On 8/22/2024 7:17 AM, WM wrote:
Le 21/08/2024 à 20:20, Jim Burns a écrit :
On 8/21/2024 6:43 AM, WM wrote:
Le 20/08/2024 à 22:05, Jim Burns a écrit :
No min.⅟ℕᵈᵉᶠ exists.
>
The reason is potential infinity.
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Is potentialᵂᴹ.infinity mathematicsᵂᴹ?
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All of classical mathematics is
embedded into potential infinity.
What you didn't say is that
potentialᵂᴹ.infinity is embedded into mathematicsᵂᴹ.
Should I draw the implicature from your response that
it isn't?
Is mathematicsᵂᴹ within potentialᵂᴹ.infinity, but
potentialᵂᴹ.infinity not within mathematicsᵂᴹ?
You (WM) have a personal vocabulary,
mathematicsᵂᴹ and potentialᵂᴹ and actualᵂᴹ.
We use the same words you do,
mathematicsⁿᵒᵗᐧᵂᴹ and potentialⁿᵒᵗᐧᵂᴹ and actualⁿᵒᵗᐧᵂᴹ,
but not in the way you do.
Work has been done during
the last few thousand years
to extend mathematicsⁿᵒᵗᐧᵂᴹ over infinityⁿᵒᵗᐧᵂᴹ.
The distinction potentialⁿᵒᵗᐧᵂᴹ::actualⁿᵒᵗᐧᵂᴹ
has turned out to be less than useful
Judging from your edicts on this topic,
the answer looks to be both 'yes' and 'no'.
>
I am interested in actual infinity.
It seems that your potentialᵂᴹ.infinity is
our infinityⁿᵒᵗᐧᵂᴹ.
It seems that your actualᵂᴹ.infinity isn't
potentialᵂᴹ.infinity but it has a subset of
potentialᵂᴹ.infinity.
There is no superset of
a potentiallyᵂᴹ.infinite (infiniteⁿᵒᵗᐧᵂᴹ) set which isn't
a potentiallyᵂᴹ.infinite (infiniteⁿᵒᵗᐧᵂᴹ) superset.
There is no actuallyᵂᴹ.infinite set.
Augmenting with darkᵂᴹ.numbers cannot change
the potentiallyᵂᴹ.infinite into the actuallyᵂᴹ.infinite
because
any non.2.ended subset which a potentiallyᵂᴹ.infinite has
is also a non.2.ended subset of any superset.
It follows that ⅟ℕᵈᵉᶠ has properties that
the unit.fractionsᵈᵉᶠ > x > 0
don't have.
There is no set ℕᵈᵉᶠ because of lacking completeness.
⎛ Each non.empty S ⊆ ℕᵈᵉᶠ holds min.S in S.
⎜ Each k ∈ ℕᵈᵉᶠ has k+1 and k-1 next in ℕᵈᵉᶠ, except for
⎝ min.ℕᵈᵉᶠ = 0 which has 0+1 but not 0-1 in ℕᵈᵉᶠ
Whatever class ℕᵈᵉᶠ belongs in, the above is true of ℕᵈᵉᶠ.
It's what we mean by ℕᵈᵉᶠ.
A finite sequence of not.first.false claims
cannot have a first false claim and thus
cannot have a false claim.
It's part of what we mean by finite.
We can assemble finite sequences of not.first.false claims
which include the above claims about ℕᵈᵉᶠ.
Each of its claims about ℕᵈᵉᶠ is true.
If that's incompleteᵂᴹ,
it'll do quite well until completeᵂᴹ comes along.
Sets of set theory are complete.
A setⁿᵒᵗᐧᵂᴹ does not change.
A setⁿᵒᵗᐧᵂᴹ which has been changed is no longer
the setⁿᵒᵗᐧᵂᴹ it was.
The setⁿᵒᵗᐧᵂᴹ it was has not changed.
Re: Replacement of Cardinality
Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity)