On 02/19/2024 12:14 PM, Mild Shock wrote:
>
> Whats the strategy for writing such nonsense as below?
>

(That sort of mercurial doffed-and-donned presumed jocularity and
familiarity is about the shallowest, vainest, fakest poser's.
That sort of inconstancy isn't "making friends and influencing people",
it's "give 'em nothing to depend on and keep 'em guessing".
It's the most obvious sort of example of a "manipulator",
which is considered a particular variety of pathological.)

Try some sincerety sometime.

> What are products of omega? How are paradoxes sets?
>
> LoL
>
> Ross A. Finlayson schrieb am Samstag, 19. April 2014 um 21:18:08 UTC+2:
>> On 4/19/2014 12:50 AM, William Elliot wrote:
>>> Does the set of all ordinals exist within ZF?
>>>
>> This is "Ord", a collection of all ordinals (from among their
>> representations). The paradox of Cesare Burali-Forti is that
>> structurally, where membership is used to model order, the
>> collection itself of the ordinals would be an ordinal, thus
>> including itself. A "paradox" is not a set in ZF.
>>
>> Then there are set theories where it is a set, but those set
>> theories have anti-foundational infinities as a natural consequence
>> of definition. Russell has these kinds of sets as "extra-ordinary"
>> for ordinary.
>>
>> foundational / anti-foundational
>> regular / irregular
>> well-founded / non-well-founded
>> ordinary / extra-ordinary
>>
>>
>> These are about the same.
>>
>> There are roundabout arguments that, for example, the finite ordinals,
>> as a set, consequently contain themselves, as an element. This is a
>> direct compactness result.
>>
>> ZF defines omega as a constant thus that omega and its products are
>> well-founded.

You mean "Russell lied to you and you bought it",
"Russell's retro-thesis", "Russell's fools"?

ORD, is the order type of ordinals, it's among
maximal elements and fixed points and universals.

It's not non-sense indeed the opposite.

My slates for uncountability and paradox,
help itemize how ordinals and sets are together.
(In a theory sets for ordinal relation, uncountability,
then a theory of sets with universes, paradox.)

(There's a theory of "ubiquitous ordinals" among
all the primordial objects of mathematics a theory
of them.)

If you study Cohen's "Independence of the Continuum Hypothesis",
right about at the end he introduces a deft consequence of ordinals,
and leaves set theory open about the Continuum Hypothesis.

In case you missed it, ....

It's pure theory, all theory.

It's called foundations, maybe you want to know it.

"Conservation of truth", all there is to it.

On 02/19/2024 02:03 PM, Ross Finlayson wrote:
> On 02/19/2024 12:14 PM, Mild Shock wrote:
>>
>> Whats the strategy for writing such nonsense as below?
>>
>
>
> (That sort of mercurial doffed-and-donned presumed jocularity and
> familiarity is about the shallowest, vainest, fakest poser's.
> That sort of inconstancy isn't "making friends and influencing people",
> it's "give 'em nothing to depend on and keep 'em guessing".
> It's the most obvious sort of example of a "manipulator",
> which is considered a particular variety of pathological.)
>
> Try some sincerety sometime.
>
>> What are products of omega? How are paradoxes sets?
>>
>> LoL
>>
>> Ross A. Finlayson schrieb am Samstag, 19. April 2014 um 21:18:08 UTC+2:
>>> On 4/19/2014 12:50 AM, William Elliot wrote:
>>>> Does the set of all ordinals exist within ZF?
>>>>
>>> This is "Ord", a collection of all ordinals (from among their
>>> representations). The paradox of Cesare Burali-Forti is that
>>> structurally, where membership is used to model order, the
>>> collection itself of the ordinals would be an ordinal, thus
>>> including itself. A "paradox" is not a set in ZF.
>>>
>>> Then there are set theories where it is a set, but those set
>>> theories have anti-foundational infinities as a natural consequence
>>> of definition. Russell has these kinds of sets as "extra-ordinary"
>>> for ordinary.
>>>
>>> foundational / anti-foundational
>>> regular / irregular
>>> well-founded / non-well-founded
>>> ordinary / extra-ordinary
>>>
>>>
>>> These are about the same.
>>>
>>> There are roundabout arguments that, for example, the finite ordinals,
>>> as a set, consequently contain themselves, as an element. This is a
>>> direct compactness result.
>>>
>>> ZF defines omega as a constant thus that omega and its products are
>>> well-founded.
>
> You mean "Russell lied to you and you bought it",
> "Russell's retro-thesis", "Russell's fools"?
>
> ORD, is the order type of ordinals, it's among
> maximal elements and fixed points and universals.
>
> It's not non-sense indeed the opposite.
>
> My slates for uncountability and paradox,
> help itemize how ordinals and sets are together.
> (In a theory sets for ordinal relation, uncountability,
> then a theory of sets with universes, paradox.)
>
> (There's a theory of "ubiquitous ordinals" among
> all the primordial objects of mathematics a theory
> of them.)
>
> If you study Cohen's "Independence of the Continuum Hypothesis",
> right about at the end he introduces a deft consequence of ordinals,
> and leaves set theory open about the Continuum Hypothesis.
>
> In case you missed it, ....
>
>
> It's pure theory, all theory.
>
> It's called foundations, maybe you want to know it.
>
> "Conservation of truth", all there is to it.
>
>
>

