Refuting the Anti-Cantor Cranks

[Antiphon]

It can be proven that the only numbers which have multiple decimal representation are numbers which end in 00000... or 9999...

It's a good thing there are only a handful of such numbers and not an infinite number of them.

If we take the diagonal in Cantor's prove and change all numbers not equal to 5 to 5, and furthermore change all 5's to 6, then we get a number not on the list and the problem will not show up. That is: a number like 0.55555555... has a unique decimal representation.

It's a good thing Cantor didn't use base 6 in his proof.

Furthermore, there are versions of Cantor's proof which do not work with decimal representation at all!

Some versions work in base 10 and some don't? I may have underestimated the flexibility of this proof!

 

[micromass]

It's a good thing there are only a handful of such numbers and not an infinite number of them.

There are an infinite number of them.

 

[Antiphon]

There are an infinite number of them.

I feel guilty- you took the bait!

 

[Parlyne]

There are an infinite number of them.

Infinite, but most certainly countable. After all, such numbers are, trivially, rational.