Talk:Cantor's diagonal argument/Arguments: Difference between revisions

Revision as of 07:52, 7 March 2024

Revision as of 07:55, 7 March 2024

Line 1,328:

:::::: Exactly, that is where the contradiction is.

:::::: Cantor's argument is just a plain proof by contradiction, no more special than the proof of irrationality of the square root of 2. In the proof of irrationality of <math>\sqrt{2}</math>, you start out by making the totally wrong assumption that assume that <math>\sqrt{2}</math> can be written as a ratio of integers <math>\frac{a}{b}</math> reduced to lowest terms, which is a totally wrong assumption. Then over the course of the proof you show that the assumption hadis to be wrongself-refuting, by producing proof that <math>\frac{a}{b}</math> was not really reduced to begin with. Even though something totally wrong (the claim that <math>\sqrt{2}=\frac{a}{b}</math> for integers <math>a,b</math>) is assumed at the beginning, this does not make the proof wrong. Instead it is what makes the proof work at all, since it works by assumingasking "what if <math>\sqrt{2}</math> iswere rational,?" and then showing that that assumption wasis nonsensical, therefore showing that the only option that makes sense is for <math>\sqrt{2}</math> to be irrational.

:::::: Similarly, when proving that there are infinitely many primes by contradiction, you start the proof by making the totally wrong assumption that some finite set <math>\{p_1,\ldots,p_n\}</math> is an exhaustive list of the primes. Then the proof proceeds by showing that this assumption is absurd, by producing another prime that has to be outside of the set <math>\{p_1,\ldots,p_n\}</math>. Again, this proof starts by making the assumption that"what if there arewere finitely many primes?", then showing that this assumption is nonsense, therefore showing the only correct option is that there are infinitely many primes.

:::::: However, with Cantor's diagonal argument, often people seem to reject the argument based on the claim that the assumption is nonsense. But this is the entire point of how proof by contradiction works, you start with the assumption that there is a complete list of the real numbers indexed by natural numbers, where when written out in decimal there are equally many columns as rows, and then show that this assumption is nonsense. Then the conclusion is that the only option that makes sense is for there to be '''no''' complete list of the real numbers with as many columns as rows, and this is what Cantor proved. [[User:C7XWiki|C7XWiki]] ([[User talk:C7XWiki|talk]]) 07:48, 7 March 2024 (UTC)