lördag 19 april 2014 kl. 10:11:25 UTC+2 skrev quasi:
> quasi wrote:
> >William Elliot wrote:
> >>
> >>Does the set of all ordinals exist within ZF?
> >
> >It's too big to be a set.
> Hmmm ...
>
> It's certainly not a set in ZFC.
>
> I'm not sure if the "too big" criterion can be applied in ZF.
>
> But how would you _define_ such a set?
>
> Wouldn't the postulated existence of such a set fall victim to
> Russell's paradox?
>
> quasi
No set can contain itself. The set of all ordinals would be a new ordinal, and thus contain itself. Ergo, there cannot be a set of all ordinals.

The phrase the "The set of all ordinals" is meaningless if ordinals
are not sets itself. At the time of Burali-Forti set theory was
not that evolved. And the proof at that time didn't use regularity axiom.

So the set of all ordinals was a notion of naive set theory, and not
formulated in modern set theory ordinal terminology, but as a
question about transfinite numbers:

Una questione sui numeri transfiniti
https://zenodo.org/records/2362091/files/article.pdf

As one can see from the paper the proof proceeded by establishing:

Ω + 1 > Ω and Ω + 1 < Ωmarkus...@gmail.com schrieb:

> No set can contain itself. The set of all ordinals would be a new ordinal
and thus contain itself. Ergo, there cannot be a set of all ordinals.
>

On 02/20/2024 11:35 AM, markus...@gmail.com wrote:
> lördag 19 april 2014 kl. 10:11:25 UTC+2 skrev quasi:
>> quasi wrote:
>>> William Elliot wrote:
>>>>
>>>> Does the set of all ordinals exist within ZF?
>>>
>>> It's too big to be a set.
>> Hmmm ...
>>
>> It's certainly not a set in ZFC.
>>
>> I'm not sure if the "too big" criterion can be applied in ZF.
>>
>> But how would you _define_ such a set?
>>
>> Wouldn't the postulated existence of such a set fall victim to
>> Russell's paradox?
>>
>> quasi
> No set can contain itself. The set of all ordinals would be a new ordinal, and thus contain itself. Ergo, there cannot be a set of all ordinals.
>

"... in set-theories like ZF
that are ordinary/well-founded,
according to an axiom like Regularity
of restriction of comprehension."

There are others, ..., "Mengenlehre(n)".

On 02/20/2024 12:15 PM, Ross Finlayson wrote:
> On 02/20/2024 11:35 AM, markus...@gmail.com wrote:
>> lördag 19 april 2014 kl. 10:11:25 UTC+2 skrev quasi:
>>> quasi wrote:
>>>> William Elliot wrote:
>>>>>
>>>>> Does the set of all ordinals exist within ZF?
>>>>
>>>> It's too big to be a set.
>>> Hmmm ...
>>>
>>> It's certainly not a set in ZFC.
>>>
>>> I'm not sure if the "too big" criterion can be applied in ZF.
>>>
>>> But how would you _define_ such a set?
>>>
>>> Wouldn't the postulated existence of such a set fall victim to
>>> Russell's paradox?
>>>
>>> quasi
>> No set can contain itself. The set of all ordinals would be a new
>> ordinal, and thus contain itself. Ergo, there cannot be a set of all
>> ordinals.
>>
>
> "... in set-theories like ZF
> that are ordinary/well-founded,
> according to an axiom like Regularity
> of restriction of comprehension."
>
> There are others, ..., "Mengenlehre(n)".
>

(The set of all ordinals has a name, it's "ORD",
the order type of ordinals, and set of ordinals.)

(One time I wrote a couple different ways to
define, the, "group of all groups", for algebra,
like "GRP".)

(There are wide varieties of, "mothers of all wavelets".)

Mostly these sorts considerations are
called "ZF with Classes" or "ZFC with Classes",
that the Classes or Klassen, if that's right,
are sets, when, you know, they're not sets.

I called it the "Group-Noun Game", because,
it eventually runs out of Group Nouns.

Someone like Quine calls the classes that aren't
sets, "ultimate" classes, while usually the name
for the classes that aren't sets are "proper", classes,
while in some considerations there can only be one,
"proper" class, because, it's as an "absolute", class.

