On 2/15/2025 7:56 AM, WM wrote:
On 14.02.2025 20:48, Jim Burns wrote:
On 2/14/2025 8:02 AM, WM wrote:
On 13.02.2025 20:47, Jim Burns wrote:
>
>
swaps such that, after all swaps,
Bob is not anywhere he has swapped to.
>
Then your logic allows
lossless exchanges with losses.
Not acceptable.
ST+F does not discuss infinite sets.
F denies any infinite set W exists.
¬∃W ∃w∈W ∃g: W→Wᐠʷ: x≠y⇒g(x)≠g(y)
ST+F discusses all hereditarily finite sets.
⎛ ST claims {} and adjunction '&' exist
⎜ If hereditarily finite sets x,y,z exist, then
⎜ there is a provable finite sequence of claims
⎜ that their adjunction to {} exists
⎜ {}&x&y&z = {x}&y&z = {x,y}&z = {x,y,z}
⎝ thus the set of x,y,z exists.
ST+F discusses all finite von Neumann ordinals.
They are hereditarily finite sets.
Their successors k+1 := k&k
are hereditarily finite sets.
Ordered pairs ⟨x,y⟩ of hereditarily finite sets
are hereditarily finite sets.
⟨x,y⟩ = {{x},{x,y}} = {}&({}&x)&({}&x&y)
By 'swap', I mean
an ordered pair ⟨k,k+1⟩ of
a finite ordinal and its successor.
ST+F discusses all swaps,
but not the infinite set of all swaps.
Order the swaps.
⟨j,j+1⟩ before ⟨k,k+1⟩ ⇔ j ⊆ k
According to ST+F,
there is no place k+1 with a swap.in
(Bob was in k, Bob is in k+1)
but not a later swap.out
(Bob is in k+1, Bob will be in k+2)
If Bob is somewhere with a swap.in,
the later swap.out hasn't happened.
It isn't after all these swaps.
Contrapositively,
if it is after all these swaps,
then Bob isn't anywhere with a swap.in.
swaps such that, after all swaps,
Bob is not anywhere he has swapped to.
>
Then your logic allows
lossless exchanges with losses.
Not acceptable.
If you accept ST+F, then
not.accepting the consequences of ST+F
is not an option.
If you don't accept ST+F,
what is it you don't accept?
Isn't there an empty set?
Do X and y exist such that X∪{y} doesn't exist?
----
The set of rows and the set of columns
are both the size of ⋃{F} and each other.
>
Yes, that is potentially infinite.
>
The set of rows, the set of columns, and ⋃{F}
do not change.
>
Then it would need to contain ℕ,
as an infinite FISON.
Infinite FISONs
(finite initial segments of naturals)
are not acceptable.
You (WM) are mistaken about 'infinite'.
finiteᵂᴹ infiniteᵂᴹ matheologicalᵂᴹ
↕ ↕ ↕
finite finite infinite
The set of necessary FISONs
Re: The set of necessary FISONs