[FOM] Suarez on Proper Subsets and Actual Infinities

[William Tait]

On Mar 11, 2006, at 6:23 PM, John Baldwin wrote:

> Buckner remarked on Suarez distinguishing between containment and
> 1-1 correspondence as notions of `smaller than'. similar by earlier is
> the work of Robert Grossteste in 13th century Oxford.
>
> Here is a quote (translated)
>
>   It is possible, however, that an infinite sum of number is  
> related to an
> infinite sum in every proportion, numerical and non-numerical. And  
> some
> infinites are larger than other infinites, and some are smaller.  
> Thus the
> sum of all numbers both even and odd is infinite. It is at the same  
> time
> greater than the sum of all the even numbers although this is likewise
> infinite, for it exceeds it by the sum of all the odd numbers. The  
> sum,
> too, of all numbers starting with one and continuing by doubling each
> successive number is infinite, and similarly the sum of all the halves
> corresponding to the doubles is infinite. The sum of these halves  
> must be
> half of the sum of their doubles. In the same way the sum of all  
> numbers
> starting with one and multiplying by three successively is three  
> times the
> sum of all the thirds corresponding to these triples. It is  
> likewise clear
> in regard to all kinds of numerical proportion that there can be a
> proportion of finite to infinite according to each of them.

This came up earlier in FOM (I think also in connection with Suarez).  
That there are 'unequal infinities' was known much earlier than 13th  
century. For example, Proclus (early fifth century), in his  
commentary on Book 1 of Euclid's Elements, notes in his discussion of  
Definition 17 that the diameter produces two semicircles and so "the  
number of semicircles is twice infinity." Proclus, like most of his  
successors, felt that this apparent paradox was solved by rejecting  
the actual infinite---i.e. (roughly) infinite sets. The first person,  
as far as I know, to both hold that there are actual infinities and  
confront the paradox was Henry of Harclay, who was chancellor of  
Oxford in 1312. The 'paradox' is ultimately that a proper subset of a  
set M can be equal to M. This seems to contradict Euclid's Common  
Notion 5 that the whole is greater than the part. Harclay said that  
there are two notions of equality and less-than involved: one  
concerns sets under the  subset relation and the other concerns sets  
under (what we would call) under the relation of cardinality. For the  
first, Common Notion 5 is true; but, in the case of infinite sets, it  
is false in the sense of cardinal number. This distinction was taken  
up by Rimini later in the fourteenth century (who, however, rejected  
the actual infinite), but it seems eventually to have been lost until  
reaffirmed by Cantor in 1878.

One of the difficulties with taking the step that Harclay and,  
eventually, Cantor took had to do with the lack of a clear  
distinction between (what we would call) a structure and its  
underlying set. Thus, even Bolzano, who made this distinction (sort  
of), rejected the definition of 'having the same cardinal number' in  
terms of one-to-one correspondence because it made the line segment  
equal to its double---clearly confusing the geometric object with its  
underlying set of points. Even Cantor was not immune from the  
confusion: When he proved that R^n has the same power as R, he wrote  
Dedekind (in June 1887) worrying that the result destroyed the notion  
of dimension. (Dedekind wrote back to point out that the one-to-one  
correspondence need not be continuous!)

Bill Tait