(Maybe that's just me.)

On 02/19/2024 05:01 PM, Mild Shock wrote:
> The contradiction is very easy:
>
> Lets say X is the set of all finite ordinals.
>
> - observe that X is an infinite ordinal.
> - observe that if Y in X, then Y is a finite ordinal.
> - hence if X in X it would be an infinite and finite ordinal at the same time.
> - an X cannot be infinite and finite at the same time.
> Q.E.D:
>
> Ross Finlayson schrieb am Dienstag, 20. Februar 2024 um 00:04:06 UTC+1:
>>>>> There are roundabout arguments that, for example, the finite ordinals,
>>>>> as a set, consequently contain themselves, as an element. This is a
>>>>> direct compactness result.
>> (Maybe that's just me.)

Imagine if ordinals' proper model was that the successor
was powerset, instead of just any old ordered pair.

So, those together are the "sets that don't contain themselves",
the sets of ordinals.

Quantifying over those, results the "Russell set the ordinal",
it contains itself.

So here Y isn't necessarily a finite ordinal.

Q.E.R.

You only make it worse!

> There are roundabout arguments that, for example,
> the FINITE ORDINALS, as a set, consequently contain
> themselves, as an element. This is a direct
> compactness result.

If you want to have ordinals that contain themselves,
you need to mention an encoding. Because per se,
we understand by ordinal an order type.

There ware various encodings for finite ordinals around:
1) von Neuman encoding, based on succ(X) = X u {X} and 0 = {}
2) Zermelo encoding, bsaed on succ(X) = {X} and 0 = {}
3) Your Powerset idea, based on succ(X) = P(X) and 0 = {}

All 3 have the property that:

/* provable */
n in n+1 and n is finite

Proof:
case 1): n+1 = n u {n}, n in n+1 because n in {n}.
further succ(X) sendes an already finite set into a finite set.
case 2): n+1 = {n}, n in n+1 because n in {n}.
further succ(X) sendes an already finite set into a finite set.
case 3): n+1 = P(n), n in n+1 because n in P(n).
further succ(X) sendes an already finite set into a finite set.
Q.E.D.

But none has the property that omega = { n } contains
itself, the proof of contradiction applies irrelevant
of the encoding, it only makes use of the

notion finite and infinite:

/* provable */
~(omega in omega) & (Y in omega => Y finite)

Proof:
(Y in omega => Y finite) follows by the claim that
omega = { n }, i.e. the least set that contains all finite
ordinals in the corresponding encoding. If it would
contain something infinite it would not be the least

set that contains all finite ordinals, would have some
extra in it. Violating the very construction of omega from
the finite ordinals.
Q.E.D.