So, after ZFC there's things like NBG, "Neumann-Bernays-Goedel",
or GBN, "Goedel-Bernays-Neumann", who, depending on who you
ask and how formalist they are that day, are or aren't,
ZFC with classes and/or a conservative extension of ZFC.

On 2/20/2024 3:15 PM, Ross Finlayson wrote:
> On 02/20/2024 11:35 AM, markus...@gmail.com wrote:

>> No set can contain itself.
>> The set of all ordinals would be a new ordinal,
>> and thus contain itself.
>> Ergo, there cannot be a set of all ordinals.
>
> "... in set-theories like ZF
> that are ordinary/well-founded,
> according to an axiom like Regularity
> of restriction of comprehension."
>
> There are others, ..., "Mengenlehre(n)".

However,
whatever sets might be,
ordinals would not be ordinals
if they weren't well.ordered by ∈

In any theory in which ordinals are ordinals,
at least the ordinals have finite.descent,
whatever might be true of other sets.

A proposed set.of.all.ordinals which
held itself would not have finite descent.

Ordinals are well.ordered.
Well.ordered.ness can be re.phrased as
transfinite.induction.ness.
(∀α:(∀β<α:P(β))⇒P(α)) ⟹ ∀γ:P(γ)

FD(γ) == "γ has finite descent"

| Assume each ordinal β < α has finite descent.
| ∀β<α:FD(β)
| | ⟨ α β δ ε ... ⟩ is a strictly.descending sequence
| α > β
| β has finite descent.
| ⟨ β δ ε ... ⟩ is finite
| ⟨ α β δ ε ... ⟩ is finite
| Generalizing over sequences,
| α has finite descent.

Therefore, generalizing over ordinals,
∀α:(∀β<α:FD(β))⇒FD(α)

By transfinite.induction (by well.order),
∀γ:FD(γ)
Each ordinal has finite descent.

Therefore,
the ordinal(?) holding all(?) ordinals
does not hold itself.

On 02/20/2024 02:02 PM, Jim Burns wrote:
> On 2/20/2024 3:15 PM, Ross Finlayson wrote:
>> On 02/20/2024 11:35 AM, markus...@gmail.com wrote:
>
>>> No set can contain itself.
>>> The set of all ordinals would be a new ordinal,
>>> and thus contain itself.
>>> Ergo, there cannot be a set of all ordinals.
>>
>> "... in set-theories like ZF
>> that are ordinary/well-founded,
>> according to an axiom like Regularity
>> of restriction of comprehension."
>>
>> There are others, ..., "Mengenlehre(n)".
>
> However,
> whatever sets might be,
> ordinals would not be ordinals
> if they weren't well.ordered by ∈
>
> In any theory in which ordinals are ordinals,
> at least the ordinals have finite.descent,
> whatever might be true of other sets.
>
> A proposed set.of.all.ordinals which
> held itself would not have finite descent.
>
>
> Ordinals are well.ordered.
> Well.ordered.ness can be re.phrased as
> transfinite.induction.ness.
> (∀α:(∀β<α:P(β))⇒P(α)) ⟹ ∀γ:P(γ)
>
> FD(γ) == "γ has finite descent"
>
> | Assume each ordinal β < α has finite descent.
> | ∀β<α:FD(β)
> |
> | ⟨ α β δ ε ... ⟩ is a strictly.descending sequence
> | α > β
> | β has finite descent.
> | ⟨ β δ ε ... ⟩ is finite
> | ⟨ α β δ ε ... ⟩ is finite
> | Generalizing over sequences,
> | α has finite descent.
>
> Therefore, generalizing over ordinals,
> ∀α:(∀β<α:FD(β))⇒FD(α)
>
> By transfinite.induction (by well.order),
> ∀γ:FD(γ)
> Each ordinal has finite descent.
>
> Therefore,
> the ordinal(?) holding all(?) ordinals
> does not hold itself.
>
>

ORD, the order type of ordinals?
The antinomy of Cesare Burali-Forti?

When you theory has a universe,
it's sort of a singular entity,
it is its own powerset and all, ....

If you stick with bounded theories
and adopt an ultra-finitist formalism,
then you might wonder sometime,
where exactly it is all, at?

It's a usual idea for sorts
of "dualist monism",
since for example
Heraklites or Zen Buddhism,
that the universe really is a thing,
and we are in it,
and that the void really is a thing,
and we are in it,
about the same thing.

Because it's a tautology, ....

