On 10/24/2024 10:29 AM, WM wrote:
On 24.10.2024 16:05, Jim Burns wrote:
The all.the.doubles _claim_
follows the
well.ordered.successored.non.0.predecessored
claim not.first.falsely,
so it can't be false.
>
It is false.
⎛ For finite sequence A
⎜ A = ⟨a₁,…,aₙ⟩
⎜⎛ either A holds first.A a₁ and last.A aₙ
⎜⎝ or else A = {}
⎜ and,
⎜ for each sub.sequence B ⊂ A
⎜ B = ⟨aₕ,aᵢ,aⱼ,…,aₓ⟩ = ⟨b₁,b₂,b₃,…,bₘ⟩
⎜⎛ either B holds first.B b₁ and last.B bₘ
⎝⎝ or else B = {}
It is the more usual thing to be concerned with
the finiteness of sequences of numbers.
Here, though, we are concerned with
the finiteness of sequences of 𝗰𝗹𝗮𝗶𝗺𝘀.
Let 𝗔 = ⟨𝗮₁,…,𝗮ₙ⟩ be a finite sequence of 𝗰𝗹𝗮𝗶𝗺𝘀.
For each sub.sequence 𝗕 = ⟨𝗯₁,…,𝗯ₘ⟩ ⊂ 𝗔
⎛ either 𝗕 holds first.𝗕 𝗯₁ and last.𝗕 𝗯ₘ
⎝ or 𝗕 = {}
Consider the sub.sequence 𝗕 of false 𝗰𝗹𝗮𝗶𝗺𝘀.
𝗕 is a sub.sequence of finite sequence 𝗔
Either 𝗕 begins and ends, or 𝗕 is empty.
Consider finite sequence 𝗔′ of 𝗰𝗹𝗮𝗶𝗺𝘀 such that
each 𝗰𝗹𝗮𝗶𝗺 is not.first.false.
Consider its sub.sequence 𝗕′ of false 𝗰𝗹𝗮𝗶𝗺𝘀:
Either 𝗕′ begins and ends, or 𝗕′ is empty.
If 𝗕′ begins, it begins at the first.false 𝗰𝗹𝗮𝗶𝗺.
However,
there is no first.false 𝗰𝗹𝗮𝗶𝗺 in 𝗔′
since they are all not.first.false.
There is no first.false 𝗰𝗹𝗮𝗶𝗺 in 𝗔′ or in 𝗕′
The sub.sequence 𝗕′ of false 𝗰𝗹𝗮𝗶𝗺𝘀 doesn't begin.
The sub.sequence 𝗕′ of false 𝗰𝗹𝗮𝗶𝗺𝘀 is empty.
The original sequence 𝗔′ holds no false 𝗰𝗹𝗮𝗶𝗺𝘀.
The original sequence 𝗔′ holds only true 𝗰𝗹𝗮𝗶𝗺𝘀.
A finite sequence of 𝗰𝗹𝗮𝗶𝗺𝘀 such that
each 𝗰𝗹𝗮𝗶𝗺 is not.first.false
holds only true 𝗰𝗹𝗮𝗶𝗺𝘀.
The all.the.doubles _claim_
follows the
well.ordered.successored.non.0.predecessored
claim not.first.falsely,
so it can't be false.
>
It is false.
A finite sequence of 𝗰𝗹𝗮𝗶𝗺𝘀 such that
each 𝗰𝗹𝗮𝗶𝗺 is not.first.false
holds only true 𝗰𝗹𝗮𝗶𝗺𝘀.
The set, when existing completely,
covers an interval, namely (0, ω).
When its density is halved
while the number of elements is constant,
then its extension is doubled.
No.
⟨0,1,...,n-1,n,n+1,...,n+n-1,n+n⟩ is finite.
…