Ross Finlayson schrieb:
> On 02/19/2024 05:01 PM, Mild Shock wrote:
>> The contradiction is very easy:
>>
>> Lets say X is the set of all finite ordinals.
>>
>> - observe that X is an infinite ordinal.
>> - observe that if Y in X, then Y is a finite ordinal.
>> - hence if X in X it would be an infinite and finite ordinal at the
>> same time.
>> - an X cannot be infinite and finite at the same time.
>> Q.E.D:
>>
>> Ross Finlayson schrieb am Dienstag, 20. Februar 2024 um 00:04:06 UTC+1:
>>>>>> There are roundabout arguments that, for example, the finite
>>>>>> ordinals,
>>>>>> as a set, consequently contain themselves, as an element. This is a
>>>>>> direct compactness result.
>>> (Maybe that's just me.)
>
> Imagine if ordinals' proper model was that the successor
> was powerset, instead of just any old ordered pair.
>
> So, those together are the "sets that don't contain themselves",
> the sets of ordinals.
>
> Quantifying over those, results the "Russell set the ordinal",
> it contains itself.
>
> So here Y isn't necessarily a finite ordinal.
>
> Q.E.R.
>
>

You could use an encoding of finite ordinals
into infinite objects, like:

0 = omega, 1 = omega+1, etc..

Then my proof doesn't work so easily. You can then
use the regularity axiom, to show:

/* provable */
~(omega in omega)

Axiom of regularity
https://en.wikipedia.org/wiki/Axiom_of_regularity

Is this your A "paradox" is not a set in ZF?
In non-ZF you could aim at making omega a Quine atom:
https://en.wikipedia.org/wiki/Urelement#Quine_atoms

Or any other construction and encoding where you
would sneak in a set into itself.

Mild Shock schrieb:
> You only make it worse!
>
> > There are roundabout arguments that, for example,
> > the FINITE ORDINALS, as a set, consequently contain
> > themselves, as an element. This is a direct
> > compactness result.
>
> If you want to have ordinals that contain themselves,
> you need to mention an encoding. Because per se,
> we understand by ordinal an order type.
>
> There ware various encodings for finite ordinals around:
> 1) von Neuman encoding, based on succ(X) = X u {X} and 0 = {}
> 2) Zermelo encoding, bsaed on succ(X) = {X} and 0 = {}
> 3) Your Powerset idea, based on succ(X) = P(X) and 0 = {}
>
> All 3 have the property that:
>
> /* provable */
> n in n+1 and n is finite
>
> Proof:
> case 1): n+1 = n u {n}, n in n+1 because n in {n}.
> further succ(X) sendes an already finite set into a finite set.
> case 2): n+1 = {n}, n in n+1 because n in {n}.
> further succ(X) sendes an already finite set into a finite set.
> case 3): n+1 = P(n), n in n+1 because n in P(n).
> further succ(X) sendes an already finite set into a finite set.
> Q.E.D.
>
> But none has the property that omega = { n } contains
> itself, the proof of contradiction applies irrelevant
> of the encoding, it only makes use of the
>
> notion finite and infinite:
>
> /* provable */
> ~(omega in omega) & (Y in omega => Y finite)
>
> Proof:
> (Y in omega => Y finite) follows by the claim that
> omega = { n }, i.e. the least set that contains all finite
> ordinals in the corresponding encoding. If it would
> contain something infinite it would not be the least
>
> set that contains all finite ordinals, would have some
> extra in it. Violating the very construction of omega from
> the finite ordinals.
> Q.E.D.
>
>
> Ross Finlayson schrieb:
>> On 02/19/2024 05:01 PM, Mild Shock wrote:
>>> The contradiction is very easy:
>>>
>>> Lets say X is the set of all finite ordinals.
>>>
>>> - observe that X is an infinite ordinal.
>>> - observe that if Y in X, then Y is a finite ordinal.
>>> - hence if X in X it would be an infinite and finite ordinal at the
>>> same time.
>>> - an X cannot be infinite and finite at the same time.
>>> Q.E.D:
>>>
>>> Ross Finlayson schrieb am Dienstag, 20. Februar 2024 um 00:04:06 UTC+1:
>>>>>>> There are roundabout arguments that, for example, the finite
>>>>>>> ordinals,
>>>>>>> as a set, consequently contain themselves, as an element. This is a
>>>>>>> direct compactness result.
>>>> (Maybe that's just me.)
>>
>> Imagine if ordinals' proper model was that the successor
>> was powerset, instead of just any old ordered pair.
>>
>> So, those together are the "sets that don't contain themselves",
>> the sets of ordinals.
>>
>> Quantifying over those, results the "Russell set the ordinal",
>> it contains itself.
>>
>> So here Y isn't necessarily a finite ordinal.
>>
>> Q.E.R.
>>
>>
>