It's a sort of brachistology.

ORD: that's its name.

Le 20/02/2024 à 23:02, Jim Burns a écrit :

> In any theory in which ordinals are ordinals,
> at least the ordinals have finite.descent,

That proves finite ascend too, because otherwise every ordinal could be
ascended and then the way upstairs could be gone back downstairs. Finite
ascend and descend prove that most ordinals are dark.

> Ordinals are well.ordered.

Only those which can be specified.

Regards, WM

On 2/21/2024 3:33 AM, WM wrote:
> Le 20/02/2024 à 23:02, Jim Burns a écrit :

>> Ordinals are well.ordered.
>
> Only those which can be specified.

No.
All of them are well.ordered.
Anything else wouldn't be the ordinals.

Anything else would be like declaring
that only specifiableᵂᴹ right.triangles
have three corners.

>> In any theory in which ordinals are ordinals,
>> at least the ordinals have finite.descent,
>
> That proves finite ascend too,
> because
> otherwise
> every ordinal could be ascended
> and then
> the way upstairs could be gone back downstairs.

The ordinals' descents and ascents are not the same.

If any of an ordinal's descents is infinite,
the ordinal doesn't have finite.descent.

If any of an ordinal's ascents is infinite,
the ordinal doesn't have finite.ascent.

Each ordinal α has a successor α+1
α+1 has α+2, etc.

For ordinal a
⟨ α α+1 α+2 α+3 ... ⟩ is an infinite ascent.
α doesn't have finite.ascent.

Generalizing over ordinals,
no ordinal a has finite.ascent.

For each ordinal ψ
if ψ has any infinite descent,
then, because well.order,
an ordinal χ exists first with any infinite descent.

However,
one step down from χ to any ordinal β < χ is to
β with only finite descents,
and finite plus one is finite.
First χ doesn't have any infinite descent.
Contradiction.

ψ doesn't have an infinite descent.

Generalizing over ordinals,
each ordinal ψ has finite.descent.

If ORD involves class/set distinction,
and a set-theory can also be written as a part-theory,
then what's part/particle distinction/

If set theory's relation is "elt", element-of, "in"
and class theory's relation is "members", "contains", "has",
then, is :
class/set theory
set/part theory?

Here that "numbering" and "counting" are two different things,
one for ordering theory the other for collection,
ordinals and sets, numbering and counting,
what about
set/class distinction and
set/part distinction and
part/class distinction?

See, this is among reasons why
I've been way both ahead of
and on top of this for a long time,
and trying to tell you so all the time.

I told you, ..., I told you.

Mostly is for understanding that
"numbering" and "counting" are
two different things, and they
involve each other in their resources.

Seriously, you don't know what classes are?

The membership relation is the same
for members of classes and for members of sets.
Since members of classes are sets just like

the members of sets are sets, in ZF. And there is
only one membership relation ∈ between sets. The
distinction between classes and sets was described

in the past as:

sets: includes collections of sizes from the numbers to
the transfinite numbers
classes: includes collections that Cantor called
NCONSISTENT MULTIPLICITIES

You had them somewhere in one of your random posts:

Ross Finlayson schrieb:
> Of course, the goal is "there are no paradoxes at all",
> then what seem "inconsistent multiplicities", just don't relate.

But this below is awful gibberish:

Ross Finlayson schrieb:
> If ORD involves class/set distinction,
> and a set-theory can also be written as a part-theory,
> then what's part/particle distinction/
>
> If set theory's relation is "elt", element-of, "in"
> and class theory's relation is "members", "contains", "has",
> then, is :
> class/set theory
> set/part theory?
>
> Here that "numbering" and "counting" are two different things,
> one for ordering theory the other for collection,
> ordinals and sets, numbering and counting,
> what about
> set/class distinction and
> set/part distinction and
> part/class distinction?
>
> See, this is among reasons why
> I've been way both ahead of
> and on top of this for a long time,
> and trying to tell you so all the time.
>
> I told you, ..., I told you.
>
> Mostly is for understanding that
> "numbering" and "counting" are
> two different things, and they
> involve each other in their resources.
>
>

In the philosophy of mathematics, specifically the philosophical
foundations of set theory, limitation of size is a concept developed by
Philip Jourdain and/or Georg Cantor to avoid Cantor's paradox. It
identifies certain "inconsistent multiplicities", in Cantor's
terminology, that cannot be sets because they are "too large". In modern
terminology these are called proper classes.
https://en.wikipedia.org/wiki/Limitation_of_size

