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

Revision as of 07:48, 7 March 2024

Revision as of 07:52, 7 March 2024

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:::::: 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 there are 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 thehow 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)

[[Category:Wikipedia mathematical arguments]]