On 02/19/2024 11:57 PM, Mild Shock wrote:
> You could use an encoding of finite ordinals
> into infinite objects, like:
>
> 0 = omega, 1 = omega+1, etc..
>
> Then my proof doesn't work so easily. You can then
> use the regularity axiom, to show:
>
> /* provable */
> ~(omega in omega)
>
> Axiom of regularity
> https://en.wikipedia.org/wiki/Axiom_of_regularity
>
> Is this your A "paradox" is not a set in ZF?
> In non-ZF you could aim at making omega a Quine atom:
> https://en.wikipedia.org/wiki/Urelement#Quine_atoms
>
> Or any other construction and encoding where you
> would sneak in a set into itself.
>
> Mild Shock schrieb:
>> You only make it worse!
>>
>> > There are roundabout arguments that, for example,
>> > the FINITE ORDINALS, as a set, consequently contain
>> > themselves, as an element. This is a direct
>> > compactness result.
>>
>> If you want to have ordinals that contain themselves,
>> you need to mention an encoding. Because per se,
>> we understand by ordinal an order type.
>>
>> There ware various encodings for finite ordinals around:
>> 1) von Neuman encoding, based on succ(X) = X u {X} and 0 = {}
>> 2) Zermelo encoding, bsaed on succ(X) = {X} and 0 = {}
>> 3) Your Powerset idea, based on succ(X) = P(X) and 0 = {}
>>
>> All 3 have the property that:
>>
>> /* provable */
>> n in n+1 and n is finite
>>
>> Proof:
>> case 1): n+1 = n u {n}, n in n+1 because n in {n}.
>> further succ(X) sendes an already finite set into a finite set.
>> case 2): n+1 = {n}, n in n+1 because n in {n}.
>> further succ(X) sendes an already finite set into a finite set.
>> case 3): n+1 = P(n), n in n+1 because n in P(n).
>> further succ(X) sendes an already finite set into a finite set.
>> Q.E.D.
>>
>> But none has the property that omega = { n } contains
>> itself, the proof of contradiction applies irrelevant
>> of the encoding, it only makes use of the
>>
>> notion finite and infinite:
>>
>> /* provable */
>> ~(omega in omega) & (Y in omega => Y finite)
>>
>> Proof:
>> (Y in omega => Y finite) follows by the claim that
>> omega = { n }, i.e. the least set that contains all finite
>> ordinals in the corresponding encoding. If it would
>> contain something infinite it would not be the least
>>
>> set that contains all finite ordinals, would have some
>> extra in it. Violating the very construction of omega from
>> the finite ordinals.
>> Q.E.D.
>>
>>
>> Ross Finlayson schrieb:
>>> On 02/19/2024 05:01 PM, Mild Shock wrote:
>>>> The contradiction is very easy:
>>>>
>>>> Lets say X is the set of all finite ordinals.
>>>>
>>>> - observe that X is an infinite ordinal.
>>>> - observe that if Y in X, then Y is a finite ordinal.
>>>> - hence if X in X it would be an infinite and finite ordinal at the
>>>> same time.
>>>> - an X cannot be infinite and finite at the same time.
>>>> Q.E.D:
>>>>
>>>> Ross Finlayson schrieb am Dienstag, 20. Februar 2024 um 00:04:06 UTC+1:
>>>>>>>> There are roundabout arguments that, for example, the finite
>>>>>>>> ordinals,
>>>>>>>> as a set, consequently contain themselves, as an element. This is a
>>>>>>>> direct compactness result.
>>>>> (Maybe that's just me.)
>>>
>>> Imagine if ordinals' proper model was that the successor
>>> was powerset, instead of just any old ordered pair.
>>>
>>> So, those together are the "sets that don't contain themselves",
>>> the sets of ordinals.
>>>
>>> Quantifying over those, results the "Russell set the ordinal",
>>> it contains itself.
>>>
>>> So here Y isn't necessarily a finite ordinal.
>>>
>>> Q.E.R.
>>>
>>>
>>
>

Thanks for writing, as you explore the issues involved with
quantification about ordinals and sets, it helps clarify
that "set theory" and "ordinal theory" are two different
theories, where being fundamental, each gets a very direct
model of the other in the respective theories.

