Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 11/7/24 4:13 AM, WM wrote:

On 07.11.2024 03:27, Richard Damon wrote:

On 11/6/24 10:15 AM, WM wrote:

On 06.11.2024 12:46, Richard Damon wrote:

On 11/5/24 11:45 AM, WM wrote:

On 05.11.2024 04:08, Richard Damon wrote:

On 11/4/24 12:11 PM, WM wrote:

On 04.11.2024 13:14, Richard Damon wrote:

On 11/4/24 6:07 AM, WM wrote:

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By induction you can prove the sum n(n+1)/2 for every initial segment 1+2+3+...+n. But not for all natural numbers.

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But all Natural Numbers can be defined.

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All defined numbers can be summed. Not all natural numbers can be summed.

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Why not?

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because most cannot be defined.

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But they ARE defined.

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Then sum all of them.

 

They will sum to Aleph_0

 That is not a sum. It is just another name for infinity.

Note, Addition on Natural Numbers is closed for FINITE sums (the sum of a finite number of numbers), not necessarily for infinite series of them.

 That is because infinitely many always contain dark numbers. All not dark numbers can be summed.
 Regards, WM
 

There are no dark numbers. You can't actually define what you mean by a "dark number" except in ways that admit that you logic can't create the actual infinite sets you want to talk about.
Sorry, logic based on the presumption of a lie is just broken.