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# Ordinals

138 views

### William Elliot

Apr 19, 2014, 3:50:34 AM4/19/14
to
Does the set of all ordinals exist within ZF?

### quasi

Apr 19, 2014, 3:59:24 AM4/19/14
to
William Elliot wrote:
>
>Does the set of all ordinals exist within ZF?

It's too big to be a set.

quasi

### quasi

Apr 19, 2014, 4:11:25 AM4/19/14
to
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

quasi

### Peter Percival

Apr 19, 2014, 5:51:35 AM4/19/14
to
Burali-Forti rather than Russell. If there was a set of all ordinals it
would be an ordinal and thus a member of itself, and thus greater than
itself.

--
...if someone seduced my daughter it would be damaging and horrifying
but not fatal. She would recover, marry and have lots of children...
On the other hand, if some elderly, or not so elderly, schoolmaster
seduced one of my sons and taught him to be a homosexual, he would ruin
him for life. That is the fundamental distinction. -- Lord Longford

### Peter Percival

Apr 19, 2014, 5:55:48 AM4/19/14
to
William Elliot wrote:
> Does the set of all ordinals exist within ZF?
>
No. In ZF there is a sort of counterpart of NBG's classes: consider the
formula with one free variable x, phi(x), that says "x is an ordinal",
then phi(x) can be treated in some ways as the class (it's a proper
class, not a set) of all ordinals. Also, if you're working with
ordinals, you might prefer ZFC to ZF.

### Peter Percival

Apr 19, 2014, 6:09:47 AM4/19/14
to
Peter Percival wrote:
> William Elliot wrote:
>> Does the set of all ordinals exist within ZF?
>>
> No. In ZF there is a sort of counterpart of NBG's classes: consider the
> formula with one free variable x, phi(x), that says "x is an ordinal",

"x is an ordinal" is (according to von Neumann)

"x is transitive" & (Uy,z in x)(y=x \/ y in x \/ x in y)

"x is transitive" is (Uy in x)(y subset x)

### Jim Burns

Apr 19, 2014, 7:18:23 AM4/19/14
to
On 4/19/2014 5:55 AM, Peter Percival wrote:
> William Elliot wrote:

>> Does the set of all ordinals exist within ZF?
>>
> No. In ZF there is a sort of counterpart of NBG's classes:
> consider the formula with one free variable x, phi(x),
> that says "x is an ordinal", then phi(x) can be treated
> in some ways as the class (it's a proper class, not a set)
> of all ordinals.

What is it that distinguishes sets from formulae?

Set notation, of course -- but that seems unimportant
and easily fixable without changing anything fundamental
in ZF.

And also sets can be elements of sets or proper classes.
What distinguishes "being an element of this set" from
"satisfying this formula"?

Speculating here, sets can act like both a formula itself
and like an object of a formula, of a variable within a
formula, the same set at the same time -- by which I mean,
contain and be contained.

This level-jumping can be done without sets, Goedel showed
us that, but sets do it promiscuously.

> Also, if you're working with ordinals,
> you might prefer ZFC to ZF.

Why would that be? Aren't ordinals well-ordered?

I would expect (without much thought) that ZF would be
fine since, if we are restricting our attention to
ordinals, we already have Choice as a theorem, call it
Choice Over Ordinals.

### Peter Percival

Apr 19, 2014, 7:26:29 AM4/19/14
to
Yes. I was just thinking that assuming choice might simplify things.

> I would expect (without much thought) that ZF would be
> fine since, if we are restricting our attention to
> ordinals, we already have Choice as a theorem, call it
> Choice Over Ordinals.
>
>

### zuhair

Apr 19, 2014, 1:46:06 PM4/19/14
to
No

### Ross A. Finlayson

Apr 19, 2014, 3:18:08 PM4/19/14
to
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

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.