Then, where "ordering theory" is about orderings, kind of
like category theory relating [0,1] to things as by functions,
that it's a function theory, ordering theory, here is that ordinal
theory is like set theory. There are "arithmetizations" of any thing
as there are "set-like associativities" of any thing, that's
the descriptive theory.

These kinds of ideas then get into that there is a theory of
mathematical objects, and these objects are same, whether
ordinals or sets (or parts, or differences, or otherwise
fundamental relations of utterly simple character that in
their classifications, effect relations, of other mathematical
objects of other mathematical object's theories, all in one theory.

So, when you look at something like Cohen's Independence of
the Continuum Hypothesis, it's very telling that it's a result
in ordinals, about cardinals, or here vice-versa.

You may be on the way to learning something.

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

(... That function theory effects "relations" that logic is
a theory of relations.)

On 02/20/2024 11:23 AM, Mild Shock wrote:
>
> To do some of Cohens work, you first have to accept
> the Skolem Paradox, i.e. that ZF has countable models.
>
> The Skolem Paradox is the thing that shattered shock
> waves through Mückenheims brain, what does it do to
>
> Rossy Boys brain? Oh, I forget Rossy Boy has no brain...
>
> https://math.stackexchange.com/a/4027015
>
> Mild Shock schrieb am Dienstag, 20. Februar 2024 um 20:10:53 UTC+1:
>> Ordinals and Sets were developed hand in hand by Cantor
>> and Zermelo. But quasi, William Elliot, Peter Percival and
>> Jim Burns did already most of the explanations.
>>
>> The problem starts with innocent formulations such as your:
>>> There are roundabout arguments that, for example, the finite ordinals,
>>> as a set, consequently contain themselves, as an element. This is a
>>> direct compactness result.
>> So you want to form a set of ordinals, the finite ones. Before
>> set theory ordinals were order types. This means they were
>> equivalence classes. Already the equivalence class of the
>>
>> ordinal 1, is too big to be a set. Since it basically contains
>> all singleton sets {X}. And if you project the singleton you
>> get the universal class, which we know since Russell,
>>
>> and even proved by Dan-O-Matik, isn't set.So you form
>> a collection of classes, sometimes called a conglomerate,
>> but you talk about it about as if it were a set.
>>
>> So how can you make it a set? Well here is the receipt:
>>
>> - Step 1: Start talking about numbers and transfinite numbers
>> Cantor 1895
>> - Step 2: Start mapping numbers [and transfinite numbers] to sets
>> Zermelo 1908
>>
>> This was refined by von Neuman. Which gives the most useful
>> encoding of ordinals. Unless you want to go with Dana Scotts
>> trick. von Neuman ordinals not only have the property that
>>
>> they are well ordered sets, their well ordering is the set
>> membership itself, they are hereditarily transitive sets.
>> You can construct inner models.
>> Ross Finlayson schrieb am Dienstag, 20. Februar 2024 um 18:53:56 UTC+1:
>>> Thanks for writing, as you explore the issues involved with
>>> quantification about ordinals and sets, it helps clarify
>>> that "set theory" and "ordinal theory" are two different
>>> theories, where being fundamental, each gets a very direct
>>> model of the other in the respective theories.

Consider about "sets in ordinals" instead of
"ordinals in sets". There's a usual idea then
that ordinals besides reflecting an inductive set,
and a well-ordering in any mapping from them,
also that the whole numbers or "number theory's
numbers", so sit with them.

So, consider prime numbers with unique
factorization, and also a subset of prime
numbers that only have at most one power
of each prime factor. This is for the "fundamental
theorem of arithmetic", numbers (natural integers,
with regards to the cases for 0 and 1), that
numbers have unique prime factorizations,
and, of those, some have unique instances
of factors.

numbers: 1 2 3 4 5 6 ...
primes: 2 3 5 7 11 13 ...
composite: 4 6 8 9 10 12 14 15 ...
"uniposite": 2 3 5 6 7 10 11 13 ...

So, when modeling "sets", in "numbers",
there is a default model giving each element
in the universe of sets or the domain of discourse,
a prime number assignment, then a given set,
is just the multiple of those in these "uniposites",
where for example prime numbers generally
model "multi-sets", quite directly.

