# [FOM] Suarez on Proper Subsets and Actual Infinities

John Baldwin jbaldwin at uic.edu
Sat Mar 11 19:23:17 EST 2006

```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.

There is more at
http://evans-experientialism.freewebspace.com/grosseteste.htm

and  the piece is in Anne Freemantle's the medieval philosophers.
Does anyone else have a copy bought for 50 cents in the 1950's?

John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics,
and Computer Science  M/C 249
jbaldwin at uic.edu
312-413-2149
Room 327 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607

Assistant to the director
Jan Nekola: 312-413-3750
```