Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.mathDate : 15. Nov 2024, 21:06:12
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <84d9831f-d23a-4937-8333-4029c6c1f4a9@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
User-Agent : Mozilla Thunderbird
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.