- #36

JeffJo

- 143

- 25

Here is an outline of Cantor's Diagonal Argument (CDA), as published by Cantor. I'll apply it to an undefined set that I will call

**(consistent with the notation in Wikipedia). One important property of the elements**

*T***of**

*t***, is that each is a surjection from the natural numbers**

*T***to some set of characters**

*N***.**

*C*- Cantor used the set of two characters
*C***=**{'m','w'} - Wikipedia used the set of two characters
*C***=**{'0','1'} - When taught to teenagers, it is usually the decimal representation of real numbers, so it uses
*C***=**{'0','1','2','3','4','5','6','7','8','9'}. With some additional steps.

**of**

*S***, if**

*T***is countable, then**

*S***is a proper subset of**

*S***" (see note). Proof:**

*T*- Let
be any subset of*S*, proper or improper, that is countable.*T* - So a bijection
**b(n)**:->*N*exists.*S* - Construct a new function
**s0:**->*N*such that that*C***s0(n)**is not equal to**b(n)(n)**. - Prove that
**s0**is in, but not in*T*.*S* - QED.

From Part A, it follows immediately that

**is uncountable. Otherwise we would have the contradiction, that an object***T***s0**would be both an element of**, but also not an element of***T***.***T*Part B, as I would word it:

The contrapositive of the proposition is Part A is "For any subset

**of***S***, if***T***is not a proper subset of***S***, then***T***is not countable.***S*

This is applicable here, because the new proposition Warp is trying to prove doesn't fit step A4. The elements of his set **always degenerate to a repeating sequence (i.e., are rational numbers). To apply CDA, he has to prove that his**

*T***s0**also degenerates like that, so it is in

**, and also is not in the set**

*T*

*S.*Note: What Cantor actually wrote is closer to "If

**s1**,

**s2**, …,

**sn**, … is any simply infinite series of elements of

**, then there always exists an element**

*T***s0**of

**, which cannot be connected with any element**

*T***sn**."

+++++

The problem Warp has with CDA,

*as it is usually taught*, is that it is indeed logically invalid. I'm not saying the conclusion is wrong, just that the one-part CDA that is usually taught is invalid.

To apply proof by contradiction, one must use everything in the assumption to establish the contradiction. If I assume that the moon in made of individually wrapped packets of green and bleu cheeses, in a ratio that equals the square root of two, I can derive the contradiction that an even number must be equal to an odd number. But I only use the assumption about the ratio, not the kinds of cheeses that comprise the moon. So all I prove is that the square root of two is irrational.

Similarly, as CDA is usually taught, only the assumption "there is a list" is used to derive the alleged contradiction. Never the assumption that this list is complete. What Part A proves,

*directly*, is negation of the unused part. Which is why proof-by-contraposition is logically more valid than proof-by-contradiction. This is actually an age-old criticism of proof-by-contradition.