Proof - There Are More Real Numbers Than Natural Numbers

We present Cantors sleek and simple diagonal argument proof (originally 1874; refined to this version in 1891) that the infinite set of Real Numbers (all the rational and irrational numbers together) has a strictly greater cardinality (number of elements) than the infinite set of Natural Numbers (1, 2, 3, ).

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[rentsboy91]

Great explanation man, great explanation

[nevershoutsamitang]

so helpful, thanks !

[aepassion]

Where's the previous proof???????????

[whatstheoutcomeof]

@kaostwenty2 but if universe is infinite then you can match 0,9999... until the numbers behind decimal fulfilling the universe, with 1. lol

[whatstheoutcomeof]

@kaostwenty2 you can match 0,99999...infinte with 1. :p

[kaostwenty2]

I still like to say that (without being arrogant) it is obvious that in the interval between 0 to 1 there an infinite set of numbers that you would spend the life of many universes to count them with the natural numbers! And yet you will not move from that interval between 0 to 1. But again... are you sure that you are counting a trascendental number? Because I'm telling you..once you start to match 1 with 0,000000 (infinite universes lives)000 and yet 1 hasn't come!! :) u're stuck there!

[kaostwenty2]

@whatstheoutcomeof in conclusion: If you want to match any number with any other number you can surely do it because there are infinite both of them (1&0,1 2&0,2 3&0,3 etc etc) but if you talk about a trascendental number... well you are in a paradox because you cannot match a nuber that is continuosly running!! The moment that you cut it for the match it is a common number like any other number therefore you obviously can match it.

PS: end of story? :)

[kaostwenty2]

@whatstheoutcomeof ok then, you can match 1 with 0,9999 because you have cut the rest of the decimals and now you have this number which is 0,9999 but still I don't understand how cantor has proved that you cannot match all the remaining trascendentals numbers!!? Can't you see the paradox? If you match them (like we have just done with 1 and 0,9999) then 0,9999 it is not a trascendental number because the real trascendental number runs forever therefore my friend..not match it is possibile.

[whatstheoutcomeof]

@kaostwenty2 if you cut 0,999999...infinite, it will be less than 1.

[automaticSOM]

You could also use the term "bijection" for a one-to-one correspondence. Brilliant video.