Re: Incompleteness of Cantor's enumeration of the rational numbers

On 11/27/24 12:13 PM, WM wrote:

On 27.11.2024 13:32, Richard Damon wrote:

On 11/27/24 5:12 AM, WM wrote:

 

Of course. |{1, 2, 3, 4, ...}| = |ℕ| and |{2, 3, 4, ...}| = |ℕ| - 1 is consistent.

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So you think, but that is because you brain has been exploded by the contradiction.
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We can get to your second set two ways, and the set itself can't know which.
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We could have built the set by the operation of removing 1 like your math implies, or we can get to it by the operation of increasing each element by its successor, which must have the same number of elements,

 Yes, the same number of elements, but not the same number of natural numbers.

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Of course they are, if n is a Natural Number, Sn (S being the Successor operator) is also one.

Hint: Decreasing every element in the real interval (0, 1] by one point yields the real interval [0, 1). The set of points remains the same, the set of positive points decreases by 1.

But what number changes "natural number" status?

 Replacing every element of the set {0, 1, 2, 3, ...} by its successor yields {1, 2, 3, ..., ω}. The number of ordinals remains the same, the number of finite ordinals decreases.

Nope, because omega is NOT the successor for any natural number, the successor of EVERY Natural Number is a Natural Number.
Defintions you know.
It it the successor for the SET of natural numbers.

 Regards, WM

So, you are just showing your ignorance of the definitions things you are talking about, because you are using a logic that has gone inconsistent and blown up your brain.