FOM: Response to adverse comment on the Arche project
William Tait
wwtx at midway.uchicago.edu
Thu Mar 15 11:41:31 EST 2001
Neil Tennant (3/14/01) wrote
> (An axiom of infinity usually states only that a countable
>infinity exists, and it is left to the powerset axiom to "blow it up" to
>get even higher infinities, via Cantor's Theorem.)
I have just an historical (and non-critical) remark: Cantor
introduced the transfinite numbers and the alephs in his *Grundlagen*
in 1883 (actually in 1882 in a letter to Dedekind). The step here
from one power (of M) to the next (of Omega(M)) is obtained by the
principle:
If X is a subset of Omega(M) of power <= M,then S(X) \in Omega(M)
where S(M) is the least number < every element of X. (The ordering is
defined inductively in the obvious way.)
It was in 1890/1 that he showed that a hierarchy of powers could also
be obtained by iterating the operation of passing from a set M to the
set of 2-valued functions on M, So historically, it was the operation
from M to Omega(M), not to powerset(M), that first generated
uncountable infinities.
>Hume's Principle immediately implies---very ambitiously---that for every
>property (or open sentence) F(x) there exists *the number of Fs*, denoted
>(say) by #xF(x). The axiom of infinity is hardly being *disguised*
>here---rather, it is being shouted from the rooftops, for all to hear.
Another historical remark, which *is* critical: Hume did not believe
that there are infinite sets, nor did anyone else who formulated the
criterion of equipollence for `having the same cardinal number' prior
to Cantor. Bolzano, who did believe that there are infinite sets,
rejected that criterion because of the `paradox' of infinite sets
equipollent to proper subsets. It was CANTOR who took that bull by
the horns and understood that there was no paradox. Notice that Frege
accepted the criterion for arbitrary,including INFINITE, sets. Only
after defining the notion of cardinal number in general did he define
the subclass of finite cardinals. So Frege owed his definition, not
to Hume or any of the other earlier people he mentioned, but to
Cantor. Frege did not like to pay his debts; but I think that it is
about time that we started paying them for him. A good way to start
would be dropping the expression ``Hume's Principle''.
With apologies to those (many) who have heard my complaint (many
times expressed) before, and with best wishes to all,
Bill Tait
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