Re: Replacement of Cardinality

Am 25.08.2024 um 23:57 schrieb Jim Burns:

On 8/25/2024 3:41 PM, WM wrote:

For a distance to exist in every case
mathematics is sufficicient:
∀n ∈ ℕ: 1/n - 1/(n+1) > 0 .

 For each n ∈ ℕ:
1/(n+1) is a counter.example disproving the claim
that 1/n is the smallest unit.fraction:
1/n - 1/(n+1) > 0

In dsm I just told him (Mückenheim):
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Unlike you, however, some people understand that your formula IMPLIES that there is no smallest unit fraction. 🙂
Proof: From ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 it follows: ∀n ∈ ℕ: 1/n > 1/(n+1). Because Ax(x e SB <-> En e IN: x = 1/n) it follows (easily): As e SB: Es' e SB: s' < s. qed
For every unit fraction there is a smaller one (hence there can't be a smallest unit fraction).
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