Now, I don't know too much who talks about
"primes and composites with only unique
factors in their decomposition, 'uniposites'",
but it would be interesting to really that a
very natural model of this sort _arithmetization_,
of sets, is exhibited _naturally_ because the
usual operation of union is taking the product
of the numbers and the usual operation of
membership is a divisibility test, and these kinds
of things.

union: product (least common multiple)
membership: divisibility test
intersection: greatest common denominator
disjoint: dividing out members

So, this introduces the usual notion of
"arithmetization", that then the operations
of arithmetic implement the set-theoretic
operations, then for such notions as
"geometrization", the sort of continuous analog,
of "arithmetization".

This sort of "composite numbers are a natural model
of a multi-set", can go a long way helping show that
the fundamental relations model and model and
model each other again, helping show why and how
it's simple that "resources in relations" establish
their orders, ..., of complexity in relation in type.

Here it's introduced the utility of
"uniposites: a sub-class of numbers
whose unique prime factorization has
unique elements".

2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47,...

https
oeis.org/search?q=2%2C3%2C5%2C6%2C7%2C10%2C11%2C13%2C14%2C15%2C17%2C19%2C21%2C23%2C26%2C29%2C31%2C33%2C34%2C35%2C37%2C39%2C41%2C42%2C43%2C46%2C47&language=english&go=Search

https
oeis.org/search?q=2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47&language=english&go=Search

Hmm, these aren't in the Online Encyclopedia of Integer Sequences?

https
oeis.org/A000040
https
oeis.org/A002808

The idea is simple that multi-sets are modeled by numbers ,
and, sets modeled by these "uniposite", numbers. Surely,
or rather, Shirley almost certainly, these already are,
"known".

A most usual sort of modeling a set as a number
is a "bit-map", for a word as wide as
the domain-of-discourse.

On 02/20/2024 11:23 AM, Mild Shock wrote:
>
> To do some of Cohens work, you first have to accept
> the Skolem Paradox, i.e. that ZF has countable models.
>
> The Skolem Paradox is the thing that shattered shock
> waves through Mückenheims brain, what does it do to
>
> Rossy Boys brain? Oh, I forget Rossy Boy has no brain...
>
> https://math.stackexchange.com/a/4027015
>
> Mild Shock schrieb am Dienstag, 20. Februar 2024 um 20:10:53 UTC+1:
>> Ordinals and Sets were developed hand in hand by Cantor
>> and Zermelo. But quasi, William Elliot, Peter Percival and
>> Jim Burns did already most of the explanations.
>>
>> The problem starts with innocent formulations such as your:
>>> There are roundabout arguments that, for example, the finite ordinals,
>>> as a set, consequently contain themselves, as an element. This is a
>>> direct compactness result.
>> So you want to form a set of ordinals, the finite ones. Before
>> set theory ordinals were order types. This means they were
>> equivalence classes. Already the equivalence class of the
>>
>> ordinal 1, is too big to be a set. Since it basically contains
>> all singleton sets {X}. And if you project the singleton you
>> get the universal class, which we know since Russell,
>>
>> and even proved by Dan-O-Matik, isn't set.So you form
>> a collection of classes, sometimes called a conglomerate,
>> but you talk about it about as if it were a set.
>>
>> So how can you make it a set? Well here is the receipt:
>>
>> - Step 1: Start talking about numbers and transfinite numbers
>> Cantor 1895
>> - Step 2: Start mapping numbers [and transfinite numbers] to sets
>> Zermelo 1908
>>
>> This was refined by von Neuman. Which gives the most useful
>> encoding of ordinals. Unless you want to go with Dana Scotts
>> trick. von Neuman ordinals not only have the property that
>>
>> they are well ordered sets, their well ordering is the set
>> membership itself, they are hereditarily transitive sets.
>> You can construct inner models.
>> Ross Finlayson schrieb am Dienstag, 20. Februar 2024 um 18:53:56 UTC+1:
>>> Thanks for writing, as you explore the issues involved with
>>> quantification about ordinals and sets, it helps clarify
>>> that "set theory" and "ordinal theory" are two different
>>> theories, where being fundamental, each gets a very direct
>>> model of the other in the respective theories.

