FOM: Response to adverse comment on the Arche project

William Tait wwtx at
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 

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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