I was in a discussion today with a philosopher about the merit of the technique of "proof by contradiction." He mentioned the Law of Excluded Middle, wherein we (typically as mathematicians) assume that we either have P or Not P. That is, if we show not "Not P", then we must have P.
Further along in the conversation, he mentions Kolmogorov's tendencies toward Intuitionist Logic (where the Law of Excluded middle does not hold; i.e. we cannot infer p from not not p). I located the source material of Kolmogorov, "On the Principle of Excluded Middle," wherein he states:
...it is illegitimate to use the principle of excluded middle in the domain of transfinite arguments.
Only the finitary conclusions of mathematics can have significance in applications. But the transfinite arguments are often used to provide a foundation for finitary conclusions.
We shall prove that all the finitary conclusions obtained by means of a transfinite use of the principle of excluded middle are correct and can be proved even without its help.
My question: What does Kolmogorov mean when he differentiates finitary conclusions from transfinite arguments? Namely, what is an example of a finite conclusion, and what is an example of a corresponding transfinite argument?
(Source material cited in Wikipedia: http://en.wikipedia.org/wiki/Andrey_Kolmogorov#Bibliography)