Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 12/22/2024 6:32 AM, WM wrote:

On 22.12.2024 11:16, Jim Burns wrote:

What does 'finite' mean?

Inserting 'ᵂᴹ' marks your use of private meanings.

A set is finiteᵂᴹ if it contains
a visibleᵂᴹ naturalᵂᴹ number of elements.
All naturalᵂᴹ numbers are finiteᵂᴹ but
the realm of darkᵂᴹ numbers
appears like infinityᵂᴹ.

Set A is finiteⁿᵒᵗᐧᵂᴹ if A∪{x} is larger, for A∌x
Set Y is infiniteⁿᵒᵗᐧᵂᴹ if Y∪{x} isn't larger, for Y∌x
ℕⁿᵒᵗᐧᵂᴹ is the set of finiteⁿᵒᵗᐧᵂᴹ.cardinals.
  which means
ℕⁿᵒᵗᐧᵂᴹ is the set of cardinals #A of
  sets A smaller.than A∪{x} for A∌x
For each finiteⁿᵒᵗᐧᵂᴹ set A,
its cardinality #A is in ℕⁿᵒᵗᐧᵂᴹ
For each set Y without #Y in ℕⁿᵒᵗᐧᵂᴹ
Y is infiniteⁿᵒᵗᐧᵂᴹ, and
Y∪{x} isn't larger than Y, for Y∌x
Eⁿᵒᵗᐧᵂᴹ(n+2) is
the set of all finiteⁿᵒᵗᐧᵂᴹ.cardinals > n+2
Eⁿᵒᵗᐧᵂᴹ(n+1)  =  Eⁿᵒᵗᐧᵂᴹ(n+2)∪{n+2}
Eⁿᵒᵗᐧᵂᴹ(n+2)∪{n+2} isn't larger.than Eⁿᵒᵗᐧᵂᴹ(n+2)
because
⎛ Eⁿᵒᵗᐧᵂᴹ(n+2) is without #Eⁿᵒᵗᐧᵂᴹ(n+2) in ℕⁿᵒᵗᐧᵂᴹ
⎜ because,
⎜⎛ for each finiteⁿᵒᵗᐧᵂᴹ.cardinal j in ℕⁿᵒᵗᐧᵂᴹ
⎜⎜ #Eⁿᵒᵗᐧᵂᴹ(n+2) isn't j
⎜⎜ because
⎜⎜⎛ Eⁿᵒᵗᐧᵂᴹ(n+2) contains a larger.than.j subset
⎝⎝⎝ Eⁿᵒᵗᐧᵂᴹ(n+2)\Eⁿᵒᵗᐧᵂᴹ(n+j+3)
#Eⁿᵒᵗᐧᵂᴹ(n+2) isn't any of
the finiteⁿᵒᵗᐧᵂᴹ.cardinals in ℕⁿᵒᵗᐧᵂᴹ
Eⁿᵒᵗᐧᵂᴹ(n+2) isn't any of the sets
smaller.than fuller.by.one sets.
Eⁿᵒᵗᐧᵂᴹ(n+2) is emptier.by.one than Eⁿᵒᵗᐧᵂᴹ(n+1), but
Eⁿᵒᵗᐧᵂᴹ(n+2) isn't smaller.by.one than Eⁿᵒᵗᐧᵂᴹ(n+1).

The function Eᵂᴹ(n) decreases
from infinityᵂᴹ to zero
by single steps of height 1
like the function NUFᵂᴹ(x) increases
by single steps of height 1. Set theorists must accept
magic steps of infiniteᵂᴹ size or
refuse to describe these transitions at all.

Alternatively,
set theorists could continue to talk about
what set theorists talk about, instead of
what you (WM) talk about.