Actually the Paul Cohen's "Independence of the Continuum,
Hypothesis, 1 and 2" that I read was from the original as
I found a copy on the National Institute of Health's web-page,
so, I read it from there, instead of taking it second-hand
from a conservative crowd of reputation-mongers.

So, when you read it, at the very end, it's like,
"surprise: ordinal's bigger".

On sci.logic one time there's a thread called
"Few questions on forcing, large cardinals".

https
groups.google.com/g/sci.logic/c/sIvO0bJ7gPY/m/VBUICn3tBAAJ

It appears that what the Burse-a-tron emitted on 1/24/2020
was "That's quite amazing!".

So, anyways, about Cohen and "Independence of the Continuum
Hypothesis", it's not necessarily easy to find a copy, but,
you want it through your own lens.

Here's a few more through mine:

https
groups.google.com/g/sci.logic/search?q=Cohen%20author%3AFinlayson

There are a few more on sci.math, alzo.

On 02/20/2024 12:13 PM, Ross Finlayson wrote:
> On 02/20/2024 11:23 AM, Mild Shock wrote:
>>
>> To do some of Cohens work, you first have to accept
>> the Skolem Paradox, i.e. that ZF has countable models.
>>
>> The Skolem Paradox is the thing that shattered shock
>> waves through Mückenheims brain, what does it do to
>>
>> Rossy Boys brain? Oh, I forget Rossy Boy has no brain...
>>
>> https://math.stackexchange.com/a/4027015
>>
>> Mild Shock schrieb am Dienstag, 20. Februar 2024 um 20:10:53 UTC+1:
>>> Ordinals and Sets were developed hand in hand by Cantor
>>> and Zermelo. But quasi, William Elliot, Peter Percival and
>>> Jim Burns did already most of the explanations.
>>>
>>> The problem starts with innocent formulations such as your:
>>>> There are roundabout arguments that, for example, the finite ordinals,
>>>> as a set, consequently contain themselves, as an element. This is a
>>>> direct compactness result.
>>> So you want to form a set of ordinals, the finite ones. Before
>>> set theory ordinals were order types. This means they were
>>> equivalence classes. Already the equivalence class of the
>>>
>>> ordinal 1, is too big to be a set. Since it basically contains
>>> all singleton sets {X}. And if you project the singleton you
>>> get the universal class, which we know since Russell,
>>>
>>> and even proved by Dan-O-Matik, isn't set.So you form
>>> a collection of classes, sometimes called a conglomerate,
>>> but you talk about it about as if it were a set.
>>>
>>> So how can you make it a set? Well here is the receipt:
>>>
>>> - Step 1: Start talking about numbers and transfinite numbers
>>> Cantor 1895
>>> - Step 2: Start mapping numbers [and transfinite numbers] to sets
>>> Zermelo 1908
>>>
>>> This was refined by von Neuman. Which gives the most useful
>>> encoding of ordinals. Unless you want to go with Dana Scotts
>>> trick. von Neuman ordinals not only have the property that
>>>
>>> they are well ordered sets, their well ordering is the set
>>> membership itself, they are hereditarily transitive sets.
>>> You can construct inner models.
>>> Ross Finlayson schrieb am Dienstag, 20. Februar 2024 um 18:53:56 UTC+1:
>>>> Thanks for writing, as you explore the issues involved with
>>>> quantification about ordinals and sets, it helps clarify
>>>> that "set theory" and "ordinal theory" are two different
>>>> theories, where being fundamental, each gets a very direct
>>>> model of the other in the respective theories.
>
> Consider about "sets in ordinals" instead of
> "ordinals in sets". There's a usual idea then
> that ordinals besides reflecting an inductive set,
> and a well-ordering in any mapping from them,
> also that the whole numbers or "number theory's
> numbers", so sit with them.
>
> So, consider prime numbers with unique
> factorization, and also a subset of prime
> numbers that only have at most one power
> of each prime factor. This is for the "fundamental
> theorem of arithmetic", numbers (natural integers,
> with regards to the cases for 0 and 1), that
> numbers have unique prime factorizations,
> and, of those, some have unique instances
> of factors.
>
> numbers: 1 2 3 4 5 6 ...
> primes: 2 3 5 7 11 13 ...
> composite: 4 6 8 9 10 12 14 15 ...
> "uniposite": 2 3 5 6 7 10 11 13 ...
>
> So, when modeling "sets", in "numbers",
> there is a default model giving each element
> in the universe of sets or the domain of discourse,
> a prime number assignment, then a given set,
> is just the multiple of those in these "uniposites",
> where for example prime numbers generally
> model "multi-sets", quite directly.
>
> Now, I don't know too much who talks about
> "primes and composites with only unique
> factors in their decomposition, 'uniposites'",
> but it would be interesting to really that a
> very natural model of this sort _arithmetization_,
> of sets, is exhibited _naturally_ because the
> usual operation of union is taking the product
> of the numbers and the usual operation of
> membership is a divisibility test, and these kinds
> of things.
>
> union: product (least common multiple)
> membership: divisibility test
> intersection: greatest common denominator
> disjoint: dividing out members
>
> So, this introduces the usual notion of
> "arithmetization", that then the operations
> of arithmetic implement the set-theoretic
> operations, then for such notions as
> "geometrization", the sort of continuous analog,
> of "arithmetization".
>
>
> This sort of "composite numbers are a natural model
> of a multi-set", can go a long way helping show that
> the fundamental relations model and model and
> model each other again, helping show why and how
> it's simple that "resources in relations" establish
> their orders, ..., of complexity in relation in type.
>
> Here it's introduced the utility of
> "uniposites: a sub-class of numbers
> whose unique prime factorization has
> unique elements".
>
> 2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47,...
>
>
> https
> oeis.org/search?q=2%2C3%2C5%2C6%2C7%2C10%2C11%2C13%2C14%2C15%2C17%2C19%2C21%2C23%2C26%2C29%2C31%2C33%2C34%2C35%2C37%2C39%2C41%2C42%2C43%2C46%2C47&language=english&go=Search
>
>
> https
> oeis.org/search?q=2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47&language=english&go=Search
>
>
> Hmm, these aren't in the Online Encyclopedia of Integer Sequences?
>
> https
> oeis.org/A000040
> https
> oeis.org/A002808
>
> The idea is simple that multi-sets are modeled by numbers ,
> and, sets modeled by these "uniposite", numbers. Surely,
> or rather, Shirley almost certainly, these already are,
> "known".
>
> A most usual sort of modeling a set as a number
> is a "bit-map", for a word as wide as
> the domain-of-discourse.
>

