Refuting the Anti-Cantor Cranks

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The discussion centers around the challenges of debating anti-Cantor cranks who reject the validity of Cantor's diagonalization proof, which demonstrates the uncountability of real numbers. The cranks often argue that the constructed number from the diagonalization process is not well-defined, leading to circular reasoning about the completeness of real number lists. Despite attempts to clarify that the contradiction arises from their assumption of a complete list, many remain unconvinced, viewing the definition as contradictory. Participants express frustration over the futility of these debates, suggesting that true understanding may be unattainable for some individuals. Ultimately, engaging with these cranks is seen as more of an exercise in amusement than a productive exchange of ideas.
  • #91
micromass said:
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!
 
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  • #92
Antiphon said:
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.
 
  • #93
micromass said:
There are an infinite number of them.

I feel guilty- you took the bait!
 
  • #94
micromass said:
There are an infinite number of them.

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

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