On 08/08/2024 03:30 AM, FromTheRafters wrote:
on 8/8/2024, WM supposed :
Le 08/08/2024 à 00:17, Moebius a écrit :
Actually, his "thinking process" is simple:
"Since there is a gap (space) between
adjacent unit fractions and
all unit fractions are in the interval (0, 1],
there must be FINITELY MANY of them
(i.e. a first/smallest one)."
No, that is nonsense.
There are not finitely many unit fractions.
Then stop assuming that
there is a first and last element.
Of course, you can start with a first and last element,
then make infinitely-many in the middle.
0 ... ( ... infinitely-many ... ) ... infinity
On 8/8/2024 8:26 PM, Ross Finlayson wrote:
On 08/08/2024 03:30 AM, FromTheRafters wrote:
on 8/8/2024, WM supposed :
Le 08/08/2024 à 00:17, Moebius a écrit :
Actually, his "thinking process" is simple:
"Since there is a gap (space) between
adjacent unit fractions and
all unit fractions are in the interval (0, 1],
there must be FINITELY MANY of them
(i.e. a first/smallest one)."
No, that is nonsense.
There are not finitely many unit fractions
On 08/09/2024 03:25 AM, FromTheRafters wrote:
Ross Finlayson explained :
On 08/08/2024 03:30 AM, FromTheRafters wrote:
on 8/8/2024, WM supposed :
There are not finitely many unit fractions.
Then stop assuming that
there is a first and last element.
Of course, you can start with a first and last element,
then make infinitely-many in the middle.
0 ... ( ... infinitely-many ... ) ... infinity
Sometimes you are as bad as he is. :)
Or, where do you think you're counting, to?
Then, drawing the ends apart with infinite in the middle,
has a little extra work and book-keeping to begin
instead of a usual "next",
yet it well expresses any matters of the "bounded",
for example, in any matters of the "unbounded".
On 8/9/2024 3:17 PM, Ross Finlayson wrote:Well, except you are Chuck Norris.
Or, where do you think you're counting, to?
Not to infinity. Whatever one counts to is not infinity.
Am 10.08.2024 um 20:02 schrieb Jim Burns:
On 8/9/2024 3:17 PM, Ross Finlayson wrote:
Or, where do you think you're counting, to?
Not to infinity. Whatever one counts to is not infinity.
Well, except you are Chuck Norris.
Remember, Chuck Norris counted to infinity - twice!
On 8/10/2024 3:46 PM, Moebius wrote:
Am 10.08.2024 um 20:02 schrieb Jim Burns:
On 8/9/2024 3:17 PM, Ross Finlayson wrote:
Or, where do you think you're counting, to?
Not to infinity. Whatever one counts to is not infinity.
Well, except you are Chuck Norris.
Remember, Chuck Norris counted to infinity - twice!
I'm using ZFC-Norris
Am 10.08.2024 um 23:22 schrieb Jim Burns:
On 8/10/2024 3:46 PM, Moebius wrote:
Am 10.08.2024 um 20:02 schrieb Jim Burns:
On 8/9/2024 3:17 PM, Ross Finlayson wrote:
Or, where do you think you're counting, to?
Not to infinity. Whatever one counts to is not infinity.
Well, except you are Chuck Norris.
Remember, Chuck Norris counted to infinity - twice!
I'm using ZFC-Norris
I see. What a pitty!
Am 10.08.2024 um 23:24 schrieb Moebius:
Am 10.08.2024 um 23:22 schrieb Jim Burns:
On 8/10/2024 3:46 PM, Moebius wrote:
Am 10.08.2024 um 20:02 schrieb Jim Burns:
On 8/9/2024 3:17 PM, Ross Finlayson wrote:
Or, where do you think you're counting, to?
Not to infinity. Whatever one counts to is not infinity.
Well, except you are Chuck Norris.
Remember, Chuck Norris counted to infinity - twice!
I'm using ZFC-Norris
I see. What a pitty!
For Chuck Norris it's a simple (super) task. At t = 0 he counts 1, at t
= 1/2 he counts 2, at t = 1/4 he counts 3, etc. (ad infinitum). At t = 1
he counts omega! You see: Hence he counted to infinity! (After all,
omega is the smallest _infinite_ ordinal number!)
On 08/10/2024 11:02 AM, Jim Burns wrote:
On 8/9/2024 3:17 PM, Ross Finlayson wrote:
On 08/09/2024 03:25 AM, FromTheRafters wrote:
Ross Finlayson explained :
On 08/08/2024 03:30 AM, FromTheRafters wrote:
on 8/8/2024, WM supposed :
There are not finitely many unit fractions.
Then stop assuming that
there is a first and last element.
Of course, you can start with a first and last element,
then make infinitely-many in the middle.
0 ... ( ... infinitely-many ... ) ... infinity
Sometimes you are as bad as he is. :)
Or, where do you think you're counting, to?
Not to infinity.
Whatever one counts to is not infinity.
That one can count to it means it is not infinity.
Consider
0, 1, 2, 3, ..., ℬ-3, ℬ-2, ℬ-1, ℬ
There are two cases to consider.
1.
There is a split
{0,1,2,3,...} ᵉᵃᶜʰ<ᵉᵃᶜʰ {...,ℬ-3,ℬ-2,ℬ-1,ℬ}
without any α in {0,1,2,3,...}
with α+1 in {...,ℬ-3,ℬ-2,ℬ-1,ℬ}
and
you can't count
from {0,1,2,3,...} to {...,ℬ-3,ℬ-2,ℬ-1,ℬ}
2.
Not 1.
There is no split
{0,1,2,3,...} ᵉᵃᶜʰ<ᵉᵃᶜʰ {...,ℬ-3,ℬ-2,ℬ-1,ℬ}
without any α in {0,1,2,3,...}
with α+1 in {...,ℬ-3,ℬ-2,ℬ-1,ℬ}
and
ℬ is finite.
Finite.
Not "it will take until the stars die to express".
"Finite", in its purposeful indefiniteness,
encompasses "until the stars die"
and more, some of which make that look small.
Then, drawing the ends apart with infinite in the middle,
has a little extra work and book-keeping to begin
instead of a usual "next",
yet it well expresses any matters of the "bounded",
for example, in any matters of the "unbounded".
