Mostowski Collapse wrote:
> Text books use also classes. Example:
>
> Provability, Computability and Reflection
> https://www.amazon.com/dp/0444854029

Another well-known example is Kelley's book 'General Topology', the
appendix of which contains an account of Morse/Kelley set theory.

Morse/Kelley set theory is used in Jean Rubin's 'Set Theory for the
Mathematician'. Note, Dan, if you're reading this, the 'for the
Mathematician'.
>
> Dan Christensen schrieb am Montag, 1. November 2021 um 15:35:51 UTC+1:
>> On Monday, November 1, 2021 at 10:14:10 AM UTC-4, Peter wrote:
>>> Dan Christensen wrote:
>>>> On Sunday, October 31, 2021 at 11:43:01 AM UTC-4, Peter wrote:
>>>>> Dan Christensen wrote:
>>>>>> On Sunday, October 31, 2021 at 4:41:53 AM UTC-4, Mostowski Collapse wrote:
>>>>>>> The phrase "X is a proper class" translates to:
>>>>>>>
>>>>>>> ~EXIST(a):ALL(b):[b e a <=> X(b)]
>>>>>>>
>>>>>>> The above is a ZFC formula, if X is a ZFC formula.
>>>>>>>
>>>>>>> It similar like translating Russells "the X is Y", you also
>>>>>>> called me first names, and said I am using non-classical
>>>>>>> logic whatever. But classes are only syntactic
>>>>>>>
>>>>>>> sugar, see Basic Set Theory by Levy:
>>>>>>> https://www.amazon.com/dp/0486420795
>>>>>>>
>>>>>>> Since you want to prove "non-existence of a set such
>>>>>>> that X", using the established phrase "X is a proper class"
>>>>>>> instead is the right thing to do.
>>>>>>
>>>>>> Too much hand waving for my liking. Will stick to ZFC or my own ZFC-like set theory and regular textbook math (without "classes") for now.
>>>>> In ZF {x:phi(x)} [phi a formula in Zf's language] is a class.
>>>>
>>>> It can't be a set, so maybe we can call it be another name! Let's see, what COULD we call it so that can't obtain arbitrary subsets of it? Hmmm....
>>
>>> What makes you think that classes can't have subsets?
>> You will have to make up your own axioms for "classes" that will hopefully avoid any RP-like inconsistencies. Since the vast majority of math textbooks are based on regular set theory, I have very little interest in incorporating classes into my DC Proof system.
>> Dan
>>
>> Download my DC Proof 2.0 freeware at http://www.dcproof.com
>> Visit my Math Blog at http://www.dcproof.wordpress.com

--
The world will little note, nor long remember what we say here
Abraham Lincoln at Gettysburg

Looks like Dan-O-Matik doesn't understand syntactic sugar:

> > What makes you think that classes can't have subsets?
> You will have to make up your own axioms for "classes" that will hopefully avoid any RP-like inconsistencies.

Syntactic sugar doessn't generate axioms. What axioms
do you want to have for "classes", if the syntactic sugar
already eliminates them?

Some phantom of the opera axioms, maybe somewhere
in the cellar underneeth the ZFC axioms, there are some
class axioms. Did you have a look?

