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

On 10/23/2024 08:05 AM, Jim Burns wrote:

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.
>
>

Ah, the main thing I want you to notice,
is, that the usual definition of "function",
in the usual descriptive milieu of functions
according to ZF set theory, is: a sub-set
of the Cartesian product of left-hand-side
and right-hand-side, or "Cartesian functions".
That is to say, the Cartesian product (usually
"X" or "x") of LHS X RHS, is a set, of tuples
(lhs, rhs), and any given subset of LHS x RHS,
has all these tuples (lhs, rhs), and so this
is a "Cartesian function", the _usual_ definition.
I.e., that is to say, it's _always_ been the case
that this is the definition of the usual descriptive
set theory's descriptive milieu of "functions", as
with regards to logic's usual terms of "terms,
propositions|predicates, relations".
Then: the "anti-diagonal" being about thusly
"Cartesian functions", where it results all
the transitive so results about pair-wise
interchange and composing functions with
the compositions of functions F1, F2, F3
as like F1 o F2 o F3 for the composition
of functions usually "o" like "f o g" for
f(g(x) or g(f(x)) respectively as a convention
may be, that these are "Cartesian functions"
courtesy the existence the "Cartesian product".
So, then this "only diagonal", has that this
function EF the n/d n->d d-> oo is NOT a
Cartesian function: it simply does NOT
have the same definition of function as being
a Cartesian function, so all that results
from Cantor-Schroeder-Bernstein about the
transitivity of functions, that's built
upon Cartesian functions: does not hold.
This way then the "anti-diagonal" has
"there are no _Cartesian_ bijections
between set and powerset", and that all
our triangles are trilateral polygons,
then the "only-diagonal" has that
"the function EF is a bijection and
not a _Cartesian_ bijection, between
naturals and unit interval [0,1]".
So, what otherwise you'd see apply,
Cantor-Schroeder-Bernstein and after
the naive composition of Cartesian functions,
does not apply, so that thusly your mathematical
conscience can be assuaged and your mathematical
wondrance can be impressed, that the most
direct mapping between discrete domain and
continuous range is this totally simple
continuum limit of n/d for natural integers
as only d is not finite and furthermore
is constant monotone strictly increasing
with a bounded range in [a,b], an infinite domain.