Non.{} set C is bounded by
non.0 ordinal k
which has last.before k-1 and
which each non.0 j < k has j-1
Set C must have least.upper.C.bound m
m-1 not.upper.C.bound
least.upper.C.bound m in C
max.C m
C is two.ended and
each non.{} subset of C, also bounded,
is two.ended.
Non.{} set C, bounded by
non.0 ordinal k
which has last.before k-1 and
which each non.0 j < k has j-1
is finite.
0, 1, 2, 3, ..., ℬ-3, ℬ-2, ℬ-1, ℬ
with the ends drawn apart and infinity in the middle
fails at having last.before somewhere,
otherwise, there isn't infinity in the middle.
So, do the rationals fill out?
I'm not talking about rationals.
Consider
0, 1, 2, 3, ..., ℬ-3, ℬ-2, ℬ-1, ℬ
Then, drawing the ends apart with infinite in the middle,
has a little extra work and book-keeping to begin
instead of a usual "next",
yet it well expresses any matters of the "bounded",
for example, in any matters of the "unbounded".
On 8/8/2024 5:26 PM, Ross Finlayson wrote:
0 ... ( ... infinitely-many ... ) ... infinity
Sure. Think of two points, and draw a line between them.
On 8/10/2024 3:43 PM, Moebius wrote:
Hint: Let's "consider" the real line:
|-----|-----|-----|-----|--..
0 1 2 3 4
[Now] omega is not a point on this line. :-P
"Out of scope", perhaps? Is that okay?
On 8/10/2024 3:43 PM, Moebius wrote:
Hint: Let's "consider" the real line:
...|-----|-----|-----|-----|--..
0 1 2 3 4
[Now] omega is not a point on this line. :-P
"Out of scope", perhaps? Is that okay?
On 8/8/2024 5:26 PM, Ross Finlayson wrote:
0 ... ( ... infinitely-many ... ) ... infinity
Sure. Think of two points, and draw a line between them.
On 8/10/2024 4:05 PM, Moebius wrote:
I guess, JB would say: "If we don't consider omega, we don't consider
omega." :-P
For some reason I am thinking of salad here...? ;^)
On 8/10/2024 6:19 PM, Moebius wrote:
Am 11.08.2024 um 02:55 schrieb Chris M. Thomasson:
On 8/10/2024 4:05 PM, Moebius wrote:
I guess, JB would say: "If we don't consider omega, we don't
consider omega." :-P
For some reason I am thinking of salad here...? ;^)
You won't agree? :-)
I can see a line, comprised of two n-ary points. They have a p0 and a
p1, yet there is an infinity between then. [etc. etc.]
Am 11.08.2024 um 00:47 schrieb Chris M. Thomasson:
On 8/10/2024 3:43 PM, Moebius wrote:
Hint: Let's "consider" the real line:
...|-----|-----|-----|-----|--..
0 1 2 3 4
[Now] omega is not a point on this line. :-P
"Out of scope", perhaps? Is that okay?
Somehow. :-P
I guess, JB would say:
"If we don't consider omega,
we don't consider omega."
:-P
On 8/11/2024 9:10 AM, Moebius wrote:
Am 11.08.2024 um 06:20 schrieb Chris M. Thomasson:
On 8/10/2024 6:19 PM, Moebius wrote:
Am 11.08.2024 um 02:55 schrieb Chris M. Thomasson:
On 8/10/2024 4:05 PM, Moebius wrote:
I guess, JB would say: "If we don't consider omega, we don't
consider omega." :-P
For some reason I am thinking of salad here...? ;^)
You won't agree? :-)
I can see a line, comprised of two n-ary points. They have a p0 and a
p1, yet there is an infinity between then. [etc. etc.]
And where do you see omega on that line? (Very ... VERY ... far away?)
https://www.youtube.com/watch?v=jvQ6dasK614
I guess its basically "out of scope" in a sense?
On 08/11/2024 09:10 AM, Moebius wrote:
[...]
How do you see omega
as the second constant after empty set
an inductive set in ZF?
It's definitely not "all" infinity.
How do you see omega
On 8/11/2024 2:10 PM, Ross Finlayson wrote:
How do you see omega
ω is [...] is for each inductive set,
the intersection of inductive subsets.
On 08/11/2024 02:38 PM, Jim Burns wrote:
[...]
Starting with a theory _without_
the constant introduced named omega,
i.e., finite sets,
given that there's axiomatized well-foundedness
when otherwise
simple comprehension would make the "omega" into
an extra-ordinary or non-well-founded or
inconsistent-multiplicity of a set,
starting _without_ omega,
the finite sets like ordinals, are, exactly
those sets that don't contain themselves.
Then, omega, as you've defined it,
contains itself,
again just quantifying over
the specification of what omega purports to be,
I'm curious, now that you have
a beginning and an end of
the finite, or 0 and omega in ZF,
On 08/11/2024 11:30 AM, Jim Burns wrote:
Do you think that would aggravate him as much as that
the Continuum Hypothesis being broken both ways
put him in the madhouse?
Do you think that would aggravate him as much as that
the Continuum Hypothesis being broken both ways
put him in the madhouse?
Cantor believed the continuum hypothesis to be true
and tried for many years to prove it, in vain.
Am 12.08.2024 um 18:39 schrieb Jim Burns:
Cantor believed the continuum hypothesis to be true
and tried for many years to prove it, in vain.
Yeah, though today we are ably to "understand" (to a certain extend, it seems) the (possibe) reasons fo t/his inability.
But the question is by no means settled.
See: https://en.wikipedia.org/wiki/Continuum_hypothesis#Arguments_for_and_against_the_continuum_hypothesis
On 8/12/2024 11:22 AM, Moebius wrote:
Am 12.08.2024 um 18:39 schrieb Jim Burns:
Cantor believed the continuum hypothesis to be true
and tried for many years to prove it, in vain.
Yeah, though today we are ably to "understand" (to a certain extend,
it seems) the (possibe) reasons fo t/his inability.
But the question is by no means settled.
See: https://en.wikipedia.org/wiki/
Continuum_hypothesis#Arguments_for_and_against_the_continuum_hypothesis
It will never be more settled than it is today. In ZF and it's variants
you can take it or leave it as you probably well know. And it isn't
going to be more settled than that.
Ah.
I've seen this one before.
Your tacit thesis is that
it is preferable to disagree with the Old Ones
even at the cost of being wrong.
Well, it's a choice.
