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Cantor and the axioms (4)
Posted:
Feb 23, 2014 6:12 AM


Only the recognition of inconsistent sets led Cantor to the consideration of axioms of transfinite arithmetic.
The fact of the "consistency" of finite multitudes is a simple unprovable truth. It is "the axiom of arithmetic (in the old meaning of the word)". And similarly is the "consistency" of the multitudes to which I attach the alephs "the axiom of the extended, the transfinite arithmetic". [Cantor to Dedekind, 28 Aug. 1899]
That the "countable" multitudes are completed sets seems to me to be an "axiomatically certain" theorem which the whole theory of functions rests upon. [Cantor to Hilbert, 10 Oct. 1898]
It is our common conviction that the "arithmetic continuum" in this sense is a "set". The question is whether this truth is a provable one or whether it is an axiom. Currently I tend more towards the latter alternative but I am open to be convinced by you of the other. [Cantor to Hilbert, 9 May 1899]
Regards, WM