Mostowski Collapse schrieb am Montag, 1. November 2021 um 16:13:21 UTC+1:
> Oops, there is something wrong with google books,
> it shows me strange preview, a different book.
>
> Equivalents of the Axiom of Choice
> https://www.elsevier.com/books/equivalents-of-the-axiom-of-choice-ii/rubin/978-0-444-87708-6
>
> Also some Rubin, are they all closely related by family?
>
> LoL
> Mostowski Collapse schrieb am Montag, 1. November 2021 um 16:05:13 UTC+1:
> > Text books use also classes. Example:
> >
> > Provability, Computability and Reflection
> > https://www.amazon.com/dp/0444854029
> > Dan Christensen schrieb am Montag, 1. November 2021 um 15:35:51 UTC+1:
> > > On Monday, November 1, 2021 at 10:14:10 AM UTC-4, Peter wrote:
> > > > Dan Christensen wrote:
> > > > > On Sunday, October 31, 2021 at 11:43:01 AM UTC-4, Peter wrote:
> > > > >> Dan Christensen wrote:
> > > > >>> On Sunday, October 31, 2021 at 4:41:53 AM UTC-4, Mostowski Collapse wrote:
> > > > >>>> The phrase "X is a proper class" translates to:
> > > > >>>>
> > > > >>>> ~EXIST(a):ALL(b):[b e a <=> X(b)]
> > > > >>>>
> > > > >>>> The above is a ZFC formula, if X is a ZFC formula.
> > > > >>>>
> > > > >>>> It similar like translating Russells "the X is Y", you also
> > > > >>>> called me first names, and said I am using non-classical
> > > > >>>> logic whatever. But classes are only syntactic
> > > > >>>>
> > > > >>>> sugar, see Basic Set Theory by Levy:
> > > > >>>> https://www.amazon.com/dp/0486420795
> > > > >>>>
> > > > >>>> Since you want to prove "non-existence of a set such
> > > > >>>> that X", using the established phrase "X is a proper class"
> > > > >>>> instead is the right thing to do.
> > > > >>>
> > > > >>> Too much hand waving for my liking. Will stick to ZFC or my own ZFC-like set theory and regular textbook math (without "classes") for now.
> > > > >> In ZF {x:phi(x)} [phi a formula in Zf's language] is a class.
> > > > >
> > > > > It can't be a set, so maybe we can call it be another name! Let's see, what COULD we call it so that can't obtain arbitrary subsets of it? Hmmm....
> > >
> > > > What makes you think that classes can't have subsets?
> > > You will have to make up your own axioms for "classes" that will hopefully avoid any RP-like inconsistencies. Since the vast majority of math textbooks are based on regular set theory, I have very little interest in incorporating classes into my DC Proof system.
> > > Dan
> > >
> > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > Visit my Math Blog at http://www.dcproof.wordpress.com

Dan Christensen explained :
> On Monday, November 1, 2021 at 10:14:10 AM UTC-4, Peter wrote:
>> Dan Christensen wrote:
>>> On Sunday, October 31, 2021 at 11:43:01 AM UTC-4, Peter wrote:
>>>> Dan Christensen wrote:
>>>>> On Sunday, October 31, 2021 at 4:41:53 AM UTC-4, Mostowski Collapse
>>>>> wrote:
>>>>>> The phrase "X is a proper class" translates to:
>>>>>>
>>>>>> ~EXIST(a):ALL(b):[b e a <=> X(b)]
>>>>>>
>>>>>> The above is a ZFC formula, if X is a ZFC formula.
>>>>>>
>>>>>> It similar like translating Russells "the X is Y", you also
>>>>>> called me first names, and said I am using non-classical
>>>>>> logic whatever. But classes are only syntactic
>>>>>>
>>>>>> sugar, see Basic Set Theory by Levy:
>>>>>> https://www.amazon.com/dp/0486420795
>>>>>>
>>>>>> Since you want to prove "non-existence of a set such
>>>>>> that X", using the established phrase "X is a proper class"
>>>>>> instead is the right thing to do.
>>>>>
>>>>> Too much hand waving for my liking. Will stick to ZFC or my own ZFC-like
>>>>> set theory and regular textbook math (without "classes") for now.
>>>> In ZF {x:phi(x)} [phi a formula in Zf's language] is a class.
>>>
>>> It can't be a set, so maybe we can call it be another name! Let's see, what
>>> COULD we call it so that can't obtain arbitrary subsets of it? Hmmm....
>
>> What makes you think that classes can't have subsets?
>
> You will have to make up your own axioms for "classes" that will hopefully
> avoid any RP-like inconsistencies. Since the vast majority of math textbooks
> are based on regular set theory, I have very little interest in incorporating
> classes into my DC Proof system.

The cardinality of a subset is less than or equal to the cardinality of
its superset.

Can your system prove that all subsets have supersets?

Would that exclude 'classes' from being also supersets since classes
aren't under restricted comprehension.

Would that then exclude 'classes' from having subsets?

Every class has the empty class as a subclass.
And it happens that the empty class is a set,
by ZFC comprehension axiom.

Every non-empty class, has a least one element x
which is a set. Then the singleton class {x}, is a
subclass of that non-empty class, and it happens,

that the singleton class {x} is a set, by ZFC pairing axiom.
In general there is a class P(C), for class C, which
is the class of all subsets of the class C. It is

defined as follows D = P(C) iff:

forall x (D(x) <=> forall y(y e x => C(y))).