### William Elliot

Apr 19, 2014, 10:27:09 PM4/19/14
to
On Sat, 19 Apr 2014, Peter Percival wrote:
> William Elliot wrote:
> > Does the set of all ordinals exist within ZF?
> >
> No. In ZF there is a sort of counterpart of NBG's classes: consider the
> formula with one free variable x, phi(x), that says "x is an ordinal", then
> phi(x) can be treated in some ways as the class (it's a proper class, not a
> set) of all ordinals.

By transfinite induction is it possible (without AxC) to define
a set A_xi for every ordinal xi? With that can one proclaim
by replacement, that \/{ A_xi | xi an ordinal } is a set?

> Also, if you're working with ordinals, you might prefer ZFC to ZF.

Not if you're proving propositions equivalent to AxC.

### Peter Percival

Apr 20, 2014, 3:12:55 AM4/20/14
to
William Elliot wrote:
> On Sat, 19 Apr 2014, Peter Percival wrote:
>> William Elliot wrote:
>>> Does the set of all ordinals exist within ZF?
>>>
>> No. In ZF there is a sort of counterpart of NBG's classes: consider the
>> formula with one free variable x, phi(x), that says "x is an ordinal", then
>> phi(x) can be treated in some ways as the class (it's a proper class, not a
>> set) of all ordinals.
>
> By transfinite induction is it possible (without AxC) to define
> a set A_xi for every ordinal xi? With that can one proclaim

Yes, in ZF every ordinal is a set, so let A_xi be xi.

> by replacement, that \/{ A_xi | xi an ordinal } is a set?
>
>> Also, if you're working with ordinals, you might prefer ZFC to ZF.
>
> Not if you're proving propositions equivalent to AxC.
>

### Mild Shock

Feb 19, 2024, 3:14:27 PMFeb 19
to

Whats the strategy for writing such nonsense as below?

What are products of omega? How are paradoxes sets?

LoL

### Mild Shock

Feb 19, 2024, 3:27:48 PMFeb 19
to
BTW: The finite ordinals as a set is omega. Omega does not
contain omega. Now it becomes clear what the strategy of
Rossy Boy is to write such nonsense.

Its a form of trolling based on:

Drunken Fist

Zui Quan

Drunken boxing (Chinese: 醉拳; pinyin: zuì quán) also
known as Drunken Fist, is a general name for all styles of
Chinese martial arts that imitate the movements of a
drunk person.[1] It is an ancient style and its origins are
mainly traced back to the Buddhist and Daoist religious
communities. The Buddhist style is related to the Shaolin
temple while the Daoist style is based on the Daoist tale of
the drunken Eight Immortals. Zui quan has the most unusual
body movements among all styles of Chinese martial arts.
Hitting, grappling, locking, dodging, feinting, ground and
aerial fighting and all other sophisticated methods of
combat are incorporated.

### Ross Finlayson

Feb 19, 2024, 5:02:44 PMFeb 19
to
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.
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.

### Ross Finlayson

Feb 19, 2024, 6:04:06 PMFeb 19
to
(Maybe that's just me.)

### Mild Shock

Feb 19, 2024, 8:01:16 PMFeb 19
to

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.

### Ross Finlayson

Feb 19, 2024, 10:47:53 PMFeb 19
to
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.

### Mild Shock

Feb 20, 2024, 2:32:21 AMFeb 20
to Ross Finlayson
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:

### Mild Shock

Feb 20, 2024, 2:57:43 AMFeb 20
to
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

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:

### Ross Finlayson

Feb 20, 2024, 12:53:56 PMFeb 20
to
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.)

### Mild Shock

Feb 20, 2024, 2:10:53 PMFeb 20
to
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.

### Mild Shock

Feb 20, 2024, 2:23:49 PMFeb 20
to

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

### markus...@gmail.com

Feb 20, 2024, 2:35:25 PMFeb 20
to
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.

### Mild Shock

Feb 20, 2024, 2:55:03 PMFeb 20
to
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

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

### mitchr...@gmail.com

Feb 20, 2024, 2:59:43 PMFeb 20
to
Man is making more out of ordinals than is there.
He even did it with the Calculus. Einstein pointed it out.
It was part of what he wrote on his death bed.

