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On 11/9/2024 6:45 AM, WM wrote:
It cannot do so because the reality of the rationals is much larger than the reality of the naturals.Everybody who believes that the intervalsOur sets do not change.
I(n) = [n - 1/10, n + 1/10]
could grow in length or number
to cover the whole real axis
is a fool or worse.
The set
{[n-⅒,n+⅒]: n∈ℕ⁺}
with the midpoints at
⟨ 1, 2, 3, 4, 5, ... ⟩
does not _change_ to the set
{[iₙ/jₙ-⅒,iₙ/jₙ+⅒]: n∈ℕ⁺}
with the midpoints at
⟨ 1/1, 1/2, 2/1, 1/3, 2/2, ... ⟩
----But it will never complete an infinite set of claims. It will forever remain in the status nascendi. Therefore irrelevant for actual or completed infinity.
Either
all instances of a 𝗰𝗹𝗮𝗶𝗺 about a set
are _only_ true or _only_ false
or
a set changes.
In the first case, with the not.changing sets,
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 which
has only true.or.not.first.false 𝗰𝗹𝗮𝗶𝗺𝘀
has only true 𝗰𝗹𝗮𝗶𝗺𝘀.
In the second case, with the changing sets,Certainly not. The intervals can neither grow in size nor in multitude.
who knows?
Perhaps something else could be done,
but not that.
Infinite sets can correspond toBut they cannot become such sets.
other infinite sets which,
without much thought about infinity,
would seem to be a different "size".
Consider again the two sets of midpointsBut they cannot be completely transformed into each other. That is prohibited by geometry. It is possible for every finite initial segment of the above sequence, but not possible to replace all the given intervals to cover all rational midpoints.
⟨ 1, 2, 3, 4, 5, ... ⟩ and
⟨ 1/1, 1/2, 2/1, 1/3, 2/2, ... ⟩
They both _are_
And their points correspond
by i/j ↦ n = (i+j-1)(i+j-2)/2+i
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