The ZFC power set axioms says, that if C is a set,
then P(C) is also a set. Cantors theorem holds for
sets, i.e. there is no surjective mapping from C to P(C),

Cantors theorem doesn't hold for proper class,
for example V = P(V). Mappings can be also proper
classes or sets.

Peter schrieb am Montag, 1. November 2021 um 16:01:38 UTC+1:
> Dan Christensen wrote:
> > On Monday, November 1, 2021 at 10:14:10 AM UTC-4, Peter wrote:
> >> Dan Christensen wrote:
> >>> On Sunday, October 31, 2021 at 11:43:01 AM UTC-4, Peter wrote:
> >>>> Dan Christensen wrote:
> >>>>> On Sunday, October 31, 2021 at 4:41:53 AM UTC-4, Mostowski
> >>>>> Collapse wrote:
> >>>>>> The phrase "X is a proper class" translates to:
> >>>>>>
> >>>>>> ~EXIST(a):ALL(b):[b e a <=> X(b)]
> >>>>>>
> >>>>>> The above is a ZFC formula, if X is a ZFC formula.
> >>>>>>
> >>>>>> It similar like translating Russells "the X is Y", you
> >>>>>> also called me first names, and said I am using
> >>>>>> non-classical logic whatever. But classes are only
> >>>>>> syntactic
> >>>>>>
> >>>>>> sugar, see Basic Set Theory by Levy:
> >>>>>> https://www.amazon.com/dp/0486420795
> >>>>>>
> >>>>>> Since you want to prove "non-existence of a set such that
> >>>>>> X", using the established phrase "X is a proper class"
> >>>>>> instead is the right thing to do.
> >>>>>
> >>>>> Too much hand waving for my liking. Will stick to ZFC or my
> >>>>> own ZFC-like set theory and regular textbook math (without
> >>>>> "classes") for now.
> >>>> In ZF {x:phi(x)} [phi a formula in Zf's language] is a class.
> >>>
> >>> It can't be a set, so maybe we can call it be another name! Let's
> >>> see, what COULD we call it so that can't obtain arbitrary subsets
> >>> of it? Hmmm....
> >
> >> What makes you think that classes can't have subsets?
> >
> > You will have to make up your own axioms for "classes"
> There is no need, it has already been done, and more than once. But
> back to the question: what makes you think that classes can't have subsets?
> > that will hopefully avoid any RP-like inconsistencies. Since the vast
> > majority of math textbooks are based on regular set theory, I have
> > very little interest in incorporating classes into my DC Proof
> > system.
> >
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com Visit my
> > Math Blog at http://www.dcproof.wordpress.com
> >
> --
> The world will little note, nor long remember what we say here
> Abraham Lincoln at Gettysburg

Cantors theorem becomes true, if we only look
at set like mappings, because the identity map:

V = P(V)

Is not set like anymore, is a proper class. This
class here, the identity map:

{ <x,x> | true }

Is a proper class. Its too big for a set.