On 08/11/2024 09:44 PM, Jim Burns wrote:
On 8/11/2024 7:39 PM, Ross Finlayson wrote:
Starting with a theory _without_
the constant introduced named omega,
i.e., finite sets,
For P(z),
use a description 𝕆ᶠⁱⁿ(z) of a finite ordinal,
and ω := {z:𝕆ᶠⁱⁿ(z)} exists
For example, use
𝕆ᶠⁱⁿ(z) ⇔
(z ∋ {} ∧ ∀y ∈ z+1: y≠{} ⇒ ∃x∈z: x+1=y)
∨ (z = {})
z+1 = z∪{z}
Then, omega, as you've defined it,
ω := {z:𝕆ᶠⁱⁿ(z)}
contains itself,
I'm curious, now that you have
a beginning and an end of
the finite, or 0 and omega in ZF,
ω is the least.upper.bound of the finites.
ω is not a finite.
ω is not the upper.end of the finites.
The upper.end of the finites doesn't exist.
Here though
it's beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.
Then you claim to have
an axiom of restriction of comprehension of the finites
unless Russell grants you
a dispensation of Russell's retro-thesis,
and say it's always so for others, too,
congratulations,
you claim to have invented a mathematics
where you = Russell.
Am 12.08.2024 um 20:37 schrieb Jeff Barnett:
On 8/12/2024 11:22 AM, Moebius wrote:
Am 12.08.2024 um 18:39 schrieb Jim Burns:
Cantor believed the continuum hypothesis to be true
and tried for many years to prove it, in vain.
Yeah, though today we are ably to "understand" (to a certain extend,
it seems) the (possibe) reasons fo t/his inability.
But the question is by no means settled.
See: https://en.wikipedia.org/wiki/
Continuum_hypothesis#Arguments_for_and_against_the_continuum_hypothesis
It will never be more settled than it is today. In ZF and it's
variants you can take it or leave it as you probably well know. And it
isn't going to be more settled than that.
So you didn't follow the link? What a pitty. (->ignorance)
Seriously Ross. What's the point of posting nonsense from
start to finish?
Am 13.08.2024 um 02:44 schrieb Python:
Seriously Ross. What's the point of posting nonsense from
start to finish?
Imho, he just should check his medication. It might help. (I'm serious.)
On 08/12/2024 04:06 PM, Jim Burns wrote:
On 8/12/2024 4:59 PM, Ross Finlayson wrote:
On 08/11/2024 09:44 PM, Jim Burns wrote:
On 8/11/2024 7:39 PM, Ross Finlayson wrote:
Starting with a theory _without_
the constant introduced named omega,
i.e., finite sets,
For P(z),
use a description 𝕆ᶠⁱⁿ(z) of a finite ordinal,
and ω := {z:𝕆ᶠⁱⁿ(z)} exists
For example, use
𝕆ᶠⁱⁿ(z) ⇔
(z ∋ {} ∧ ∀y ∈ z+1: y≠{} ⇒ ∃x∈z: x+1=y)
∨ (z = {})
z+1 = z∪{z}
Then, omega, as you've defined it,
ω := {z:𝕆ᶠⁱⁿ(z)}
contains itself,
I'm curious, now that you have
a beginning and an end of
the finite, or 0 and omega in ZF,
ω is the least.upper.bound of the finites.
ω is not a finite.
ω is not the upper.end of the finites.
The upper.end of the finites doesn't exist.
Here though
_Where_ though?
it's beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.
For ω as I've defined it, no upper.end exists.
for each k ∈ ω
𝕆ᶠⁱⁿ(k)
𝕆ᶠⁱⁿ(k+1)
k+1 ∈ ω
k is not the upper end of ω
for each k ∉ ω
k is not the upper end of ω
Then you claim to have
an axiom of restriction of comprehension of the finites
To review:
What I claim is
⎛ ∃{}
⎜ ∀x∀y∃z=x∪{y}
⎝ and extensionality
⎛ ∃∃xx={z:P(z)}: ∀y: y ∈ {z:P(z)} ⇔ P(y)
⎝ and extensionality
∃∃{z:P(z)} is unrestricted comprehension.
Unless we are no longer uninterested in what words mean.
unless Russell grants you
a dispensation of Russell's retro-thesis,
and say it's always so for others, too,
congratulations,
you claim to have invented a mathematics
where you = Russell.
Ah.
I've seen this one before.
Your tacit thesis is that
it is preferable to disagree with the Old Ones
even at the cost of being wrong.
Well, it's a choice.
Oh, I have the entire canon here along.
It's like yesterday, in this thread with the subject
of it talking about "infinite in the middle and always
with both ends", or, "here...", pointing out that some
modern philosophers with their Ph.D.s. resuscitate a
metaphysics that Compte and Boole and Russell and Carnap
made so nice for Marx and nihilism and extistentialism
and the sort of post-modern that begets itself.
Now, I confiscate logical positivism from Compte and
brush off Boole for De Morgan and point out Russell
and for example Whitehead suffer their own arguments
and Carnap was quite a pleasant fellow and I like Quine
yet I'm not a nominalist fictionalist. So, a stronger
logical positivism and the ontological is kept with
a strong mathematical platonism and teleological.
Talking about "the Old Ones", you still got Zeno
shaking his head and pointing at his watch.
Furthermore, I'm a constructivist and agree with
notions like infinite induction already and as there's
already, for example a sort of ubiquitous ordinals,
and even a sort of axiomless natural deduction seated
in reason.
The, "material implication", or, "ex falso quodlibet",
has that material implication is neither material nor implication,
and ex falso is mistakes or lies.
Some kinds of strong constructivists don't accept
non-constructive proofs, for example via contradiction,
de dicto, at all.
Here though it's just "modular: always modular,
of integral wholes, infinite in the middle, modular",
just so different from "and a 1 and on down and a 2
and on down and a 3 and on down and ... an omega and
on down", or, you know, not so.
See, "modularity" is regular, rulial, in both
increment and dispersion.
... Which most have as properties of integers
as with regards to associates with magnitudes,
or measures.
Heh, you brought up "The Old Ones", it's like,
what did the librarian or book-keeper say
when the paranoiac asked for self-help books,
"they're right behind you".
So, for example, Anderson's Relevance Logic many
have as more relevant than the quasi-modal, which
is neither temporal nor modal, like De Morgan's is,
with direct implication, there are some fans of Dana Scott
and not for his coat-tailing and wall-papering,
the theory of types is often attributed to Peirce,
the completeness theorems of arithmetic often to Frege,
the extra-ordinary of set theory to Mirimanoff and also
a bit to Quine about ultimate classes, you keep the
Vienna Circle, and, I'll stick with the larger, fuller canon.