You might like this book:

Cantor's ideas formed the basis for set theory and also for the
mathematical treatment of the concept of infinity. The philosophical and
heuristic framework he developed had a lasting effect on modern
mathematics, and is the recurrent theme of this volume. Hallett explores
Cantor's ideas and, in particular, their ramifications for
Zermelo-Frankel set theory.
https://academic.oup.com/pq/article-abstract/36/144/429/1567519

Mild Shock schrieb:
> Seriously, you don't know what classes are?
>
> The membership relation is the same
> for members of classes and for members of sets.
> Since members of classes are sets just like
>
> the members of sets are sets, in ZF. And there is
> only one membership relation ∈ between sets. The
> distinction between classes and sets was described
>
> in the past as:
>
> sets: includes collections of sizes from the numbers to
>   the transfinite numbers
> classes: includes collections that Cantor called
>   NCONSISTENT MULTIPLICITIES
>
> You had them somewhere in one of your random posts:
>
> Ross Finlayson schrieb:
> > Of course, the goal is "there are no paradoxes at all",
> > then what seem "inconsistent multiplicities", just don't relate.
>
> But this below is awful gibberish:
>
> Ross Finlayson schrieb:
>> If ORD involves class/set distinction,
>> and a set-theory can also be written as a part-theory,
>> then what's part/particle distinction/
>>
>> If set theory's relation is "elt", element-of, "in"
>> and class theory's relation is "members", "contains", "has",
>> then, is :
>> class/set theory
>> set/part theory?
>>
>> Here that "numbering" and "counting" are two different things,
>> one for ordering theory the other for collection,
>> ordinals and sets, numbering and counting,
>> what about
>> set/class distinction and
>> set/part distinction and
>> part/class distinction?
>>
>> See, this is among reasons why
>> I've been way both ahead of
>> and on top of this for a long time,
>> and trying to tell you so all the time.
>>
>> I told you, ..., I told you.
>>
>> Mostly is for understanding that
>> "numbering" and "counting" are
>> two different things, and they
>> involve each other in their resources.
>>
>>
>

Le 21/02/2024 à 18:59, Jim Burns a écrit :
> On 2/21/2024 3:33 AM, WM wrote:
>> Le 20/02/2024 à 23:02, Jim Burns a écrit :
>
>>> Ordinals are well.ordered.
>>
>> Only those which can be specified.
>
> No.
> All of them are well.ordered.

How do you know?

> Anything else wouldn't be the ordinals.

In fact, not these ordinals.
>
> Anything else would be like declaring
> that only specifiableᵂᴹ right.triangles
> have three corners.

That is too drastic. Natnumbers keep almost all of their properties.
>
> The ordinals' descents and ascents are not the same.

Every way up can be reversed. That proves that also the ascents are
finite.
>
> For each ordinal ψ
> if ψ has any infinite descent,
> then, because well.order,
> an ordinal χ exists first with any infinite descent.
>
> However,
> one step down from χ to any ordinal β < χ is to
> β with only finite descents,
> and finite plus one is finite.
> First χ doesn't have any infinite descent.
> Contradiction.
>
> ψ doesn't have an infinite descent.

And one step upwards is finite too. Finite plus one is finite.a

ψ doesn't have an infinite ascent (for every visible predecessor).
>
> Generalizing over ordinals,
> each ordinal ψ has finite.descent.

Each ordinal has finite ascent.

Regards, WM

On 2/22/2024 8:00 AM, WM wrote:
> Le 21/02/2024 à 18:59, Jim Burns a écrit :
>> On 2/21/2024 3:33 AM, WM wrote:
>>> Le 20/02/2024 à 23:02, Jim Burns a écrit :

>>>> Ordinals are well.ordered.
>>>
>>> Only those which can be specified.
>>
>> No.
>> All of them are well.ordered.
>
> How do you know?

In the same way that I know
that right.triangles have three corners.
Not by looking at ordinals.
By knowing what "ordinal" means.

>> Anything else wouldn't be the ordinals.
>
> In fact, not these ordinals.

Then "these ordinals" are like
four.cornered right.triangles.

