On 06/29/2024 06:24 AM, Jim Burns wrote:
On 6/28/2024 9:50 AM, WM wrote:
Le 28/06/2024 à 06:31, Jim Burns a écrit :
On 6/27/2024 2:37 PM, WM wrote:
Le 27/06/2024 à 18:30, Jim Burns a écrit :
>
Cardinalities which can grow by 1 are finite.
Cardinalities which cannot grow by 1 are infinite.
>
Cardinalities are useless.
Sets can grow by 1 element.
>
Sets can have elements inserted,
which makes them different sets.
The effect on size of inserting 1
is not the same for all sets.
>
It is changing the infinite set
but not its cardinality.
>
A set with a cardinal.growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
cannot change without its cardinality changing.
>
Not all cardinals are growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
A set with a cardinal.not.growable.by.1 (infiniteⁿᵒᵗᐧᵂᴹ)
can change without its cardinality changing.
>
⎛ ...not that setsⁿᵒᵗᐧᵂᴹ changeⁿᵒᵗᐧᵂᴹ
⎜ Adjunct X∪{Y} isn't X
⎜( Sets can grow by 1 element.
⎜ means
⎝( For each X,Y X∪{Y} exists
>
The cardinal ℵ₀ for
all cardinals.growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
is a cardinal.not.growable.by.1 (infiniteⁿᵒᵗᐧᵂᴹ)
>
| Assume otherwise.
| Assume ℵ₀ is growable.by.1
|
| If
| the cardinal ℵ₀ of
| all cardinals.growable.by.1
| is growable.by.1
| then
| the cardinal ℵ₀ᐡᴮᵒᵇ for
| all cardinals.growable.by.1
| and Bob too
| is growable.by.1 too
| and
| there are more.than.ℵ₀.many
| cardinals.growable.by.1
|
| However,
| there are ℵ₀.many
| cardinalities growable.by.1
| Contradiction.
>
Therefore,
the cardinal ℵ₀ for
all cardinals.growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
is a cardinal.not.growable.by.1 (infiniteⁿᵒᵗᐧᵂᴹ)
>
Not all cardinals are growable.by.1 (finiteⁿᵒᵗᐧᵂᴹ)
A set with a cardinal.not.growable.by.1 (infiniteⁿᵒᵗᐧᵂᴹ)
can change without its cardinality changing.
>
Therefore cardinality is useless for my proof.
>
if we stop _saying_
( cardinal of {k……} ̊≤ cardinal of {k⁺¹……}
k ↦ k⁺¹ doesn't stop _being_
1.to.1 from {k……} to {k⁺¹……}
>
For any difficulty which cardinality presents,
not.saying 'cardinal' not.resolves the difficulty.
>
>
Yet, 'density', is its own thing, ....
Then you either got two 'not.resolves' or none,
was the idea, ....
(This was from the other day when I remarked on
the essential lack of discernability, or discernibility,
of two propositions which confound each other,
thus requiring a deconstructive account, where
the joke went 'pick one' and the joke was
'ha, I put them together, it's both or none'.)
Somehow I went all through school and calculus and
econometrics and statistics and with real analysis,
and transfinite cardinals were never introduced, ....
Then later I was interested in what I learned was
foundations most particularly about the continuous
and discrete, and I learned about the Galilean notion
of that countably infinite sets are equivalent, while
then there's duBois-Reymond with the diagonal argument,
that nested intervals is the usual old infinite-divisibility,
while there's then Cantor and the m-w proof and in the
language of set theory one of the consequences of
comprehension or the powerset result, then as with
regards to that making an inequality in cardinals,
that the real numbers were a rather usual construction
axiomatically with axiomatizing least-upper-bound
and axiomatizing measure 1.0, yet, I also learned
about the 'density' asymptotically or Schnirelmann,
about Katz' OUTPACING and similar notions, then as
well that the notion of continuous domain was largely
un-defined yet as with regards to Jordan measure,
Lebesgue measure, and Shannon/Nyquist supersampling,
that there's at least three.
The "extra-ordinary" then or Mirimanoff, who is to
be thanked for the introduction of both the "ordinary"
and "extra-ordinary" in what resulted ordinary regular
set theory, Zermelo-Fraenkel set theory most usually
in a model theory, though with of course "naive set-builder"
having been ubiquitous since forever, really gets into
the logical foundations, then for why dually-self-infraconsistency
is a thing and that the more one studies foundations
classically and modernly, that it all results sort of one thing,
a platonist's ideal strong mathematical platonism, of
a universe of mathematical objects like a "Hilbert's
Infinite Living, Working Museum of Mathematics, Now Open".
Does the number of nines increase?
Re: Does the number of nines increase?