Not that there's anything wrong with that, .....
On 08/12/2024 05:44 PM, Python wrote:
Le 13/08/2024 à 02:28, Ross Finlayson a écrit :
On 08/12/2024 04:06 PM, Jim Burns wrote:
On 8/12/2024 4:59 PM, Ross Finlayson wrote:
On 08/11/2024 09:44 PM, Jim Burns wrote:
On 8/11/2024 7:39 PM, Ross Finlayson wrote:
Starting with a theory _without_
the constant introduced named omega,
i.e., finite sets,
For P(z),
use a description 𝕆ᶠⁱⁿ(z) of a finite ordinal,
and ω := {z:𝕆ᶠⁱⁿ(z)} exists
For example, use
𝕆ᶠⁱⁿ(z) ⇔
(z ∋ {} ∧ ∀y ∈ z+1: y≠{} ⇒ ∃x∈z: x+1=y)
∨ (z = {})
z+1 = z∪{z}
Then, omega, as you've defined it,
ω := {z:𝕆ᶠⁱⁿ(z)}
contains itself,
I'm curious, now that you have
a beginning and an end of
the finite, or 0 and omega in ZF,
ω is the least.upper.bound of the finites.
ω is not a finite.
ω is not the upper.end of the finites.
The upper.end of the finites doesn't exist.
Here though
_Where_ though?
it's beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.
For ω as I've defined it, no upper.end exists.
for each k ∈ ω
𝕆ᶠⁱⁿ(k)
𝕆ᶠⁱⁿ(k+1)
k+1 ∈ ω
k is not the upper end of ω
for each k ∉ ω
k is not the upper end of ω
Then you claim to have
an axiom of restriction of comprehension of the finites
To review:
What I claim is
⎛ ∃{}
⎜ ∀x∀y∃z=x∪{y}
⎝ and extensionality
⎛ ∃∃xx={z:P(z)}: ∀y: y ∈ {z:P(z)} ⇔ P(y)
⎝ and extensionality
∃∃{z:P(z)} is unrestricted comprehension.
Unless we are no longer uninterested in what words mean.
unless Russell grants you
a dispensation of Russell's retro-thesis,
and say it's always so for others, too,
congratulations,
you claim to have invented a mathematics
where you = Russell.
Ah.
I've seen this one before.
Your tacit thesis is that
it is preferable to disagree with the Old Ones
even at the cost of being wrong.
Well, it's a choice.
Oh, I have the entire canon here along.
It's like yesterday, in this thread with the subject
of it talking about "infinite in the middle and always
with both ends", or, "here...", pointing out that some
modern philosophers with their Ph.D.s. resuscitate a
metaphysics that Compte and Boole and Russell and Carnap
made so nice for Marx and nihilism and extistentialism
and the sort of post-modern that begets itself.
Now, I confiscate logical positivism from Compte and
brush off Boole for De Morgan and point out Russell
and for example Whitehead suffer their own arguments
and Carnap was quite a pleasant fellow and I like Quine
yet I'm not a nominalist fictionalist. So, a stronger
logical positivism and the ontological is kept with
a strong mathematical platonism and teleological.
Talking about "the Old Ones", you still got Zeno
shaking his head and pointing at his watch.
Furthermore, I'm a constructivist and agree with
notions like infinite induction already and as there's
already, for example a sort of ubiquitous ordinals,
and even a sort of axiomless natural deduction seated
in reason.
The, "material implication", or, "ex falso quodlibet",
has that material implication is neither material nor implication,
and ex falso is mistakes or lies.
Some kinds of strong constructivists don't accept
non-constructive proofs, for example via contradiction,
de dicto, at all.
Here though it's just "modular: always modular,
of integral wholes, infinite in the middle, modular",
just so different from "and a 1 and on down and a 2
and on down and a 3 and on down and ... an omega and
on down", or, you know, not so.
See, "modularity" is regular, rulial, in both
increment and dispersion.
... Which most have as properties of integers
as with regards to associates with magnitudes,
or measures.
Heh, you brought up "The Old Ones", it's like,
what did the librarian or book-keeper say
when the paranoiac asked for self-help books,
"they're right behind you".
So, for example, Anderson's Relevance Logic many
have as more relevant than the quasi-modal, which
is neither temporal nor modal, like De Morgan's is,
with direct implication, there are some fans of Dana Scott
and not for his coat-tailing and wall-papering,
the theory of types is often attributed to Peirce,
the completeness theorems of arithmetic often to Frege,
the extra-ordinary of set theory to Mirimanoff and also
a bit to Quine about ultimate classes, you keep the
Vienna Circle, and, I'll stick with the larger, fuller canon.
Not that there's anything wrong with that, .....
Seriously Ross. What's the point of posting nonsense from
start to finish?
If you don't know your history
someone's bound to try and re-write it for you.
On 08/12/2024 06:06 PM, Ross Finlayson wrote:
On 08/12/2024 05:44 PM, Python wrote:
Le 13/08/2024 à 02:28, Ross Finlayson a écrit :
On 08/12/2024 04:06 PM, Jim Burns wrote:
On 8/12/2024 4:59 PM, Ross Finlayson wrote:
On 08/11/2024 09:44 PM, Jim Burns wrote:
On 8/11/2024 7:39 PM, Ross Finlayson wrote:
Starting with a theory _without_
the constant introduced named omega,
i.e., finite sets,
For P(z),
use a description 𝕆ᶠⁱⁿ(z) of a finite ordinal,
and ω := {z:𝕆ᶠⁱⁿ(z)} exists
For example, use
𝕆ᶠⁱⁿ(z) ⇔
(z ∋ {} ∧ ∀y ∈ z+1: y≠{} ⇒ ∃x∈z: x+1=y)
∨ (z = {})
z+1 = z∪{z}
Then, omega, as you've defined it,
ω := {z:𝕆ᶠⁱⁿ(z)}
contains itself,
I'm curious, now that you have
a beginning and an end of
the finite, or 0 and omega in ZF,
ω is the least.upper.bound of the finites.
ω is not a finite.
ω is not the upper.end of the finites.
The upper.end of the finites doesn't exist.
Here though
_Where_ though?
it's beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.
