Cantor's Discovery

[SLD]

The set of real numbers cannot be put into one-to-one correspondence with the integers, yet there is a one-to-one correspondence between the points of a line and the points of R^n. Cantor proved it but stated, “I see it but I do not believe it.”

In a nutshell, what’s the proof?

 

[Bomb#20]

The set of real numbers cannot be put into one-to-one correspondence with the integers, yet there is a one-to-one correspondence between the points of a line and the points of R^n. Cantor proved it but stated, “I see it but I do not believe it.”

In a nutshell, what’s the proof?

Of which part? That there's a one-to-one correspondence between the points of a line and the points of R^n is pretty easy to see. A = 0.1234567891011... <=> X,Y,Z = (0.14711..., 0.2580..., 0.3691...). That's more or less how the proof goes for n=3; it works the same way for other n except you take every nth digit instead of every 3rd digit. There are some gotchas you have to work around. You have to turn all the terminating reals like 0.25 into the equivalent nonterminating 0.249999...; and the way I showed it only works for positive numbers, so you have to finesse it a little to get from one sign-bit to n sign-bits, but that's not hard.

That the set of real numbers cannot be put into one-to-one correspondence with the integers is trickier; that's where Cantor proved he was a genius. I'll leave that half as an exercise for the reader.

 

[SLD]

The set of real numbers cannot be put into one-to-one correspondence with the integers, yet there is a one-to-one correspondence between the points of a line and the points of R^n. Cantor proved it but stated, “I see it but I do not believe it.”

In a nutshell, what’s the proof?

Of which part? That there's a one-to-one correspondence between the points of a line and the points of R^n is pretty easy to see. A = 0.1234567891011... <=> X,Y,Z = (0.14711..., 0.2580..., 0.3691...). That's more or less how the proof goes for n=3; it works the same way for other n except you take every nth digit instead of every 3rd digit. There are some gotchas you have to work around. You have to turn all the terminating reals like 0.25 into the equivalent nonterminating 0.249999...; and the way I showed it only works for positive numbers, so you have to finesse it a little to get from one sign-bit to n sign-bits, but that's not hard.

That the set of real numbers cannot be put into one-to-one correspondence with the integers is trickier; that's where Cantor proved he was a genius. I'll leave that half as an exercise for the reader.

That does seem rather simple. But it took Cantor three years to figure it out. Perhaps the details are more involved, but I see what you did.

The first theorem seems more intuitive at first glance, but maybe not.

 

[fromderinside]

Gnurs.

 

[lpetrich]

 Cantor's diagonal argument presents the proof.

It shows that R[0,1] is uncountable, and from there, that R is uncountable. It proceeds by contradiction. Suppose that R[0,1] is countable. Then one can take every member of that set with 1, 2, 3, ..., One can line them up in sequence. Now take the first trailing digit of the first number, the second trailing digit of the second number, and so forth. Change each digit, and assemble a member of R[0,1] from them. It is not in that list, thus making such a list is not possible.