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Le 30/07/2024 à 03:18, Richard Damon a écrit :And where is that in {1, 2, 3, ... w} ?On 7/29/24 9:11 AM, WM wrote:ω/2But what number became ω when doubled?
Which is what number?>ω/2 and larger.
No, that is w double, what number in the first set became w?
>>>>
Every natural number when doubled is a Natural Number.
No.
Why not? WHich ones don't?
But you combined two different sets, so why can't there be a gap?>Because otherwise there was a gap below ω.>>Note, ω-1 doesn't exist in the base transfinite numbers, just as -1 doesn't exist in the Natural Numbers, you can't go below the first element.>
If all natural numbers exist, then ω-1 exists.
Why?
Right, so we can say that ∀n ∈ ℕ: 1/n > 1/(n+1), so that for every unit fraction 1/n, there exists another unit fraction smaller than itself. The only way to have a smallest unit fraction is to have a largest natural number, at which point they aren't even "potentially infinite" as you have established a finite limit to them.∀n ∈ ℕ: 1/n - 1/(n+1) > 0. Note the u niversal quantifier.That is unavoidable. You believe in the magical appearance of infinitely many unit fractions. That breaks logic and mathematics.>
Nope,
Regards, WM
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