For ω as I've defined it, no upper.end exists.
for each k ∈ ω
𝕆ᶠⁱⁿ(k)
𝕆ᶠⁱⁿ(k+1)
k+1 ∈ ω
k is not the upper end of ω
for each k ∉ ω
k is not the upper end of ω
Then you claim to have
an axiom of restriction of comprehension of the finites
To review:
What I claim is
⎛ ∃{}
⎜ ∀x∀y∃z=x∪{y}
⎝ and extensionality
⎛ ∃∃xx={z:P(z)}: ∀y: y ∈ {z:P(z)} ⇔ P(y)
⎝ and extensionality
∃∃{z:P(z)} is unrestricted comprehension.
Unless we are no longer uninterested in what words mean.
unless Russell grants you
a dispensation of Russell's retro-thesis,
and say it's always so for others, too,
congratulations,
you claim to have invented a mathematics
where you = Russell.
Ah.
I've seen this one before.
Your tacit thesis is that
it is preferable to disagree with the Old Ones
even at the cost of being wrong.
Well, it's a choice.
Oh, I have the entire canon here along.
It's like yesterday, in this thread with the subject
of it talking about "infinite in the middle and always
with both ends", or, "here...", pointing out that some
modern philosophers with their Ph.D.s. resuscitate a
metaphysics that Compte and Boole and Russell and Carnap
made so nice for Marx and nihilism and extistentialism
and the sort of post-modern that begets itself.
Now, I confiscate logical positivism from Compte and
brush off Boole for De Morgan and point out Russell
and for example Whitehead suffer their own arguments
and Carnap was quite a pleasant fellow and I like Quine
yet I'm not a nominalist fictionalist. So, a stronger
logical positivism and the ontological is kept with
a strong mathematical platonism and teleological.
Talking about "the Old Ones", you still got Zeno
shaking his head and pointing at his watch.
Furthermore, I'm a constructivist and agree with
notions like infinite induction already and as there's
already, for example a sort of ubiquitous ordinals,
and even a sort of axiomless natural deduction seated
in reason.
The, "material implication", or, "ex falso quodlibet",
has that material implication is neither material nor implication,
and ex falso is mistakes or lies.
Some kinds of strong constructivists don't accept
non-constructive proofs, for example via contradiction,
de dicto, at all.
Here though it's just "modular: always modular,
of integral wholes, infinite in the middle, modular",
just so different from "and a 1 and on down and a 2
and on down and a 3 and on down and ... an omega and
on down", or, you know, not so.
See, "modularity" is regular, rulial, in both
increment and dispersion.
... Which most have as properties of integers
as with regards to associates with magnitudes,
or measures.
Heh, you brought up "The Old Ones", it's like,
what did the librarian or book-keeper say
when the paranoiac asked for self-help books,
"they're right behind you".
So, for example, Anderson's Relevance Logic many
have as more relevant than the quasi-modal, which
is neither temporal nor modal, like De Morgan's is,
with direct implication, there are some fans of Dana Scott
and not for his coat-tailing and wall-papering,
the theory of types is often attributed to Peirce,
the completeness theorems of arithmetic often to Frege,
the extra-ordinary of set theory to Mirimanoff and also
a bit to Quine about ultimate classes, you keep the
Vienna Circle, and, I'll stick with the larger, fuller canon.
Not that there's anything wrong with that, .....
Seriously Ross. What's the point of posting nonsense from
start to finish?
If you don't know your history
someone's bound to try and re-write it for you.
I released a new pod-cast the other day, you could
put a voice-stress analysis on it and see if it's
perceived veracity.
https://www.youtube.com/@rossfinlayson
Wouldn't that be nice, an un-obtrusive red or green
dot on your screen indicating when someone's either
lying to your face or entirely empty?
Perhaps you might put its transcript to one of these
modern mechanical thinking apparatuses and see whether
it finds anything of value or exactly what it doesn't agree.
Though I suppose you could always down a bottle of Xanax
and some booze and not worry much either way except
for perhaps the headache the next day, or otherwise
how to reclaim the brain from a pool of re-uptake inhibitors.
On the idiot scale, damnit I'm not going to beat your wife
for you. And you can take the entire slate of rhetorical fallacies
and wash it on down with some "I told you so".
https://www.youtube.com/@rossfinlayson
Am 13.08.2024 um 02:44 schrieb Python:
Seriously Ross. What's the point of posting nonsense from
start to finish?
Imho, he just should check his medication. It might help. (I'm serious.)
On 08/12/2024 07:47 PM, Python wrote:
Le 13/08/2024 à 03:16, Ross Finlayson a écrit :
...
https://www.youtube.com/@rossfinlayson
Oh dear this is bad. Do you realize that you are putting
words upon words, sentences upon sentences NONE of them
making any sense. Just like what you are posting on Usenet,
but there on air for hours...
You need medical help, FAST!
Let me know when they grant your M.D.
On 08/12/2024 08:00 PM, Python wrote:
Le 13/08/2024 à 02:59, Moebius a écrit :
Am 13.08.2024 um 02:44 schrieb Python:
Seriously Ross. What's the point of posting nonsense from
start to finish?
Imho, he just should check his medication. It might help. (I'm serious.)
The guy is clearly at risk.
At risk of kicking you in the nuts.
On 08/12/2024 04:06 PM, Jim Burns wrote:
On 8/12/2024 4:59 PM, Ross Finlayson wrote:
it's
beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.
For ω as I've defined it, no upper.end exists.
for each k ∈ ω
𝕆ᶠⁱⁿ(k)
𝕆ᶠⁱⁿ(k+1)
k+1 ∈ ω
k is not the upper end of ω
for each k ∉ ω
k is not the upper end of ω
It's like yesterday,
in this thread with the subject of it
talking about
"infinite in the middle and
always with both ends",
On 08/12/2024 09:25 PM, Jim Burns wrote:
On 8/12/2024 8:28 PM, Ross Finlayson wrote:
It's like yesterday,
in this thread with the subject of it
talking about
"infinite in the middle and
always with both ends",
I have just realized that
I have been overlooking your "always".
"ALWAYS with both ends" is finite.
If it's infinite in the middle
then, the middle acts as the fixed-point,
thus augmenting automatically yon definition,
and suffering not this.
Or, for example, it's a counter-example.