>> Anything else would be like declaring
>> that only specifiableᵂᴹ right.triangles
>> have three corners.
>
> That is too drastic.
> Natnumbers keep almost all of their properties.

Four.cornered right.triangles share
many properties with other plane figures.
They aren't right.triangles.

>> The ordinals' descents and ascents are not the same.
>
> Every way up can be reversed.

An infinite way up isn't _to_ any ordinal.
Reversed, it isn't _from_ any ordinal.

> That proves that also the ascents are finite.

By excluding all infinite ascents,
we can prove that all _remaining_ ascents
are finite.

....which doesn't deny that
⟨ α α+1 α+2 α+3 ... ⟩ is an infinite ascent.

In contrast,
there is no first ordinal with
an infinite descent, so
there is no ordinal with
an infinite descent.

>> For each ordinal ψ
>> if ψ has any infinite descent,
>> then, because well.order,
>> an ordinal χ exists first with any infinite descent.
>>
>> However,
>> one step down from χ to any ordinal β < χ is to
>> β with only finite descents,
>> and finite plus one is finite.
>> First χ doesn't have any infinite descent.
>> Contradiction.
>>
>> ψ doesn't have an infinite descent.
>
> And one step upwards is finite too.
> Finite plus one is finite.a

Finite ascents exist.
Finite descents exist too.

Infinite ascents exist,
⟨ α α+1 α+2 α+3 ... ⟩
A first infinite descent is
a contradiction.

> ψ doesn't have an infinite ascent
> (for every visible predecessor).

Daek numbers wouldn't make
⟨ ψ ψ+1 ψ+2 ψ+3 ... ⟩ less infinite.

>> Generalizing over ordinals,
>> each ordinal ψ has finite.descent.
>
> Each ordinal has finite ascent.

Know what "finite ascent" means:
no infinite ascents.

On 02/22/2024 12:03 AM, Mild Shock wrote:
> Seriously, you don't know what classes are?
>
> The membership relation is the same
> for members of classes and for members of sets.
> Since members of classes are sets just like
>
> the members of sets are sets, in ZF. And there is
> only one membership relation ∈ between sets. The
> distinction between classes and sets was described
>
> in the past as:
>
> sets: includes collections of sizes from the numbers to
> the transfinite numbers
> classes: includes collections that Cantor called
> NCONSISTENT MULTIPLICITIES
>
> You had them somewhere in one of your random posts:
>
> Ross Finlayson schrieb:
> > Of course, the goal is "there are no paradoxes at all",
> > then what seem "inconsistent multiplicities", just don't relate.
>
> But this below is awful gibberish:
>
> Ross Finlayson schrieb:
>> If ORD involves class/set distinction,
>> and a set-theory can also be written as a part-theory,
>> then what's part/particle distinction/
>>
>> If set theory's relation is "elt", element-of, "in"
>> and class theory's relation is "members", "contains", "has",
>> then, is :
>> class/set theory
>> set/part theory?
>>
>> Here that "numbering" and "counting" are two different things,
>> one for ordering theory the other for collection,
>> ordinals and sets, numbering and counting,
>> what about
>> set/class distinction and
>> set/part distinction and
>> part/class distinction?
>>
>> See, this is among reasons why
>> I've been way both ahead of
>> and on top of this for a long time,
>> and trying to tell you so all the time.
>>
>> I told you, ..., I told you.
>>
>> Mostly is for understanding that
>> "numbering" and "counting" are
>> two different things, and they
>> involve each other in their resources.
>>
>>
>

Actually, for class/set distinction,
I just introduced set/part distinction,
and part/particle distinction,
and set/particle distinction.

set:class::part:particle

set:part::class:particle

This is a usual form that A:B::C:D is
that A relates to B as C relates to D,
"set is to class as part is to particle", and
"set is to part as class is to particle".

Doesn't make any sense at all.

Not a single mention of proper classes here:
https://plato.stanford.edu/ENTRIES/mereology/

Ross Finlayson schrieb:
> Actually, for class/set distinction,
> I just introduced set/part distinction,
> and part/particle distinction,
> and set/particle distinction.
>
> set:class::part:particle
>
> set:part::class:particle
>
> This is a usual form that A:B::C:D is
> that A relates to B as C relates to D,
> "set is to class as part is to particle", and
> "set is to part as class is to particle".
>
>

I guess we have reached your intellectual
boundaries, inherent in your squirell brain
sized, that of a walnut, cerebrum and cerebrellum.