Numbers with max(multiplicity(prime-factor)) = 1

> Here it's introduced the utility of
> "uniposites: a sub-class of numbers
> whose unique prime factorization has
> unique elements".
>
>
2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47,...

>
>
> https
>
oeis.org/search?q=2%2C3%2C5%2C6%2C7%2C10%2C11%2C13%2C14%2C15%2C17%2C19%2C21%2C23%2C26%2C29%2C31%2C33%2C34%2C35%2C37%2C39%2C41%2C42%2C43%2C46%2C47&language=english&go=Search

>
>
> https
>
oeis.org/search?q=2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47&language=english&go=Search

>
>
> Hmm, these aren't in the Online Encyclopedia of Integer Sequences?
>
> https
> oeis.org/A000040
> https
> oeis.org/A002808

https
oeis.org/wiki/Prime_factors

"The arithmetic function omega(n) represents
the number of distinct prime factors of n

omega(n) = Sigma_i=1^omega(n) alpha_i^0 = Sigma_i=1^omega(n) 1

if you forgive the tautological expression."

https
en.wikipedia.org/wiki/Table_of_prime_factors

"A powerful number (also called squareful) has multiplicity
above 1 for all prime factors."

Ah, "square-free integers". "A square-free integer has no
prime factor with multiplicity above 1".

https
oeis.org/A005117

So, the square-free numbers are natural representatives of
subsets in arithmetic, of natural representatives of powersets
in arithmetic, where elements have natural representations
as prime numbers, and the set of all of them is called the
"primorial" which is like "factorial" for the first n-many primes.

(Here this is "square-free numbers excluding 1".)

2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47,...

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30,
31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47

Ah, looks I omitted 22 and 38, and 30.

Yes, of course square-free numbers are very well-known.

https
mathoverflow.net/questions/16098/complexity-of-testing-integer-square-freeness

It reminds me of "Digit Summation Congruence", which is
a method for machine numbers that rapidly tests divisibility
using binary arithmetic, for various prime factors.