On 12/3/2024 8:02 AM, WM wrote:
On 03.12.2024 01:32, Jim Burns wrote:
On 12/2/2024 9:28 AM, WM wrote:
A quantifier shift tells you (WM) what you (WM) _expect_
but a quantifier shift is untrustworthy.
>
Here is no quantifier shift but an identity:
Yes, it is a true claim.
That true claim does not conflict with
⎛ an empty intersection does not require
⎝ an empty end.segment.
E(1), E(2), E(3), ...
and
E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
are identical for every n and in the limit
because
E(1)∩E(2)∩...∩E(n) = E(n).
>
No.
For the set of finite cardinals,
EVEN IF NO END.SEGMENT IS EMPTY,
the intersection of all end segments is empty.
>
You cannot read or understand the above.
⟦0,ℵ₀⦆ is the set of finite cardinals,
usually written ℕ
⎛ For each finite.cardinal k
⎜⎛ for each finite.cardinal j
⎜⎝ j is fewer than the finite cardinals from k up
⎜
⎜ ∀k ∈ ⟦0,ℵ₀⦆
⎜ ∀j ∈ ⟦0,ℵ₀⦆: j < |⟦k,k+j⟧| ≤ |⟦k,ℵ₀⦆|
⎜ |⟦k,ℵ₀⦆| ∉ ⟦0,ℵ₀⦆
⎜
⎜ Each end.segment E(k) = ⟦k,ℵ₀⦆
⎝ doesn't have any finite.cardinality j
⎛ For each non.empty set S of finite.cardinals,
⎜ among which are the non.empty end.segments ⟦k,ℵ₀⦆
⎜ that set S holds a minimum finite.cardinal.
⎜ min.⟦k,ℵ₀⦆ is the index of ⟦k,ℵ₀⦆
⎜
⎜ Each finite.cardinal k is
⎜ index min.⟦k,ℵ₀⦆ of end.segment ⟦k,ℵ₀⦆
⎜
⎜ There is a bijection between the finite.cardinals and
⎜ the end.segments of the finite cardinals.
⎝ k ⇉ ⟦k,ℵ₀⦆ ⇉ k
⎛ For each finite.cardinal k
⎜ there are fewer finite.cardinals up.to k
⎜ than there are all finite.cardinals
⎜ ==
⎜ ∀k ∈ ⟦0,ℵ₀⦆:
⎜ |⟦0,k⟧| < |⟦0,k+1⟧| ≤ |⟦0,ℵ₀⦆|
⎜
⎜ Consider the bijection
⎜ k ⇉ ⟦k,ℵ₀⦆ ⇉ k
⎜
⎜ For each finite.cardinal k
⎜ there are fewer end.segment.minima up.to k
⎜ (minima of those end.segments which hold k)
⎜ than there are all end.segment.minima
⎜ ==
⎜ For each finite.cardinal k
⎜ there are fewer end.segments which hold k
⎜ than there are all end.segments.
⎜ ==
⎜ For each finite.cardinal k
⎜ there are end.segments not.holding k
⎜ ==
⎜ For each finite.cardinal k
⎜ k is not.in the intersection of all end.segments.
⎜
⎝ The intersection of all end.segments is empty.
For the finite.cardinals,
each end.segment is infinite,
the intersection of all is empty.
E(1)∩E(2)∩...∩E(n) = E(n).
Sequences which are identical in every term
have identical limits.
An empty intersection does not require
an empty end.segment.
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