There’s no ‘best’ embedding of any countable model of the nonstandard integers into the complex numbers, and not even any computable such embedding. But there are lots of them. Given any embedding

of any field into the complex numbers, you can compose it with any automorphism

and get a new embedding

And there are vast numbers of such automorphisms! To be precise, there are of them where is the cardinality of the continuum. The reason is:

1) Any **transcendence basis** of — that is, a maximal set of numbers that obey no polynomial equations with rational coefficients — has cardinality

2) Given any transcendence basis of and any permutation there is an automorphism extending

For a bit more read Andrés Caicedo’s remark on MathOverflow. Note that in this comment I’m assuming the axiom of choice.

]]>Thanks for the answer and fixing the LaTeX. Wanted also to amend myself and add that I meant countable *exponent* ultrapower and hence countinuum power , as in parent comment, though perhaps this doesn’t makes the embedding more constructive.

Given any countable model of Peano arithmetic, there’s never a *specific* embedding of it into the complex numbers: the theorem simply says that such an embedding *exists*, and the proof gives no clue about how to build it. Basically you just choose a transcendental number for each nonstandard integer, subject to the conditions that the arithmetic operations on these numbers match those of the corresponding integers. This is done with the help of the Axiom of Choice.

If , given an ultrafilter , is a countable ultrapower of the standard model of arithmetic, how would the map look like? What complex number would correspond to, say, an hyperinteger ?

]]>Thanks for your feedback John. I referred to Rudin’s proof as long-winded only in contrast with Ax’s terse proof, which deftly reduces the theorem to the pigeonhole principle with the help of some basic theorems of model theory and algebra.

Also sorry for not having noticed the clear Latex-related instructions in bold fonts (in these situations I am reminded of how some of us resemble some of our students).

]]>Neat! I haven’t thought much about the Grothendieck–Ax theorem. It reminds me slightly of the apparently much harder

Jacobian conjecture.

The model-theoretic proof should be compared to the long-winded one using standard tools of multivariable complex variables, by the eminent Walter Rudin in the following paper:

• Rudin, W. (1995) Injective Polynomial Maps are Automorphisms, American Mathematical Monthly, 102, 6:540-543.

I’ll have to look at it! It’s rare to see a long-winded proof that’s only 3 pages long… though some of my students’ homeworks contain examples.

By the way, when posting a comment with LaTeX here, read the directions written in big black letters that appears directly over the comment box:

]]>You can use Markdown or HTML in your comments. You can also use LaTeX, like this: $latex E = m c^2 $. The word ‘latex’ comes right after the first dollar sign, with a space after it.

An exposition can be found in the Chapter 6 of the following master’s thesis by Amanda Purcell:

https://pqdtopen.proquest.com/doc/1476434703.html?FMT=AI

The model-theoretic proof should be compared to the long-winded one using standard tools of multivariable complex variables, by — guess who? — the eminent Walter Rudin in the following paper:

Rudin, W. (1995) Injective Polynomial Maps are Automorphisms, American Mathematical Monthly, 102, 6:540-543

]]>https://mathoverflow.net/questions/66146/nonstandard-reals-in-the-complex-plane

]]>silvascientist wrote:

Should we not also add, “and taking roots of polynomials”?

Yes indeed! Thanks, I’ll fix that comment of mine.

]]>Thanks for your reply! One minor thing… when you say the transcendence degree of a field is “the least number of elements of we need to throw into to generate all of using the field operations…”, should we not also add, “and taking roots of polynomials”? For instance, I’m pretty sure has transcendence degree one, even though a finite number of elements could never suffice to generate over using just the field operations…

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