Mostowski Collapse schrieb am Montag, 1. November 2021 um 19:09:19 UTC+1:
> Every class has the empty class as a subclass.
> And it happens that the empty class is a set,
> by ZFC comprehension axiom.
>
> Every non-empty class, has a least one element x
> which is a set. Then the singleton class {x}, is a
> subclass of that non-empty class, and it happens,
>
> that the singleton class {x} is a set, by ZFC pairing axiom.
> In general there is a class P(C), for class C, which
> is the class of all subsets of the class C. It is
>
> defined as follows D = P(C) iff:
>
> forall x (D(x) <=> forall y(y e x => C(y))).
>
> The ZFC power set axioms says, that if C is a set,
> then P(C) is also a set. Cantors theorem holds for
> sets, i.e. there is no surjective mapping from C to P(C),
>
> Cantors theorem doesn't hold for proper class,
> for example V = P(V). Mappings can be also proper
> classes or sets.
> Peter schrieb am Montag, 1. November 2021 um 16:01:38 UTC+1:
> > Dan Christensen wrote:
> > > On Monday, November 1, 2021 at 10:14:10 AM UTC-4, Peter wrote:
> > >> Dan Christensen wrote:
> > >>> On Sunday, October 31, 2021 at 11:43:01 AM UTC-4, Peter wrote:
> > >>>> Dan Christensen wrote:
> > >>>>> On Sunday, October 31, 2021 at 4:41:53 AM UTC-4, Mostowski
> > >>>>> Collapse wrote:
> > >>>>>> The phrase "X is a proper class" translates to:
> > >>>>>>
> > >>>>>> ~EXIST(a):ALL(b):[b e a <=> X(b)]
> > >>>>>>
> > >>>>>> The above is a ZFC formula, if X is a ZFC formula.
> > >>>>>>
> > >>>>>> It similar like translating Russells "the X is Y", you
> > >>>>>> also called me first names, and said I am using
> > >>>>>> non-classical logic whatever. But classes are only
> > >>>>>> syntactic
> > >>>>>>
> > >>>>>> sugar, see Basic Set Theory by Levy:
> > >>>>>> https://www.amazon.com/dp/0486420795
> > >>>>>>
> > >>>>>> Since you want to prove "non-existence of a set such that
> > >>>>>> X", using the established phrase "X is a proper class"
> > >>>>>> instead is the right thing to do.
> > >>>>>
> > >>>>> Too much hand waving for my liking. Will stick to ZFC or my
> > >>>>> own ZFC-like set theory and regular textbook math (without
> > >>>>> "classes") for now.
> > >>>> In ZF {x:phi(x)} [phi a formula in Zf's language] is a class.
> > >>>
> > >>> It can't be a set, so maybe we can call it be another name! Let's
> > >>> see, what COULD we call it so that can't obtain arbitrary subsets
> > >>> of it? Hmmm....
> > >
> > >> What makes you think that classes can't have subsets?
> > >
> > > You will have to make up your own axioms for "classes"
> > There is no need, it has already been done, and more than once. But
> > back to the question: what makes you think that classes can't have subsets?
> > > that will hopefully avoid any RP-like inconsistencies. Since the vast
> > > majority of math textbooks are based on regular set theory, I have
> > > very little interest in incorporating classes into my DC Proof
> > > system.
> > >
> > > Dan
> > >
> > > Download my DC Proof 2.0 freeware at http://www.dcproof.com Visit my
> > > Math Blog at http://www.dcproof.wordpress.com
> > >
> > --
> > The world will little note, nor long remember what we say here
> > Abraham Lincoln at Gettysburg

On Monday, November 1, 2021 at 11:16:20 AM UTC-4, Peter wrote:
> Mostowski Collapse wrote:
> > Text books use also classes. Example:
> >
> > Provability, Computability and Reflection
> > https://www.amazon.com/dp/0444854029
> Another well-known example is Kelley's book 'General Topology', the
> appendix of which contains an account of Morse/Kelley set theory.
>
> Morse/Kelley set theory is used in Jean Rubin's 'Set Theory for the
> Mathematician'. Note, Dan, if you're reading this, the 'for the
> Mathematician'.

You would probably have me include so-called "relevance" logic as well???

No thanks. Please note, I said, "the vast majority of math textbooks."

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

There is no other choice than use classes. Its a fundamental
principle. More fundamental than sets. The only logic you
need to deal with classes is FOL.

You find classes everywhere. Its only a paramterized FOL
formula. So whenever somebody writes A(x) or so its classes.
Already the induction axiom in Peano, uses classes:

P(0) & forall x(P(x) => P(x+1))
--------------------------------------------------
forall y P(y)

Inside a theory where the domain of discourse is N,
the P is a class. Peano axioms do use already classes.
You cannot say classes are not existent.

Peano himself had classes, for him N was a class,
which is correct, inside arithmetic, where there are only
individuals which are numbers, there is no omega.

Omega is not a natural number. So what are then
all natural numbers. Of course a class.

Dan Christensen schrieb am Montag, 1. November 2021 um 21:38:26 UTC+1:
> On Monday, November 1, 2021 at 11:16:20 AM UTC-4, Peter wrote:
> > Mostowski Collapse wrote:
> > > Text books use also classes. Example:
> > >
> > > Provability, Computability and Reflection
> > > https://www.amazon.com/dp/0444854029
> > Another well-known example is Kelley's book 'General Topology', the
> > appendix of which contains an account of Morse/Kelley set theory.
> >
> > Morse/Kelley set theory is used in Jean Rubin's 'Set Theory for the
> > Mathematician'. Note, Dan, if you're reading this, the 'for the
> > Mathematician'.
> You would probably have me include so-called "relevance" logic as well???
>
> No thanks. Please note, I said, "the vast majority of math textbooks."
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

In Peano you also find class complement. For a class C(x)
obviously ~C(x) is the class complement. Unlike for sets,
your PURPLE example, the class complement always exists.