### Ross Finlayson

Feb 20, 2024, 3:13:48 PMFeb 20
to
"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.

### Ross Finlayson

Feb 20, 2024, 3:15:43 PMFeb 20
to
"... 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)".

### Ross Finlayson

Feb 20, 2024, 3:27:22 PMFeb 20
to
(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.

### Ross Finlayson

Feb 20, 2024, 3:41:57 PMFeb 20
to
On 02/20/2024 11:23 AM, Mild Shock wrote:
>
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,
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

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

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

### Jim Burns

Feb 20, 2024, 5:02:34 PMFeb 20
to
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.

### Ross Finlayson

Feb 20, 2024, 5:12:55 PMFeb 20
to
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.

Any sorts addition formula are usually considered
very conducive to tractability.

### mitchr...@gmail.com

Feb 20, 2024, 8:21:12 PMFeb 20
to
On Tuesday, February 20, 2024 at 11:35:25 AM UTC-8, 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
> >
> > quasi
> No set can contain itself.

This is stupid. How can a set not contain itself?

### Ross Finlayson

Feb 20, 2024, 10:36:43 PMFeb 20
to
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
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,

Because it's a tautology, ....

It's a sort of brachistology.

ORD: that's its name.

### WM

Feb 21, 2024, 3:33:49 AMFeb 21
to
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

### Mild Shock

Feb 21, 2024, 8:47:14 AMFeb 21
to
**********************************************************************
Welcome to brain gymnastics about the "class" and "set" distinction.
**********************************************************************

"Ord" is the predication whether a class is transitive and is
well-ordered by the membership relation.

"On" usually denotes the class of sets that are ordinal.
On itself is ordinal, although not set-like.

Its basically the first example of an ordinal in every

⊢ ¬ On ∈ V
https://us.metamath.org/mpeuni/onprc.html

⊢ Ord On
https://us.metamath.org/mpeuni/ordon.html

And this definition here:

⊢ On = {𝑥 ∣ Ord 𝑥}
https://us.metamath.org/mpeuni/df-on.html

### Mild Shock

Feb 21, 2024, 8:50:53 AMFeb 21
to
BTW: Ord is prima facie a higher order logic predicate (HOL),
it is not from first order logic (FOL), because it takes a

class argument. But you might rewrite it to FOL for some
kind of arguments sometimes. Its defined here:

⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
https://us.metamath.org/mpeuni/df-ord.html

### Mild Shock

Feb 21, 2024, 9:16:20 AMFeb 21
to
Higher order logic (HOL) was already in use among logicians
when Gödel wrote this booklet:

Kurt Gödel: The Consistency of the Axiom of Choice and of
the Generalized Continuum Hypothesis with the Axioms of
Set Theory, Annals of Mathematical Studies, Volume 3, Princeton NJ, 1940
https://www.amazon.com/dp/0691079277

Meta math is not so open about it that it uses HOL.
Using Neumann-Bernays-Gödel-Mengenlehre (NBG)
might also not help much. Pocking into Isabelle/HOL wasn't

so satisfactory either, they often work with α set type
constructor, so that the set theory and all theorems have
a type parameter α. But Meta math looks very cute,

is less a eyesore than anything else.

### Jim Burns

Feb 21, 2024, 12:59:58 PMFeb 21
to
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.

ψ doesn't have an infinite descent.

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

### Ross Finlayson

Feb 21, 2024, 1:24:29 PMFeb 21
to
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,
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.

### Mild Shock

Feb 22, 2024, 3:04:08 AMFeb 22
to
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

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:

### Mild Shock

Feb 22, 2024, 3:11:38 AMFeb 22
to

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.

Mild Shock schrieb:

### WM

Feb 22, 2024, 8:00:16 AMFeb 22
to
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.
>
> 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

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