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

On 11/15/2024 12:06 PM, Jim Burns wrote:

On 11/15/2024 1:05 PM, Ross Finlayson wrote:

On 11/15/2024 09:55 AM, Jim Burns wrote:

On 11/15/2024 5:10 AM, WM wrote:

On 14.11.2024 19:31, Jim Burns wrote:

>

Setting aside for a moment
what you _think_ Cantor's bijection is,
what part of _that_
is impossible to represent geometrically?

>
It is impossible to cover the matrix
XOOO...
XOOO...
XOOO...
XOOO...
...
by shuffling, shifting, reordering the X,
because they are not distinguishable.

>
⟨k,1⟩ ↦ ⟨i,j⟩ ↤ ⟨k,1⟩
>
⎛ i = k-⌈(2⋅k+¼)¹ᐟ²-1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-3/2⌉/2
⎜ j = ⌈(2⋅k+¼)¹ᐟ²+1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-1/2⌉/2-1-k
⎝ k = (i+j-1)⋅(i+j-2)/2+i
>
Each ⟨k,1⟩ sends X to ⟨i,j⟩
Each ⟨i,j⟩ receives X from ⟨k,1⟩
>
According to geometry.
Which I predict makes geometry wrong[WM], too.

>
Non-standard models of integers exist.

>
Non.standard models of integers are not
standard models of integers.
>
Consider
a finite sequence of claims which begins with
a description of the standard model ℕ of integers,
a description such as
⎛ i+1≠0 ∧ j≠k⇒j+1≠k+1
⎜ 0 ∈ ℕ ∧ '+1':ℕ→ℕ
⎜ 0 ∈ S ∧ '+1':S→S  ⇒  ℕ ⊆ S
⎝ ...
>
There are
models for which that is incorrect.
>
However,
suppose we are discussing only
models for which that is correct.
>
In that case,
those are true claims,  and,
if we augment that finite sequence with only
claims which are true.or.not.first.false,
each of those augmenting claims is true
-- true about the standard model.
>
If this claim sequence,
which starts with a standard.model.description,
is read as making claims about NON.standard models,
we can't give a similar guarantee.
>
Yes,
not.first.false claims are still
not.first.false claims,
but, if they follow a false claim
(about, let's say, a non.standard model),
they can be true or false and still not.first.false.
>

Non-standard models of integers exist.

>
Yes, and,
when we discuss non.standard models,
we can assemble claim.sequences which
start with a description of a non.standard model.
And, when we do that,
augmenting true.or.not.false claims
will be true about the described non.standard models.
>
However,
non.standard models of integers are not
standard models of integers.
>
>

Ah, yet according to Mirimanoff,
there do not exist standard models of integers,
that Russell has fooled you with a contradictory statement,
and there are only fragments, if unbounded,
and extensions, sublime.
This also makes a wide variety of number-theoretic
conjectures, about numbers great and small:
independent "standard" number theory.
So, your putative model is either a mere incomplete fragment,
or, it's an extra-ordinary whole.