You find in Peano:

IV. De classibus
Sit a classis, tunc -a significatur classis individuis
consituta quae non sunt a.
46. a e K => forall x (x e -a iff x ~e a)

See for yourself:

Arithmetices principia: nova methodo
https://archive.org/details/arithmeticespri00peangoog/page/n16/mode/2up

Mostowski Collapse schrieb am Montag, 1. November 2021 um 22:13:47 UTC+1:
> There is no other choice than use classes. Its a fundamental
> principle. More fundamental than sets. The only logic you
> need to deal with classes is FOL.
>
> You find classes everywhere. Its only a paramterized FOL
> formula. So whenever somebody writes A(x) or so its classes.
> Already the induction axiom in Peano, uses classes:
>
> P(0) & forall x(P(x) => P(x+1))
> --------------------------------------------------
> forall y P(y)
>
> Inside a theory where the domain of discourse is N,
> the P is a class. Peano axioms do use already classes.
> You cannot say classes are not existent.
>
> Peano himself had classes, for him N was a class,
> which is correct, inside arithmetic, where there are only
> individuals which are numbers, there is no omega.
>
> Omega is not a natural number. So what are then
> all natural numbers. Of course a class.
> Dan Christensen schrieb am Montag, 1. November 2021 um 21:38:26 UTC+1:
> > On Monday, November 1, 2021 at 11:16:20 AM UTC-4, Peter wrote:
> > > Mostowski Collapse wrote:
> > > > Text books use also classes. Example:
> > > >
> > > > Provability, Computability and Reflection
> > > > https://www.amazon.com/dp/0444854029
> > > Another well-known example is Kelley's book 'General Topology', the
> > > appendix of which contains an account of Morse/Kelley set theory.
> > >
> > > Morse/Kelley set theory is used in Jean Rubin's 'Set Theory for the
> > > Mathematician'. Note, Dan, if you're reading this, the 'for the
> > > Mathematician'.
> > You would probably have me include so-called "relevance" logic as well???
> >
> > No thanks. Please note, I said, "the vast majority of math textbooks."
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

Peanos K is the type class. He writes a e K, to indicate
that a is a class. His Induction axiom reads:

Aociomata:
9.
k e K & 1 e k & x e N & forall x (x e k => x+1 e k) => N subset k
https://archive.org/details/arithmeticespri00peangoog/page/n22/mode/2up

So k is a class. k is what we nowadays write down as P,
Peano is more precise, he has also x e N, and he starts
indicution at 1, but basically what he says is this here:

P(0) & forall x(P(x) => P(x+1))
--------------------------------------------------
forall y P(y)

He also uses N subset k as a shorthand for forall y P(y)
in the context of N, i.e. in the context where y would
run over N. To understand the shorthand,

