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




Re: Matheology § 236
Posted:
Apr 4, 2013 7:23 AM


On Apr 4, 12:26 am, WM <mueck...@rz.fhaugsburg.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) 206213]http://www.ams.org/notices/200602/feamycielski.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>