Mild Shock schrieb:
> Doesn't make any sense at all.
>
> Not a single mention of proper classes here:
> https://plato.stanford.edu/ENTRIES/mereology/
>
> Ross Finlayson schrieb:
>> Actually, for class/set distinction,
>> I just introduced set/part distinction,
>> and part/particle distinction,
>> and set/particle distinction.
>>
>> set:class::part:particle
>>
>> set:part::class:particle
>>
>> This is a usual form that A:B::C:D is
>> that A relates to B as C relates to D,
>> "set is to class as part is to particle", and
>> "set is to part as class is to particle".
>>
>>
>

What we can say is the following:

i) Every set is a class
ii) Not every class is a set

So there is a hypernym / hyponym relationship
between the two. Here is are proof of i) and ii):

Proof i): Let s be a set. Then we can form
the class { x | x e s }. So there is an injection
from the sets to the classes.

Proof ii): Let V be the class { x | true },
this is the universal class which is provably
not a set. So there is no surjection from
the sets to the classes.

Hope this helps. Injection is usually taken
as indicative that two sets are in the
less than or equal relation ship, i.e. ⊆.
And lack of surjection indicates that there
is no bijection, i.e. ≠, so we have:

Sets ⊆ Classes and Sets ≠ Classes

Or together:

Sets ⊂ Classes

The difference Class \ Sets, those things
that are classes but not sets, are called
proper classes.

Mild Shock schrieb:
> I guess we have reached your intellectual
> boundaries, inherent in your squirell brain
> sized, that of a walnut, cerebrum and cerebrellum.
>
> Mild Shock schrieb:
>> Doesn't make any sense at all.
>>
>> Not a single mention of proper classes here:
>> https://plato.stanford.edu/ENTRIES/mereology/
>>
>> Ross Finlayson schrieb:
>>> Actually, for class/set distinction,
>>> I just introduced set/part distinction,
>>> and part/particle distinction,
>>> and set/particle distinction.
>>>
>>> set:class::part:particle
>>>
>>> set:part::class:particle
>>>
>>> This is a usual form that A:B::C:D is
>>> that A relates to B as C relates to D,
>>> "set is to class as part is to particle", and
>>> "set is to part as class is to particle".
>>>
>>>
>>
>

On 02/22/2024 10:22 AM, Mild Shock wrote:
> I guess we have reached your intellectual
> boundaries, inherent in your squirell brain
> sized, that of a walnut, cerebrum and cerebrellum.
>
> Mild Shock schrieb:
>> Doesn't make any sense at all.
>>
>> Not a single mention of proper classes here:
>> https://plato.stanford.edu/ENTRIES/mereology/
>>
>> Ross Finlayson schrieb:
>>> Actually, for class/set distinction,
>>> I just introduced set/part distinction,
>>> and part/particle distinction,
>>> and set/particle distinction.
>>>
>>> set:class::part:particle
>>>
>>> set:part::class:particle
>>>
>>> This is a usual form that A:B::C:D is
>>> that A relates to B as C relates to D,
>>> "set is to class as part is to particle", and
>>> "set is to part as class is to particle".
>>>
>>>
>>
>

Hm. "squirrel:brain::walnut:cerebrum".

(The spell-checker there seems omitted 'cerebellum',
and didn't read into the theory of parts some
'ultimate particles', perhaps it might help to
addend an entry 'atomism'. Here though it's mathematics
and we have this entire canonical exposition about
"uniform and continuous time", "Zeno's arguments".
Maybe that will add some more context for the
"brain of a squirrel" simile, metaphor.)

With regards to intensionality and extensionality,

intensionality <- structure-equals
extensionality <- duck-type-equals

simile <- declaring-relates
metaphor <- not-equals-so-relate-to-relates

and simile and metaphor, and a strong theory of types,
one can make a simile and metaphor out of pretty much
anything, that of course there's a usual idea that
"relevance logic" dictates relevance for a strong
theory of types.

My, what a rude person. It's kind of like in the old days,
when you'd see something doing something or not demonstrating
knowledge of something, one might say, "what are you doing,
you big dummy". This was like, "hey, you big dummy, what do
you think you are doing". Or, like to a dog, "aw, you big dummy".