check out his chapter IV. De classibus

Mostowski Collapse schrieb am Montag, 1. November 2021 um 22:21:23 UTC+1:
> In Peano you also find class complement. For a class C(x)
> obviously ~C(x) is the class complement. Unlike for sets,
> your PURPLE example, the class complement always exists.
>
> You find in Peano:
>
> IV. De classibus
> Sit a classis, tunc -a significatur classis individuis
> consituta quae non sunt a.
> 46. a e K => forall x (x e -a iff x ~e a)
>
> See for yourself:
>
> Arithmetices principia: nova methodo
> https://archive.org/details/arithmeticespri00peangoog/page/n16/mode/2up
> Mostowski Collapse schrieb am Montag, 1. November 2021 um 22:13:47 UTC+1:
> > There is no other choice than use classes. Its a fundamental
> > principle. More fundamental than sets. The only logic you
> > need to deal with classes is FOL.
> >
> > You find classes everywhere. Its only a paramterized FOL
> > formula. So whenever somebody writes A(x) or so its classes.
> > Already the induction axiom in Peano, uses classes:
> >
> > P(0) & forall x(P(x) => P(x+1))
> > --------------------------------------------------
> > forall y P(y)
> >
> > Inside a theory where the domain of discourse is N,
> > the P is a class. Peano axioms do use already classes.
> > You cannot say classes are not existent.
> >
> > Peano himself had classes, for him N was a class,
> > which is correct, inside arithmetic, where there are only
> > individuals which are numbers, there is no omega.
> >
> > Omega is not a natural number. So what are then
> > all natural numbers. Of course a class.
> > Dan Christensen schrieb am Montag, 1. November 2021 um 21:38:26 UTC+1:
> > > On Monday, November 1, 2021 at 11:16:20 AM UTC-4, Peter wrote:
> > > > Mostowski Collapse wrote:
> > > > > Text books use also classes. Example:
> > > > >
> > > > > Provability, Computability and Reflection
> > > > > https://www.amazon.com/dp/0444854029
> > > > Another well-known example is Kelley's book 'General Topology', the
> > > > appendix of which contains an account of Morse/Kelley set theory.
> > > >
> > > > Morse/Kelley set theory is used in Jean Rubin's 'Set Theory for the
> > > > Mathematician'. Note, Dan, if you're reading this, the 'for the
> > > > Mathematician'.
> > > You would probably have me include so-called "relevance" logic as well???
> > >
> > > No thanks. Please note, I said, "the vast majority of math textbooks."
> > > Dan
> > >
> > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > Visit my Math Blog at http://www.dcproof.wordpress.com

On Monday, November 1, 2021 at 5:13:47 PM UTC-4, Mostowski Collapse wrote:
> There is no other choice than use classes. Its a fundamental
> principle. More fundamental than sets. The only logic you
> need to deal with classes is FOL.
>
> You find classes everywhere. Its only a paramterized FOL
> formula. So whenever somebody writes A(x) or so its classes.
> Already the induction axiom in Peano, uses classes:
>
> P(0) & forall x(P(x) => P(x+1))

I prefer:

ALL(a):[Set(a) & ALL(b):[b in a => b in N] => [0 in a & ALL(b):[b in a => S(a) in b] => ALL(b):[b in N => b in a]]

No classes. Just sets. The usual setup. See: http://mathworld.wolfram.com/PeanosAxioms.html

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

I guess the problem is that your family name is Christensen,
and that you don't have the family name Peano.

This makes you extremly stubborn and idiotic. Just accept it
that grown ups in math are fluent in classes.

Dan Christensen schrieb am Dienstag, 2. November 2021 um 01:07:05 UTC+1:
> On Monday, November 1, 2021 at 5:13:47 PM UTC-4, Mostowski Collapse wrote:
> > There is no other choice than use classes. Its a fundamental
> > principle. More fundamental than sets. The only logic you
> > need to deal with classes is FOL.
> >
> > You find classes everywhere. Its only a paramterized FOL
> > formula. So whenever somebody writes A(x) or so its classes.
> > Already the induction axiom in Peano, uses classes:
> >
> > P(0) & forall x(P(x) => P(x+1))
> I prefer:
>
> ALL(a):[Set(a) & ALL(b):[b in a => b in N] => [0 in a & ALL(b):[b in a => S(a) in b] => ALL(b):[b in N => b in a]]
>
> No classes. Just sets. The usual setup. See: http://mathworld.wolfram.com/PeanosAxioms.html
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

On Tuesday, November 2, 2021 at 5:16:59 AM UTC-4, Mostowski Collapse wrote:
> I guess the problem is that your family name is Christensen,
> and that you don't have the family name Peano.
>
> This makes you extremly stubborn and idiotic. Just accept it
> that grown ups in math are fluent in classes.

It seems Terry Tao may need some "growing up." Check out his "Analysis I."

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

You are definitively a moron. You use also classes in DC
Proof. You just posted here a proof that uses classes:

32 EXIST(r):[Set(r) & ALL(a):[a in r <=> a in u & ~a in a]]
Subset, 13

ZFC has two axiom schemas that deal with classes.
One axiom schema deals with unary predicates as classes,
and one axiom schema deals with binary mappings as classes,

the axiom schemas are:

Axiom schema of specification
It says { x | P(x) & x e y } is a set, when y is a set and P is a class
https://en.wikipedia.org/wiki/Axiom_schema_of_specification

Axiom schema of replacement
It says that { F(x) | x e y } is a set, when y is a set and F is a class
https://en.wikipedia.org/wiki/Axiom_schema_of_replacement

Your Subset is just Axiom schema of specification.
The two axiom schemas are the interface between classes
and sets in ZFC.

