Re: How many different unit fractions are lessorequal than all unit fractions? (constructive)

On 09/08/2024 12:13 PM, Jim Burns wrote:

On 9/8/2024 11:39 AM, Ross Finlayson wrote:

On 09/07/2024 07:29 PM, Ross Finlayson wrote:

On 09/07/2024 05:29 PM, Jim Burns wrote:

On 9/7/2024 3:13 PM, Ross Finlayson wrote:

>

Aristotle has both _prior_ and _posterior_ analytics.

>

So, when you give him
a perfectly good syllogism with which he disagrees,
he has either of prior or posterior to deconstruct
either posterior or prior,

>
Wikipedia seems to say that
syllogisms are prior, and
use of syllogisms is posterior.
>
They don't seem to be 'either.or', but 'both.and'.

>

So, being constructive, constructive criticism,
when I look at the outcome of
otherwise a proof by contradiction to be rejected,
that a "strongly constructivist" view requires that
it's immaterial the order of the introduction of
any stipulations,
where
in the usual syllogism's proof by contradiction,
whatever non-logical term is introduced last
sort of wins,
when
if the terms are discovered and evaluated
in an arbitrary order,
it's arbitrary which decides and which is decided.

>
Formally
(which is what I think 'by prior analysis' means),
a proof by contradiction only shows that
_one of_ our assumptions is false.
>
However,
of those assumptions,
some are better supported than others.
This is when posterior analysis enters the ring.
>
Consider Boolos's ST
⎛ ∃{}
⎜ ∃z = x∪{y}
⎝ extensionality
>
Those are assumptions.
They are very un.challenging assumptions,
but that is why we select them as starters.
I am currently engaging in posterior analysis, I think.
>
We can follow those with a list of _definitions_
which describe natural and rational numbers
in the language of ST.
⎛ 0 := {}
⎜ k+1 := k∪{k}
⎜ ℕ(k) :⇔
⎜  (k = 0 ∨ k ∋ 0)  ∧
⎜  (∀j ∈ k+1: (∃i ∈ j: i+1=j) ⇔ j≠0)
⎝ ...
>
Definitions aren't assumptions.
They are more like public service announcements,
informing "the public"
⎛ let us say: the phone.booth.sized crowd
⎝ which reads my posts
what it means when I write 'foo'.
>
I'm not sure whether definitions stand
with prior or posterior analysis.
>
There is a very strong assumption that
I mean that which I say I mean,
an assumption which does not extend to
an assumption that
that which I mean is correct, or even makes sense.
>
Then there are assumptions made
for the purpose of 'proving' contradictions,
and thereby being themselves disproved.
⎛ √2 ∈ ℚ
⎝ ...
>
Assume √2 ∈ ℚ
Prove a contradiction.
Thereby prove that _one of_
⎛ ∃{}
⎜ ∃z = x∪{y}
⎜ extensionality
⎝ √2 ∈ ℚ
is false.
>
But the first three assumptions, ST, are
very un.challenging assumptions -- by design.
Is ∃{} false, or is √2 ∈ ℚ false?
Is ∃z = x∪{y} false, or is √2 ∈ ℚ false?
Is extensionality false, or is √2 ∈ ℚ false?
>
Posterior analysis(?) suggests √2 ∈ ℚ is false.
It is not as bullet.proof as prior analysis(?),
but it is damn good.
>

Then,
that Russell's retro-thesis is
simply not a fact, logically, [...]

>
What do you mean by "Russell's retro-thesis"?
>
Yes, I know about
Russell's "set" of all non.self.membered sets.
It would be helpful if you (RF) simply stated,
without embroidery, your own stance
with regard to Russell's "set".
>
>

The, "Fixx"
https://www.youtube.com/watch?v=JHYIGy1dyd8
https://www.youtube.com/watch?v=JOiZP8FS5Ww
The, "Cure"
https://www.youtube.com/watch?v=WKg_QOy6tRk&list=OLAK5uy_kVsEOBk05Q7DzR8AJImpSEvG59S2OxuFc&index=1
https://www.youtube.com/watch?v=7VgIwlFdVqw
It's already been thoroughly elaborated and as attached
to formalistic symbolry, that "Russell's thesis, of an antinomy"
is that the set of the finite sets that don't contain themselves,
exactly like the ordinals are mostly simply modeled to be,
does and doesn't contain itself, blowing wide open that there
is a class of propositions external "tertium non datur".
Then, "Russell's retro-thesis", is "forget I, Russell I, said that,
and now that Frege's out of the way, think me and Whitehead made
1 + 1 = 2, after this brief 0 = 1 as it were", asking you simply ignore
his stated antinomy, and furthermore your own conscience
as with regards to these matters.
"What, is it?"
"Superunknown"
Anyways that such a notion of "an infinite" really "is"
extra-ordinary, we've discovered, for example as Mirimanoff
describes as what went into making ZF set theory what it is,
has of course that Goedel also makes "at least two incompleteness
theorems", as with regards to the illative, univalency, and
"a theory with the strength of ZFC and at least two large cardinals",
which of course, are neither cardinals nor sets. Then Cohen
sort of neatly for "ubiquitous ordinals" has that neither CH
nor Not CH, in ZFC.
You know one can build reals from integers and vice-versa,
both ways, sort of establishing that 1 + 1 = 2 and all.
"Ken", is a name and a word in English which means "know"
and in Japanese which means "sword", "Ken: 2 + 2 = 4".
... and Russell doesn't necessarily have much to say
after taking himself back.
The idea of "Russell's retro-thesis" has already of course
been defined and exactly and as so briefly as above in
these longer, extended discussions, thorough, in a sense.