Here I don't quite get that kind of attitude in a
conversation about the fundamental objects of the
universe of the fundamental objects of mathematics,
about the theory of the objects of the fundamental
theory of the fundamental objects of mathematics.

It's like, "you're going to have to be a bigger dummy".

I.e., the universe of mathematical objects
"is what it is", it's got numbers in it, and on it,
and counting is an act, two different things.

Better symbolism would be:

Sets' ⫋ Classes

Where Sets' results from Sets by the injection
{ x | x e s } for each x e Sets. This gives a little
transfer principle. If you can prove, i.e. that

a property holds for all classes:

∀X P(X)

Then it follows, that the property holds for all sets.

∀x P(x)

Proof: In higher order logic one would probably
write λy.(x y) for { y | y e x }, by eta reduction
we have λy.(x y)= x, so one can prove:

∀X P(X)
------------ (∀ elim)
P(λy.(x y))
------------ (η-reduction)
P(x)
------------ (∀ Intro)
∀x P(x)

Q.E.D.

η-reduction expresses the idea of extensionality
https://en.wikipedia.org/wiki/Lambda_calculus#%CE%B7-reduction

Wao! Now I did a lot of cheating, sweeping a lot of
details under the rug. I guess this is not
the standard way to do these things.

Better have a look here:

Basic Set Theory - Azriel Levy
https://www.amazon.com/dp/0486420795

Mild Shock schrieb:
> What we can say is the following:
>
> i) Every set is a class
> ii) Not every class is a set
>
> So there is a hypernym / hyponym relationship
> between the two. Here is are proof of i) and ii):
>
> Proof i): Let s be a set. Then we can form
> the class { x | x e s }. So there is an injection
> from the sets to the classes.
>
> Proof ii): Let V be the class { x | true },
> this is the universal class which is provably
> not a set. So there is no surjection from
> the sets to the classes.
>
> Hope this helps. Injection is usually taken
> as indicative that two sets are in the
> less than or equal relation ship, i.e. ⊆.
> And lack of surjection indicates that there
> is no bijection, i.e. ≠, so we have:
>
> Sets ⊆ Classes and Sets ≠ Classes
>
> Or together:
>
> Sets ⊂ Classes
>
> The difference Class \ Sets, those things
> that are classes but not sets, are called
> proper classes.
>
> Mild Shock schrieb:
>> I guess we have reached your intellectual
>> boundaries, inherent in your squirell brain
>> sized, that of a walnut, cerebrum and cerebrellum.
>>
>> Mild Shock schrieb:
>>> Doesn't make any sense at all.
>>>
>>> Not a single mention of proper classes here:
>>> https://plato.stanford.edu/ENTRIES/mereology/
>>>
>>> Ross Finlayson schrieb:
>>>> Actually, for class/set distinction,
>>>> I just introduced set/part distinction,
>>>> and part/particle distinction,
>>>> and set/particle distinction.
>>>>
>>>> set:class::part:particle
>>>>
>>>> set:part::class:particle
>>>>
>>>> This is a usual form that A:B::C:D is
>>>> that A relates to B as C relates to D,
>>>> "set is to class as part is to particle", and
>>>> "set is to part as class is to particle".
>>>>
>>>>
>>>
>>
>

On 02/22/2024 10:20 AM, Mild Shock wrote:
> Doesn't make any sense at all.
>
> Not a single mention of proper classes here:
> plato.stanford.edu/ENTRIES/mereology/
>
> Ross Finlayson schrieb:
>> Actually, for class/set distinction,
>> I just introduced set/part distinction,
>> and part/particle distinction,
>> and set/particle distinction.
>>
>> set:class::part:particle
>>
>> set:part::class:particle
>>
>> This is a usual form that A:B::C:D is
>> that A relates to B as C relates to D,
>> "set is to class as part is to particle", and
>> "set is to part as class is to particle".
>>
>>
>

Hm. "squirrel:brain::walnut:cerebrum".

(The spell-checker there seems omitted 'cerebellum',
and didn't read into the theory of parts some
'ultimate particles', perhaps it might help to
addend an entry 'atomism'. Here though it's mathematics
and we have this entire canonical exposition about
"uniform and continuous time", "Zeno's arguments".
Maybe that will add some more context for the
"brain of a squirrel" simile, metaphor.)