Dan Christensen schrieb am Dienstag, 2. November 2021 um 12:06:33 UTC+1:
> On Tuesday, November 2, 2021 at 5:16:59 AM UTC-4, Mostowski Collapse wrote:
> > I guess the problem is that your family name is Christensen,
> > and that you don't have the family name Peano.
> >
> > This makes you extremly stubborn and idiotic. Just accept it
> > that grown ups in math are fluent in classes.
> It seems Terry Tao may need some "growing up." Check out his "Analysis I."
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Dan Christensen wrote:
> On Monday, November 1, 2021 at 11:16:20 AM UTC-4, Peter wrote:
>> Mostowski Collapse wrote:
>>> Text books use also classes. Example:
>>>
>>> Provability, Computability and Reflection
>>> https://www.amazon.com/dp/0444854029
>> Another well-known example is Kelley's book 'General Topology', the
>> appendix of which contains an account of Morse/Kelley set theory.
>>
>> Morse/Kelley set theory is used in Jean Rubin's 'Set Theory for the
>> Mathematician'. Note, Dan, if you're reading this, the 'for the
>> Mathematician'.
>
> You would probably have me include so-called "relevance" logic as well???

I wouldn't have you include anything, but it would be nice if you didn't
sneer at things you aren't interested in.
>
> No thanks. Please note, I said, "the vast majority of math textbooks."
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>

--
The world will little note, nor long remember what we say here
Abraham Lincoln at Gettysburg

The Z in ZFC stands for Zermelo. You can read the original
phrasing of the Axiom schema of specification:

Axiom der Aussonderung:
Ist die Klassenaussage E(x) definit für alle Elemente einer
Menge M, so besitzt M immer eine Untermenge, welche alle
diejenigen Elemente x von M, für welche E(x) wahr ist,
und nur solche als Elemente enthält.
https://de.wikipedia.org/wiki/Zermelo-Mengenlehre#Zermelos_Axiome_1907

He uses the phrase "Klassenaussage", i.e. predicate class.
I really don't know whats going on in your brain Dan-O-Matik.
Do you never read books or papers.

Classes are an established thing in mathematics.

Mostowski Collapse schrieb am Dienstag, 2. November 2021 um 13:40:57 UTC+1:
> You are definitively a moron. You use also classes in DC
> Proof. You just posted here a proof that uses classes:
>
> 32 EXIST(r):[Set(r) & ALL(a):[a in r <=> a in u & ~a in a]]
> Subset, 13
>
> ZFC has two axiom schemas that deal with classes.
> One axiom schema deals with unary predicates as classes,
> and one axiom schema deals with binary mappings as classes,
>
> the axiom schemas are:
>
> Axiom schema of specification
> It says { x | P(x) & x e y } is a set, when y is a set and P is a class
> https://en.wikipedia.org/wiki/Axiom_schema_of_specification
>
> Axiom schema of replacement
> It says that { F(x) | x e y } is a set, when y is a set and F is a class
> https://en.wikipedia.org/wiki/Axiom_schema_of_replacement
>
> Your Subset is just Axiom schema of specification.
> The two axiom schemas are the interface between classes
> and sets in ZFC.
> Dan Christensen schrieb am Dienstag, 2. November 2021 um 12:06:33 UTC+1:
> > On Tuesday, November 2, 2021 at 5:16:59 AM UTC-4, Mostowski Collapse wrote:
> > > I guess the problem is that your family name is Christensen,
> > > and that you don't have the family name Peano.
> > >
> > > This makes you extremly stubborn and idiotic. Just accept it
> > > that grown ups in math are fluent in classes.
> > It seems Terry Tao may need some "growing up." Check out his "Analysis I."
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

The word class is used when you are interested in
the extension of a concept. A predicate does not really
capture the connotation of a class. If I say I have

a predicate P(x), one usually assumes that its intention
is also given, i.e. that you have a formula which is
the predicate P(x). By class we mean only the extensional

aspect which is defined as the elements from domain
of discourse that make the predicate true, i.e. {x | P(x)}.
In as far you can have two predicates P1 and P2 which

are intensionally different, i.e. which are different formulas,
but neverless are the same class, since {x | P1(x)} = {x | P2(x)}
a typical example where different intensions give the

same extension is the universal "set", in ZFC these are all
the same class:

{ x | true }

{ x | x = x }

{ x | ~(x e x) }

You can prove {x | P1(x)} = {x | P2(x)}, in that you simply
prove forall x (P1(x) <=> P2(x)).

