Refuting the Anti-Cantor Cranks

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Discussion Overview

The discussion centers around the validity of Cantor's diagonalization proof of the uncountability of the real numbers, particularly in the context of arguments made by individuals who reject this proof, often referred to as "anti-Cantor cranks." Participants explore the challenges faced when attempting to engage with these individuals and the nature of their arguments.

Discussion Character

  • Debate/contested
  • Exploratory

Main Points Raised

  • One participant describes a typical exchange with anti-Cantor cranks, highlighting their insistence that a proposed number constructed via diagonalization is not well-defined.
  • Another participant argues that the construction of a real number differing from each number in a list is valid regardless of whether the list is complete or not.
  • Some participants assert that the contradiction arises from the assumption of a complete list of real numbers, rather than from the proposed definition itself.
  • There is a suggestion that anti-Cantor cranks may lack the mathematical background to engage with more complex concepts, leading to their focus on simpler, more accessible mathematical statements.
  • One participant expresses frustration at the inability to convince anti-Cantor cranks, suggesting that some individuals may be resistant to change their views regardless of the arguments presented.

Areas of Agreement / Disagreement

Participants generally agree on the challenges posed by anti-Cantor cranks and the nature of their arguments, but there is no consensus on how to effectively engage or convince them of the validity of Cantor's proof.

Contextual Notes

The discussion reflects a range of assumptions about the nature of mathematical understanding and the motivations behind the rejection of established mathematical proofs. There are unresolved tensions regarding the definitions and implications of Cantor's diagonalization proof.

  • #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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