K-theory and sets
Ignacio Añón
ianon at latahona.com.uy
Sun Jun 5 19:29:46 EDT 2022
Moroe Eskew wrote:
*"Could you elaborate on this claim? In what way do Todorcevic and Woodin
come from “different traditions” in set theory?"*
The following is a concrete case: to his dying day, Gödel seems to have
remained skeptical, about the existence of a consistent, arbitrary large
value for 2^Aleph_alpha(the power of the continuum). It is well documented
that he differed with Solovay on this point(*). He even went as far as
sketching a draft, where he supposedly proved that the continuum was of
power Aleph_2; the draft was sent to Tarski for publication in the PNAS(**).
In the draft Gödel uses 4 axioms: the first two are today known in the
literature as the square and rectangular axioms, and the 4th axiom is
particularly interesting. Gödel calls it "Hausdorff's continuity axiom".
Regarding this axiom, Solovay writes, in the notes to the collected works:
"I have been unable to locate this phrase in the literature", and goes on
to propose that the axiom refers to a definition in Hausdorff's "*Grundzüge*"
classic: Solovay also mentions that both Takeuti and Todorcevic said to him
that this was probably what Gödel meant...
To the few of us who have read with lovingly care Hausdorff's complete
opus, it seems quite likely that what Gödel referred to as the "Hausdorff
continuity axiom", is a particular type of extended continuous map that
Hausdorff defined in one of his late papers: "Erweiterung einer stetigen
abbildung" (Extension of a continuous map). The paper is not available in
English(I'm translating it myself), and is quite fascinating. It might be
impossible to know, for sure, what Gödel meant by that axiom, and it
remains a fascinating, crucial open problem in ZFC, whether these 4 axioms
are consistent, and if consistent, whether they imply the Aleph-2 limit for
the continuum. When I get enough time, I hope to attack this issue.
Hausdorff gaps seem to be the key difficulty here, but the known proofs of
their absoluteness, are, in my view, defective.
My intention in mentioning all this stuff, is the following: the
impeccable, detailed work done by Solovay in the collected works,
nevertheless missed this quite relevant aspect about the Hausdorff axiom:
this can hardly be blamed on Solovay, but it underlines perfectly how rich,
subtle, and complex, the literature and legacy of ideas is within set
theory.
With regard to your concrete point about Woodin and Todorcevic: it is clear
that Woodin, like Kunen and Solovay, takes for granted the results in(***),
and works with inner models as if there are arbitrary large, consistent,
values for 2^Aleph_alpha. Todorcevic, on the other hand, has investigated
with care Gödel's axioms in search for a concrete, quasi topological, limit
for the continuum(****). His work is quite relevant in complex analytic
number theory, not just in fom.
Moroe Eskew wrote:
*"What writings of Kanamori support this claim, and how?"*
My reference to Kanamori, aimed at revealing the obvious fact, that set
theory has quite antithetical legacies. To my knowledge, Kanamori has never
addressed in detail the work of Todorcevic or Woodin(I hope he does!). But
take in a row the following 4 pieces: "Bernays and set theory", "Gödel and
set theory", "Cohen and set theory", "Kunen and et theory". By reading
them, you'll get a concrete sense that set theory is a rich forest of
cultural ideas, no matter how much some specialist would like to turn it
into a poor, unified, closed academic "guild"...
Best
(*) Solovay's comments in Vol II of Gödel's collected works are a
fascinating read. In a note added to the 1965 edition of his original paper
on Con-ZF, Gödel writes: "The value that can consistently be assigned to
2^Aleph_alpha turns out to be almost completely arbitrary". It's weird, but
you can still perceive a subtle note of skepticism behind that comment...
(**) See Vol III collected works, Solovay's Comments in the intro to the
draft.
(***) Solovay. Independence results in the theory of cardinals. I, II,
Notices of the American Mathematical Society 10, 595
(****) Rectangular axioms, perfect set properties and decomposition.
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.533.834&rep=rep1&type=pdf
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