g. if you've seen the definition of limits:

$\lim_{x \to a} f(x) = L$ if and only if, for every nonzero infinitesimal $\epsilon$, you have $\mathop{\mathrm{std}} f(\epsilon) = L$

only applies to standard functions $f$ and standard extended real numbers $a$ and $L$ (and the last appearance of $f$ is meant to be its transfer to the nonstandard model). This definition doesn't work if $f$ is not a (transfer of a) standard function, or if $a$ and $L$ are not (transfers of) standard reals.

(However, the usual $\epsilon-\delta$ definition of continuity works works for all internal functions $f$ and all nonstandard real numbers $a, L$; in fact, for internal functions from the extended hyperreals to itself, internal limits are external limits)

Addenda:

  • When $\infty$ means the largest element of the extended real numbers (as it usually does in calculus and analysis), $\mathop{\mathrm{std}} x = \infty$ if and only if $x$ is a positive transfinite number. And (the transfer of) $\infty$ is greater than every hyperreal number.

  • If really desired, we can have a nonstandard model of the nonstandard model, so that nonstandard objects can be studied with nonstandard nonstandard techniques in the same way standard objects can be studied with nonstandard techniques. (e.g. internal limits can be defined using the infintiesimals of the nonstandard nonstandard model that are smaller than every infinitesimal of the nonstandard model)

  • The existence of the nonstandard model that is elementarily equivalent is easily proven by taking the theory whose axioms are every true statement about standard analysis. Then they must also be true in any nonstandard model of the theory.

  • For a more "constructive" approach, look up the term "superstructure"

  • There is a reasonable generalization of the notion of "metric" where the distance function takes values in an ordered abelian group rather than in the reals. These generalized metric spaces will have some of the nice properties of ordinary metric spaces, but not all of them.

  • If you're interested in how much can be done semialgebraically (i.e. $+,-,\times,\div,<$), the relevant notion is to study "real closed fields" rather than hyperreals and nonstandard analysis. There is even a whole field of "real algebraic geometry". You can talk about limits, derivatives, differential forms, and such using just semi-algebraic methods, although there are limits to what you can do (e.g. integration doesn't have a direct analog). I think there is a related field

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  • $\begingroup$ Thank you, I just noticed this answer. Please give me some time to read it and digest it, possibly ask a follow up. $\endgroup$ – Gary. Jul 2 '15 at 15:29
  • $\begingroup$ @Hurkyl : Sorry for the necro-post, How do we determine if *f is a transfer of a function f, or if numbers *a, *L are transfers of Standard Reals a,L respectively? $\endgroup$ – MSIS Aug 20 '19 at 20:23
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    $\begingroup$ @MSIS: Basically, (AFAIK) the question "I have some nonstandard thing; how do I tell if it's standard?" never comes up in practice. You only know and/or care about the answer when you started with something standard, or did standard calculations with standard things, or did a calculation that explicitly produces standard results (e.g. apply the "round to the standard part" function). $\endgroup$ – user14972 Aug 21 '19 at 9:54

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