FOM: Cantor and Kronecker
William Tait
wtait at ix.netcom.com
Mon Mar 15 22:44:42 EST 1999
I want to add three remarks to my reply to Alaxander Zenkin.
1) The statement that Cantor's 1874 proof that a one-to-one enumeration of
reals does not contain evry real in a given interval is constructive does
not of course mean that Kronecker would have accepted it. It is
constructive in the sense of Bishop's conception of constructive
mathematics. But Kronecker held to a strict finitist conception, which
excludes even the conception of an arbitrary one-to-one enumeration of
reals. (See the passage that Dedekind cites.)
2. None of the passages, either of Dedekind, Kronecker or Cantor so far
cited by me concern Cantor's theory of transfinite numbers and so none are
really concerned directly with Kronecker's opposition to Cantor's set
theory. (K's objections apply even to analysis as this was developed in the
19th century by e.g. Weierstrass.)
3. Cantor however does address Kronecker's finitism, fairly persuasively,
in his first paper on general set theory, the 1883 ``Ueber unendliche
lineare Punktmannigfaltigkeiten'' Nr 5. See section 4.
Bill Tait
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