Chat GPT on Cantor's diagonal argument


12 April 2023

Question: Does Cantor's famous diagonal argument rely on actual infinity or potential infinity? In case of the latter, is it possible to prove Cantor's result with potential infinity?


Cantor's diagonal argument relies on the concept of actual infinity, specifically the notion that there is an infinite set of infinite sequences. Cantor's argument involves assuming that there exists a list of all possible infinite sequences of 0's and 1's, and then constructing a new sequence that differs from every sequence in the list by at least one digit. This leads to a contradiction, showing that the assumption that such a list exists is false.

The concept of potential infinity is the idea that a process can be continued indefinitely, without actually reaching infinity. While potential infinity is a useful concept in many areas of mathematics, it is not sufficient to prove Cantor's diagonal argument. In order to construct the diagonal sequence that leads to the contradiction, one needs to be able to consider an infinite set of infinite sequences, which requires the concept of actual infinity.

Therefore, it is not possible to prove Cantor's result with potential infinity alone.