https://en.wikipedia.org/wiki/Dickson%27s_lemma

and

https://en.wikipedia.org/wiki/Buchberger%27s_algorithm

Dicksons lemma follows by a straightforward induction argument.

So no need to apply Zorns Lemma here.

]]>I think your interpretation of the ultraproduct is exactly right.

]]>But these other proofs (especially 2) are really elegant.

Proof #3 — which I don’t really understand — looks a lot like ‘the product partial field of infinitely many finite fields has a homomorphism to a field of char 0’.

]]>I think there may be an opportunity for some reverse mathematics here: which axiom scheme is necessary to prove Theorem 1?

]]>I was a bit confused by proof #2 at first, but I think you made a typo. Don’t you mean that if M has no representation in char. 0, then T + { *not* S_p : p prime } is inconsistent?

]]>that sounds like a really good idea. As you suggest, the algebraic cycle matroid looks like it is coming from the singleton partition in this way. It would be great to hear if you make any progress with this idea. For example, which bipartitions of the set of all ends induce infinite matroids in this way? What do you think?

]]>thanks for your comment. It seems to me that, at least on a finite ground set, the construction you suggest should give a matroid, namely the dual of the union of N with the dual of M. This uses the matroid union theorem.

]]>I also sent you an email applying a specific instance of this construction to study the problem of whether every nearly finitary matroid is $k$-nearly finitary.

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