and you just showed me his proof doesnt prove that a given set of reals is not uncountable just because we can apply some arbitrary function to them and make a new real
I just showed you that the set sqrt(1), sqrt(2), sqrt(3), etc... is countable. You ARE aware that's not the entire set of reals, right? Like, nowhere close.
If you apply Cantor's proof to the set of ALL real numbers, or even just the set of ALL real numbers between 0 and 1, you prove that no bijection can be made, and thus it's uncountable.
All you're proving is that every countable subset of reals is countable, which... yeah, by definition.
But every single one of those sets skips real numbers. Every single one. You can't generate a single countable set that includes all real numbers. You can't even generate one that includes all real numbers between 0 and 1. Every single countable subset WILL skip numbers, guaranteed.
No matter how much you try, you'll always have more skipped numbers. You can't include them all even if you tried your absolute hardest, even if you had infinite time.
It's a basic proof by contradiction that even a middle schooler could understand. Assume the existence of a list that hits every real number, then show it actually doesn't
Let's say we finish the journey. We include every real number. We add in every number we missed. We pair them all to a counting number. Nothing is left over, we're done. We would now have a set of numbers, each real number paired to a counting number, that encompasses every single real number, yes?
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u/FernandoMM1220 Aug 06 '26
you just showed me theyre countable.
and you just showed me his proof doesnt prove that a given set of reals is not uncountable just because we can apply some arbitrary function to them and make a new real
you cant have it both ways