On 08/13/2024 08:37 PM, Jim Burns wrote:
On 8/13/2024 9:03 PM, Ross Finlayson wrote:
On 08/12/2024 09:25 PM, Jim Burns wrote:
On 8/12/2024 8:28 PM, Ross Finlayson wrote:
It's like yesterday,
in this thread with the subject of it
talking about
"infinite in the middle and
always with both ends",
"ALWAYS with both ends" is finite.
If it's infinite in the middle
If
it's infinite in the middle and
its non.{} subsets always have both ends,
then
it's not infinite in the middle.
So, you seem to imply that
the integers by your definition,
the integers by your definition,
by simply assigning 1 and -1 to the beginning,
then interleaving them,
and filling in as like a Pascal's Triangle of sorts,
or pyramidal numbers, that
that's, not, infinite?
Or, the rationals in the usual assignment of
assigning them integer values
and all the criss-crossing and
from either end, building in the middle,
not, infinite?
On 08/17/2024 02:12 PM, Jim Burns wrote:
Lemma 1.
⎛ No set B has both
⎝ finiteᵖᵍˢˢ order ⟨B,<⟩ and infiniteᵖᵍˢˢ order ⟨B,⩹⟩.
Definition.
⎛ An order ⟨B,<⟩ of B is finiteᵖᵍˢˢ iff
⎜ each non.empty subset S ⊆ B holds
⎝ both min[<].S and max[<].S
A finiteᵖᵍˢˢ set has a finiteᵖᵍˢˢ order.
An infiniteᵖᵍˢˢ set doesn't have a finiteᵖᵍˢˢ order.
ℕ ℤ ℚ and ℝ each have infiniteᵖᵍˢˢ orders.
In the standard order,
ℕ ℤ ℚ and ℝ are subsets of ℕ ℤ ℚ and ℝ with
0 or 1 ends.
Thus, the standard order is infiniteᵖᵍˢˢ.
Thus, by lemma 1, no non.standard order is finiteᵖᵍˢˢ.
They do not have any finiteᵖᵍˢˢ order.
Whatever non.standard order you propose,
you are proposing an infiniteᵖᵍˢˢ order;
you are proposing an order with
some _subset_ with 0 or 1 ends.
One more time:
In a finiteᵖᵍˢˢ order,
_each non.empty subset_ is 2.ended.
Two ends for the set as a whole isn't enough
to make the order finiteᵖᵍˢˢ.
So, with "infinite in the middle", it's just
that the natural order
0, infinity - 0,
1, infinity - 1,
...
has pretty simply two constants "0", "infinity",
then successors,
and it has all the models where infinity equates to
one of 0's successors, and they're finite,
and a model where it doesn't, that it's infinite.
Then, also it happens that
there's the usual order of sucessors and predecessors
that happens to hold,
naturally enough those are both infinite also.
At any rate, just identifying
even if just defining
the "predecessors of a limit ordinal"
as with no other facility than
"the successors of a limit ordinal",
So, ..., "well-order the reals".
On 08/18/2024 10:50 AM, Jim Burns wrote:
On 8/18/2024 10:17 AM, Ross Finlayson wrote:
On 08/17/2024 02:12 PM, Jim Burns wrote:
Lemma 1.
⎛ No set B has both
⎝ finiteᵖᵍˢˢ order ⟨B,<⟩ and infiniteᵖᵍˢˢ order ⟨B,⩹⟩.
Definition.
⎛ An order ⟨B,<⟩ of B is finiteᵖᵍˢˢ iff
⎜ each non.empty subset S ⊆ B holds
⎝ both min[<].S and max[<].S
A finiteᵖᵍˢˢ set has a finiteᵖᵍˢˢ order.
An infiniteᵖᵍˢˢ set doesn't have a finiteᵖᵍˢˢ order.
ℕ ℤ ℚ and ℝ each have infiniteᵖᵍˢˢ orders.
In the standard order,
ℕ ℤ ℚ and ℝ are subsets of ℕ ℤ ℚ and ℝ with
0 or 1 ends.
Thus, the standard order is infiniteᵖᵍˢˢ.
Thus, by lemma 1, no non.standard order is finiteᵖᵍˢˢ.
They do not have any finiteᵖᵍˢˢ order.
Whatever non.standard order you propose,
you are proposing an infiniteᵖᵍˢˢ order;
you are proposing an order with
some _subset_ with 0 or 1 ends.
Robinson arithmetic has non.standard models
with infinite naturals.
For example, {0}×ℕ ∪ ℚ⁺×ℤ
⎛ ⟨p,j⟩ <ꟴ ⟨q,k⟩ ⇔
⎝ p < q ∨ (p = q ∧ j < k)
⎛ Numbers ⟨p,j⟩ and ⟨q,k⟩ with p<q are
⎝ infinitely.far apart.
⎛ There are splits between ⟨p,j⟩ and ⟨q,k⟩
⎝ with no step from foresplit to hindsplit.
( ⟨p,j⟩ is not countable.to ⟨q,k⟩
( Not all subsets are 2.ended.
I'm really beginning to warm up to this idea of
"finite" and "all orderings are well-orderings"
being a thing.
[...] that they're not "immediate" successors,
thus it's delineated that they're "deferred" successors.
So, ordinals less than a limit ordinal are predecessors,
So, with "infinite in the middle", it's just
that the natural order
0, infinity - 0,
1, infinity - 1,
...
has pretty simply two constants "0", "infinity",
then successors,
and it has all the models where infinity equates to
one of 0's successors, and they're finite,
and a model where it doesn't, that it's infinite.
On 08/18/2024 09:56 PM, Jim Burns wrote:
[...]
I mean it's a great definition that finite has that
there exists a normal ordering that's a well-ordering
and that all the orderings of the set are well-orderings.
That's a great definition of finite and now it stands
for itself in enduring mathematical definition in defense.
Why is it you think that Stackel's definition of finite
and "not Dedekind's definition of countably infinite"
don't agree?
[...] is another little fact of mathematics
missing from your neat little hedgerow.
On 08/19/2024 02:43 PM, Jim Burns wrote:
On 8/19/2024 3:27 PM, Ross Finlayson wrote:
On 08/18/2024 09:56 PM, Jim Burns wrote:
[...]
I mean it's a great definition that finite has that
there exists a normal ordering that's a well-ordering
...in both directions...
and that all the orderings of the set are well-orderings.
...in both directions...
That's a great definition of finite and now it stands
for itself in enduring mathematical definition in defense.
...for comfortably more than a century.
