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Topic: Matheology § 236
Replies: 5   Last Post: Apr 4, 2013 11:34 PM

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FredJeffries@gmail.com

Posts: 988
Registered: 11/29/07
Re: Matheology § 236
Posted: Apr 4, 2013 7:23 AM
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On Apr 4, 12:26 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> Tarski?s theorem: (For all infinite sets X there exists a bijection of
> X to X × X) ==> (Axiom of Choice). [...] Fréchet and Lebesgue refused
> to present it. Fréchet wrote that an implication between two well
> known propositions is not a new result. Lebesgue wrote that an
> implication between two false propositions is of no interest.
> [Jan Mycielski: "A System of Axioms of Set Theory for the
> Rationalists", Notices of the AMS 53,2 (2006) 206-213]http://www.ams.org/notices/200602/fea-mycielski.pdf
>


The entire paragraph is rather more interesting:
<quote>
The above ideas should not be construed as a
criticism of a branch of foundations called Reverse
Mathematics. In this branch one proves theorems
of the form T -> A, where T is some interesting theorem
and A is an axiom (of course A is not assumed
in the proof of T -> A). Some examples of such theorems
are the following. Tarski's theorem: (For all
infinite sets X there exists a bijection of X to X x X)
-> (Axiom of Choice). Or Sierpinski's theorem: (The
space R3 with a Cartesian coordinate system
X,Y,Z, is a union of three sets A,B, and C such that
every linear section of A parallel to X is finite,
every linear section of B parallel to Y is finite, and
every linear section of C parallel to Z is finite) ->
CH. There are many interesting theorems of Reverse
Mathematics, but some critics do not care for such
results. [Tarski told me the following story. He
tried to publish his theorem (stated above) in the
Comptes Rendus Acad. Sci. Paris but Fréchet and
Lebesgue refused to present it. Fréchet wrote that
an implication between two well known propositions
is not a new result. Lebesgue wrote that an
implication between two false propositions is of no
interest. And Tarski said that after this misadventure
he never tried to publish in the Comptes Rendus.]
</quote>




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