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

On 10/22/2024 8:49 PM, Ross Finlayson wrote:

On 10/21/2024 11:09 AM, Ross Finlayson wrote:

On 10/21/2024 09:21 AM, Jim Burns wrote:

On 10/20/2024 8:20 PM, Ross Finlayson wrote:

Anyways you still haven't picked "anti and only".

>
I vaguely recall that
you (RF) made some incorrect claims about
Cantor's  argument from anti.diagonals,
and you asked for my participation in some way.
Could you refresh my memory? TIA.

Then I suggested that I would put anti-diagonal
in one fist, only-diagonal in the other, then
hide them behind my back and perhaps exchange
them, then that you get to pick.
>
You get to pick, was the idea, then I laughed
and said that I had put them together, so,
you get both or none.
>
"You" here meaning anybody, ...,
because it's a mathematical statement
so is the same for anyone.

Below or something like below
is what I mean by
Cantor's argument from anti.diagonals.
Put d and anything else behind your back.
Swap swap swap, pull them out.
I pick both.
One of those picked is in [0,1]ᴿ\f(ℕ)
f(ℕ) ≠ [0,1]ᴿ
If you think I'm wrong, say why.
You're welcome.
What was the point of that, Ross?
⎛ ℕ and [0,1]ᴿ have different cardinalities.
⎜
⎜⎛ Assume f: ℕ → ℝ is onto [0,1]ᴿ
⎜⎜ Assume ∀ᴿx ∈ [0,1]: ∃ᴺn: f(n) = x
⎜⎜
⎜⎜⎛ x@n means 'decimal digit n of positive real x'
⎜⎜⎜ x@n = ⌊(x⋅10ⁿ⁻¹-⌊x⋅10ⁿ⁻¹⌋)⋅10⌋
⎜⎜⎜
⎜⎜⎜ For real numbers x, y
⎜⎜⎜ if y@n = (x@n+5) mod 10
⎜⎜⎝ then |y-x| > 10⁻ⁿ and y ≠ x
⎜⎜
⎜⎜ Consider d ∈ [0,1]ᴿ such that
⎜⎜ ∀ᴺn: d@n = (f(n)@n+5) mod 10
⎜⎜
⎜⎜ ∀ᴺn: f(n) ≠ d
⎜⎜
⎜⎜ ∃ᴿx ∈ [0,1]: ∀ᴺn: f(n) ≠ x
⎜⎜ Proof: Let x = d
⎜⎜
⎜⎝ ¬∀ᴿx ∈ [0,1]: ∃ᴺn: f(n) = x
⎜
⎜ Therefore,
⎜ f: ℕ → ℝ is not onto [0,1]ᴿ
⎜ f: ℕ → ℝ does not biject ℕ and [0,1]ᴿ
⎜ No f: ℕ → ℝ bijects ℕ and [0,1]ᴿ
⎜
⎝ ℕ and [0,1]ᴿ do not have the same cardinality.