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Dear philosophers,
I really appreciate your website, which I just discovered!
I'd like to make one comment regarding the recent questions about infinite sets on March 7 and March 14. In your responses (Allen Stairs and Richard Heck on March 14), you write that you do not know of any professional mathematicians who deny the existence of infinite sets. However, such mathematicians do indeed exist (although marginally). They are sometimes referred to as "ultrafinitists". One well-known living proponent of this view is Princeton mathematician Edward Nelson, also see http://en.wikipedia.org/wiki/Edward_Nelson and http://en.wikipedia.org/wiki/Ultrafinitism
Specifically, one argument an ultrafinitist might use is that formal proofs are finite. Thus, although we might use the concept of infinite sets in our reasoning, there is no need to assume that infinite sets actually exist, because any mathematical statement could be preceded by the phrase "There is a finite proof of the statement that ..."
I hope this makes for an interesting addition to your answers.
Best regards,
Sam
PhD student, mathematical logic
RU Nijmegen, the Netherlands

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