With regards to intensionality and extensionality,

intensionality <- structure-equals
extensionality <- duck-type-equals

simile <- declaring-relates
metaphor <- not-equals-so-relate-to-relates

and simile and metaphor, and a strong theory of types,
one can make a simile and metaphor out of pretty much
anything, that of course there's a usual idea that
"relevance logic" dictates relevance for a strong
theory of types.

My, what a rude person. It's kind of like in the old days,
when you'd see something doing something or not demonstrating
knowledge of something, one might say, "what are you doing,
you big dummy". This was like, "hey, you big dummy, what do
you think you are doing". Or, like to a dog, "aw, you big dummy".

Here I don't quite get that kind of attitude in a
conversation about the fundamental objects of the
universe of the fundamental objects of mathematics,
about the theory of the objects of the fundamental
theory of the fundamental objects of mathematics.

It's like, "you're going to have to be a bigger dummy".

I.e., the universe of mathematical objects
"is what it is", it's got numbers in it, and on it,
and counting is an act, two different things.

Are you pulling a John Gabriel? Back
to greek ratios. Euclids general form:

A : B = C : D

What about this form:

A : B : C = E : F : G
http://aleph0.clarku.edu/~djoyce/elements/bookV/defV3.html

I would especially recommend equations of the form:

cornet:walnut:pistachio
= cup:banana:mango
= hotday:ingest:cooling

Ross Finlayson schrieb:
> Hm.  "squirrel:brain::walnut:cerebrum".

If you follow this thinking long enough,
you will proof Mückenheims identifiable numbers.
And suddently have finite ascent in mathematics.

LoL

Ross Finlayson schrieb:
> I.e., the universe of mathematical objects
> "is what it is", it's got numbers in it, and on it,
> and counting is an act, two different things.

Wolfgang Mückenheim copy cats are rather boring.

Mild Shock schrieb:
>
> If you follow this thinking long enough,
> you will proof Mückenheims identifiable numbers.
> And suddently have finite ascent in mathematics.
>
> LoL
>
> Ross Finlayson schrieb:
>> I.e., the universe of mathematical objects
>> "is what it is", it's got numbers in it, and on it,
>> and counting is an act, two different things.
>

John Gabriel copy cats are rather boring.

Mild Shock schrieb:
> Are you pulling a John Gabriel? Back
> to greek ratios. Euclids general form:
>
> A : B = C : D
>
> What about this form:
>
> A : B : C = E : F : G
> http://aleph0.clarku.edu/~djoyce/elements/bookV/defV3.html
>
> I would especially recommend equations of the form:
>
> cornet:walnut:pistachio
>    = cup:banana:mango
>    = hotday:ingest:cooling
>
>
> Ross Finlayson schrieb:
>> Hm.  "squirrel:brain::walnut:cerebrum".

On 2/22/2024 3:03 AM, Mild Shock wrote:
> Ross Finlayson schrieb:

>> [...]
>
> Seriously, you don't know what classes are?
>
> The membership relation is the same
> for members of classes and for members of sets.
> Since members of classes are sets just like
>
> the members of sets are sets, in ZF. And there is
> only one membership relation ∈ between sets. The
> distinction between classes and sets was described
>
> in the past as:
>
> sets: includes collections of sizes from the numbers to
>   the transfinite numbers
> classes: includes collections that Cantor called
>   NCONSISTENT MULTIPLICITIES

I once had a lecturer in math who would refer to
proper classes as "syntactic sugar",
which I found out today is a _programming_ term.

https://en.wikipedia.org/wiki/Syntactic_sugar
| In computer science, syntactic sugar is syntax within
| a programming language that is designed to make things
| easier to read or to express. It makes the language
| "sweeter" for human use: things can be expressed
| more clearly, more concisely, or in an alternative style
| that some may prefer.

I take that to mean that
| α ∈ ORD
is a "sweeter" way to say something like
| α is a regular transitive set of transitive sets
| Reg(α) ∧ Trans(α) ∧ ∀β ∈ α: Trans(β)

Reg(α) ⟺
α ≠ ∅ ⟹ ∃β ∈ α: β∩α = ∅

Trans(α) ⟺
∀β,γ: γ ∈ β ∧ β ∈ α ⟹ γ ∈ α

On the one hand,
questions surrounding the existence of ORD
have a lower bar to clear,
but
some things which a set can do
proper class ORD can't do.

Or that is my impression.
Caveat lector.