Why is it you think that Stackel's definition of finite
and "not Dedekind's definition of countably infinite"
don't agree?
They mostly agree.
Given the Axiom of Choice
(let us say, if an inaccessible cardinal exists),
they completely agree.
My impression from somewhere is that,
if they disagree,
they disagree on some very weird sets.
https://en.wikipedia.org/wiki/Finite_set
[...] is another little fact of mathematics
missing from your neat little hedgerow.
I mark my neat little hedgerow, and
I describe what's true everywhere inside the hedgerow.
That allows me to learn about
what's inside the hedgerow,
even though it's infinite and I am finite.
I like doing that.
It isn't wrong for me to do that.
I will continue doing that.
I try to avoid using the word "weird",
it's associated with too much jingoism of
the knee-jerk variety, i.e. Pavlovian or
operant-conditioning the conditioned-responses,
"odd", for example, and for some time if you
recall at some point I declared that the use of
the word "but" was modally unhygienic and that
that is only "yet", so, if yet oddly, there are sets
that are totally regular as being divvied up as
by parts from part theory, or for example for
Brentano for boundaries, totally regular,
yet that for the usual regularity of a set
theory like ZF, "I am least deep", is the opposite.
It is _not_ yet oddly that that is so, that the
same underlying universe of substrate of elements
in relation has to model sets and parts, and that
the rulialities, the regularities, than in the universals
fall out not-ultimately-untrue thus having been at some
point not-first-false, eg where your limit ordinal comes
from as a matter of definition or axiomn if necessary,
yet for those for whom deductive inference is available
not necessarily so contrived, has that now that you've
definitely declared that your hedgerow has an inside
and an outside, that there is an outside, and,
it's its own inside, about complementary duals,
like parts and sets, where the comprehension either
divides or increments, like sets and and classes where
the comprehension either collects or elaborates, and
like numbering and counting where the comprehension
either enumerates as arithmetically or algebraically,
that each one of these pairs shows itself what must
be an entire meta-theory all outside your theory,
then whether you ever enjoy the light of day, as it were.
Each of these insides is finite to itself yet they
are infinite outsides to each other.
Give yourself some credit for having a perfect twin
who simply was inculcated with axioms starting from
the opposite side.
Or, you know, whether or not "complementary duals"
results "complimentary duels".
On 08/19/2024 02:43 PM, Jim Burns wrote:
[...]
[...] eg where your limit ordinal comes from
as a matter of definition or axiomn if necessary,
[...] now that you've definitely declared that
your hedgerow has an inside and an outside,
On 08/19/2024 04:18 PM, Jim Burns wrote:
[...]
Then, about that
the class of ordinal is an ordinal
and needn't be given by axiom or relation to an axiom,
yet instead as a matter of comprehension over the class,
On 08/19/2024 05:33 PM, Jim Burns wrote:
On 8/19/2024 8:08 PM, Ross Finlayson wrote:
On 08/19/2024 04:18 PM, Jim Burns wrote:
[...]
Then, about that
the class of ordinal is an ordinal
True because of what we mean by 'ordinal'
⎛ which leaves open the other question about
⎜ whether that class or finite ordinals or
⎝ inaccessible cardinal or ... _exist_
and needn't be given by axiom or relation to an axiom,
yet instead as a matter of comprehension over the class,
...if the class exists.
How we know that a class exists is by axiom.
It's an abstract object.
What other way could we know?
This need for some axiom to start off the existing
is harder to paper over in a formal language.
But, with either a formal or natural language,
it's inherent in exploring Plato's realm of Forms.
How else do we enter that realm?
You mean it's a void or a universe
and one can't know which and its
very contemplation thus inverts it
thus it's some dually-self-infraconsistent
Ding-an-Sich this primary object an ur-element?
I just call it that.
This way both "how do you get something from
nothing" and "how do you get nothing from
something" result the same answer so that
Kant's Sublime is Supreme and Hegel's
Nothing is Being.
Then, Leibnitz doesn't really refer to Plato
is his monadology, nor Wittgenstein in his
Tractatus, yet Gadamer wraps up for them "Amicus Plato".
Here it's simply that axiomless natural deduction is
this thing then axiomless geometry arrives at
it fully suffices for all the Euclidean
then the rest "must" be, "Es muss sein",
and it requires of course that one has arrived
at a theory of a Comenius-like language and then
that that there are no mathematical nor logical
paradoxes, at all, those all being resolved by
dually-self-infraconsistency.
Perhaps you've never left that realm.
On 08/19/2024 12:27 PM, Ross Finlayson wrote:
On 08/18/2024 09:56 PM, Jim Burns wrote:
Definition.
⎛ An order ⟨B,<⟩ of B is finiteᵖᵍˢˢ iff
⎜ each non.empty subset S ⊆ B holds
⎝ both min[<].S and max[<].S
Why is it you think that Stackel's definition of finite
and "not Dedekind's definition of countably infinite"
don't agree?
The entire idea here that there's a particular _regularity_
due dispersion and modularity only courtesy division down
from a fixed-point, that "Peano's axioms" don't give integers,
they only give increments, i.e. not necessarily constant increments,
that there's more than one _regularity_, REQUIRED, is another
little fact of mathematics missing from your neat little hedgerow.
..., REQUIRED, ....
On 08/29/2024 05:12 PM, Ross Finlayson wrote:
On 08/29/2024 04:32 PM, Jim Burns wrote:
On 8/29/2024 6:46 PM, Ross Finlayson wrote:
..., REQUIRED, ....
Things missing from my neat little hedgerow are
missing because I intend for them to be missing.
My neat little hedgerow has no weeds.
It has not had and will not have weeds.
And weeds would not be an improvement.
My neat little hedgerow is well.ordered;
each non.empty subset holds a minimum.
In my neat little hedgerow,
each Little Bunny Foo Foo has a successor,
scooping up the field mice and bopping them on the head,
and is a successor, except the first, named 0.
Successors are non.0 non.doppelgänger non.final.
You are welcome to talk about something else, Ross.
Note, though, that,
if you are talking about something else,
then you are talking about something else.
Non.triangles are not counter.examples to triangles.
Non.Bunny.Foo.Foos are not counter.examples to Bunny.Foo.Foos.
Have a nice day.
Sort of like you don't apply the inductive cases
that each stay "nope" and instead only affirm
that each one "goes", where, it goes.
Where it _goes_.
