Re: The set of necessary FISONs

On 1/29/2025 3:26 AM, WM wrote:

On 28.01.2025 17:29, Jim Burns wrote:

On 1/28/2025 10:29 AM, WM wrote:

On 28.01.2025 14:10, Jim Burns wrote:

A set such that a larger FISON does NOT exist
is sufficiently large in order for Bob
to disappear purely from swaps within the set.

>
He disappears from visibility.

>
There are no swaps into a room,
except for rooms with a later swap.out.

>
No.

Yes.
You are thinking of some other rooms
with other swaps.
What I mean by growable¹ and shrinkable¹:
⎛ For a growable¹ set, there are
⎜ fuller¹ (by one) sets which are larger.
⎜ For a shrinkable¹ set, there are
⎝ emptier¹ (by one) sets which are smaller.
For the rooms I refer to,
each room has
an ordinal both growable¹ and shrinkable¹.
and
each ordinal both growable¹ and shrinkable¹
has a room.
For the swaps I refer to,
each pair of
fuller¹(emptier¹), grown¹(shrunk¹) ordinals
has a swap between their rooms,
and
only pairs of
fuller¹(emptier¹), grown¹(shrunk¹) ordinals
have a swap between their respective rooms.
For these rooms and these swaps,
there are no swaps into a room,
except for rooms with a later swap.out.
⎛ Consider four sets A ≠ A∪{a},  B ≠ B∪{b}
⎜ For more comfortable reading,
⎜ I will write Aᵃ = A∪{a},  Bᵇ = B∪{b}
⎜ Aᵃ is fuller¹ than A.
⎜ B is emptier¹ than Bᵇ.
⎜
⎜⎛ A is smaller than B  iff  Aᵃ is smaller than Bᵇ
⎜⎝ |A| < |B|  ⇔  |Aᵃ| < |Bᵇ|
⎜
⎜ Consider B = Aᵃ
⎜ |A| < |Aᵃ|  ⇔  |Aᵃ| < |(Aᵃ)ᵇ|
⎜
⎜ There is no negative cardinality.
⎜ ¬(|A| > |Aᵃ|)  ∧  ¬(|Aᵃ| > |(Aᵃ)ᵇ|)
⎜
⎜ There are only sets Aᵃ
⎜ such that |A| < |Aᵃ| < |(Aᵃ)ᵇ|
⎜ or
⎜ such that |A| = |Aᵃ| = |(Aᵃ)ᵇ|
⎜
⎜ There are only sets Aᵃ
⎜ such that Aᵃ is both shrinkable¹ and growable¹
⎜ or
⎝ such that Aᵃ is both unshrinkable¹ and ungrowable¹.
----
There are no swaps into a room,
except for rooms with a later swap.out.
Consider room n with swap.in n-1⇄n
Room n-1 has swap.out n-1⇄n
Only room n-1 both growable¹ and shrinkable¹ has a swap.out.
n-1 grows¹ to fuller¹ n
n shrinks¹ to emptier¹ n-1
n is shrinkable¹, thus
n is necessarily both shrinkable¹ and growable¹.
n has both n-1⇄n and n⇄n+1,
both a swap.in and a swap.out.

Darkᵂᴹ or visibleᵂᴹ,
there are no swaps into a room,
except for rooms with a later swap.out.

>
One exception exists: ω-1.

ω is defined to be the least.upper.bound of
ordinals both growable¹ and shrinkable¹.
(0 has honorary status as shrinkable¹.)
⎛ For ordinals k and ξ,
⎜ if
⎜ k is both growable¹ and shrinkable¹, and
⎜ ξ is both ungrowable¹ and unshrinkable¹,
⎜ then
⎝ k < ξ
Each ordinal both ungrowable¹ and unshrinkable¹
is an upper bound of
all ordinals both growable¹ and shrinkable¹.
ω-1, if ω-1 existed,
cannot be both ungrowable¹ and unshrinkable¹.
If it were,
it'd be an upper bound before the first such,
contradiction.
ω-1, if ω-1 existed,
would need to be both growable¹ and shrinkable¹.
However,
in that case,
ω would need to be both growable¹ and shrinkable¹,
ω+1 would need to be both growable¹ and shrinkable¹,
and
ω would not be an upper bound of
all ordinals both growable¹ and shrinkable¹,
which ω is defined to be the first such,
contradiction.

Darkᵂᴹ or visibleᵂᴹ,
there are no swaps into a room,
except for rooms with a later swap.out.

>
One exception exists: ω-1.

Darkᵂᴹ or visibleᵂᴹ,
ω-1 does not exist.