Then a claim like "I don't pick wrong"
So, well-order the reals.
On 08/29/2024 10:24 PM, Jim Burns wrote:
[...]
Regularity of _difference_, and,
regularity of _dispersion_,
both _increment_, and _modularity_,
are examples of two various kinds of regularity,
The reals actually give a well-ordering, though,
it's their normal ordering as via a model of line-reals.
Of course
any other one you'd give would have
taking a subset of ordinals,
which of course are _always_ well-ordered,
with those being an uncountable subset's, of the reals,
_also in their normal ordering_.
On 08/30/2024 02:41 PM, Jim Burns wrote:
On 8/30/2024 4:00 PM, Ross Finlayson wrote:
The reals actually give a well-ordering, though,
it's their normal ordering as via a model of line-reals.
No.
The normal ordering of the reals
is not a well.ordering.
In a well.ordering,
each nonempty subset holds a minimum.
In the normal ordering of ℝ,
(0,1] does not hold a minimum.
The normal ordering of ℝ is not a well.ordering.
Then, here is the great example of examples
from well-ordering the reals,
because
they're given an axiom to provide least-upper-bound,
"out of induction's sake",
then on giving for the axiom a well-ordering,
what sort of makes for a total ordering in any
what's called a space,
there are these continuity criteria where
thusly,
given a well-ordering of the reals,
one provides various counterexamples
in least-upper-bound, and thus topology,
for example
the first counterexample from topology
"there is no smallest positive real number".
Then the point that induction lets out is
at the Sorites or heap,
for that Burns' "not.first.false", means
"never failing induction first thus
being disqualified arbitrarily forever",
least-upper-bound, has that
that's been given as an axiom above or "in" ZFC,
that the least-upper-bound property even exists
after the ordered field that is
"same as the rationals, models the rationals,
thus where it's the only model of the rationals
it's given the existence",
Here then this "infinite middle"
is just like "unbounded in the middle"
which is just like this
"the well-ordering of the reals up to
their least-upper-boundedness",
On 09/02/2024 02:46 PM, Jim Burns wrote:
[...]
If a well-ordering exists, then,
consider it as a bijective function from ordinal O,
and thus its "elements" or ordinals O,
to domain D.
As a Cartesian function the usual way, that's thusly
a set of ordered pairs (o, d) which then
via usual axioms and schema of comprehension and
the existence of choice,
read out in order the element (o_alpha, d).
So, a well-ordering of the reals, this function, takes
any subset of uncountably many elements (o_alpha, d, alpha).
Now, what's so is that
only countably many of the d can be in their normal order,
that alpha < beta -> d_alpha < d_beta.
This is because
there are rational numbers between any of those,
and only countably many of those.
On 09/02/2024 02:46 PM, Jim Burns wrote:
On 9/1/2024 2:44 PM, Ross Finlayson wrote:
least-upper-bound, has that
that's been given as an axiom above or "in" ZFC,
No, least.upper.bound isn't an axiom above or in ZFC.
Then,
about the least-upper-bound actually being an axiom,
it sort of is,
that Dedekind-Eudoxus-Cauchy or
"there are all the infinite sequences",
as that there are "enough" elements in Cantor space
to fulfill least-upper-bound, it's an axiom.
On 09/02/2024 02:46 PM, Jim Burns wrote:
On 9/1/2024 2:44 PM, Ross Finlayson wrote:
Then the point that induction lets out is
at the Sorites or heap,
for that Burns' "not.first.false", means
"never failing induction first thus
being disqualified arbitrarily forever",
Not.first.false is about formulas which
are not necessarily about induction.
A first.false formula is false _and_
all (of these totally ordered formulas)
preceding formulas are true.
A not.first.false formula is not.that.
not.first.false Fₖ ⇔
¬(¬Fₖ ∧ ∀j<k:Fⱼ) ⇔
Fₖ ∨ ∃j<k:¬Fⱼ ⇔
∀j<k:Fⱼ ⇒ Fₖ
A finite formula.sequence S = {Fᵢ:i∈⟨1…n⟩} has
a possibly.empty sub.sequence {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ}
of false formulas.
If {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} is not empty,
it holds a first false formula,
because {Fᵢ:i∈⟨1…n⟩} is finite.
If each Fₖ ∈ {Fᵢ:i∈⟨1…n⟩} is not.first.false,
{Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} does not hold a first.false, and
{Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} is empty, and
each formula in {Fᵢ:i∈⟨1…n⟩} is true.
And that is why I go on about not.first.false.
Then about not.first.false
thanks for writing that up a bit more,
then that also you can see what I make of it.
Not.ultimately.untrue, ..., has that
F, bears the value for all F_alpha parameterized by ordinals
(which suffice, large enough, to totally order things),
of true, and that,
there are classes of formulas F,
for example self-referential or differential formulas,
defined for example according to
"when F_alpha is not also as for an ordinal less than omega",
at least making a trivial clear example of
a definition that is for classes of these sorts formulas
where "not.ultimately.untrue" is not held by all classes
for formulas "not.first.false".
On 09/05/2024 12:57 PM, Ross Finlayson wrote:
On 09/03/2024 01:50 PM, Jim Burns wrote:
[...][...]
Back in the 80's and 90's
it was Nelson's Internal Set Theory
where it was figured that
the avenue toward true non-standard real analysis
was to result.
I.e.,
not-a-real-functions with real analytical character,
like Dirac's delta function or
here for example
the Natural/Unit Equivalency Function,
it is expected that
"foundations" _does_ formalize them, and that
what doesn't, simply, isn't,
respectively.
Then this "infinite middle" is just about
the simplest "non-Archimedean" that there is,
and in fact even simpler, than for example
axiomatizing "0" and "omega"
axiomatizing "0" and "omega"
with an infinite-middle pretty much
exactly like ZF does,
except symmmetric about the middle
instead of non-inductive yet declared fiat
(stipulated).
Then this is usually about what's called "transfer principle",
that "what's true for each is true for all", these kinds of
things, and about limits and where it does or doesn't hold,
that's what it's called and that's what it is.
| Sysop: | Keyop |
|---|---|
| Location: | Huddersfield, West Yorkshire, UK |
| Users: | 715 |
| Nodes: | 16 (2 / 14) |
| Uptime: | 17:42:59 |
| Calls: | 12,103 |
| Calls today: | 3 |
| Files: | 15,004 |
| Messages: | 6,518,073 |