Mostowski Collapse schrieb am Dienstag, 2. November 2021 um 13:48:52 UTC+1:
> The Z in ZFC stands for Zermelo. You can read the original
> phrasing of the Axiom schema of specification:
>
> Axiom der Aussonderung:
> Ist die Klassenaussage E(x) definit für alle Elemente einer
> Menge M, so besitzt M immer eine Untermenge, welche alle
> diejenigen Elemente x von M, für welche E(x) wahr ist,
> und nur solche als Elemente enthält.
> https://de.wikipedia.org/wiki/Zermelo-Mengenlehre#Zermelos_Axiome_1907
>
> He uses the phrase "Klassenaussage", i.e. predicate class.
> I really don't know whats going on in your brain Dan-O-Matik.
> Do you never read books or papers.
>
> Classes are an established thing in mathematics.
> Mostowski Collapse schrieb am Dienstag, 2. November 2021 um 13:40:57 UTC+1:
> > You are definitively a moron. You use also classes in DC
> > Proof. You just posted here a proof that uses classes:
> >
> > 32 EXIST(r):[Set(r) & ALL(a):[a in r <=> a in u & ~a in a]]
> > Subset, 13
> >
> > ZFC has two axiom schemas that deal with classes.
> > One axiom schema deals with unary predicates as classes,
> > and one axiom schema deals with binary mappings as classes,
> >
> > the axiom schemas are:
> >
> > Axiom schema of specification
> > It says { x | P(x) & x e y } is a set, when y is a set and P is a class
> > https://en.wikipedia.org/wiki/Axiom_schema_of_specification
> >
> > Axiom schema of replacement
> > It says that { F(x) | x e y } is a set, when y is a set and F is a class
> > https://en.wikipedia.org/wiki/Axiom_schema_of_replacement
> >
> > Your Subset is just Axiom schema of specification.
> > The two axiom schemas are the interface between classes
> > and sets in ZFC.
> > Dan Christensen schrieb am Dienstag, 2. November 2021 um 12:06:33 UTC+1:
> > > On Tuesday, November 2, 2021 at 5:16:59 AM UTC-4, Mostowski Collapse wrote:
> > > > I guess the problem is that your family name is Christensen,
> > > > and that you don't have the family name Peano.
> > > >
> > > > This makes you extremly stubborn and idiotic. Just accept it
> > > > that grown ups in math are fluent in classes.
> > > It seems Terry Tao may need some "growing up." Check out his "Analysis I."
> > > Dan
> > >
> > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > Visit my Math Blog at http://www.dcproof.wordpress.com

Dan Christensen wrote:
> On Monday, November 1, 2021 at 5:13:47 PM UTC-4, Mostowski Collapse wrote:
>> There is no other choice than use classes. Its a fundamental
>> principle. More fundamental than sets. The only logic you
>> need to deal with classes is FOL.
>>
>> You find classes everywhere. Its only a paramterized FOL
>> formula. So whenever somebody writes A(x) or so its classes.
>> Already the induction axiom in Peano, uses classes:
>>
>> P(0) & forall x(P(x) => P(x+1))
>
> I prefer:
>
> ALL(a):[Set(a) & ALL(b):[b in a => b in N] => [0 in a & ALL(b):[b in a => S(a) in b] => ALL(b):[b in N => b in a]]
>
> No classes. Just sets.

In the best-known theories of sets-and-classes, all sets are classes.
Though not all classes are sets.

> The usual setup. See: http://mathworld.wolfram.com/PeanosAxioms.html
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>

--
The world will little note, nor long remember what we say here
Abraham Lincoln at Gettysburg

The problem with this Peano, its not a light weight Peano.
A mathematician would probably not use it for some parts
of number theory. What makes it heavy, it imports ZFC

or some set theory. To have reasoning "b in N" etc.. But
classes are only FOL, they do not introduce a new relation "in".
For example to prove commutativity:

forall x,y (x + y = y + x)

via induction, it doesn't make sense to first form a set "a"
for the commutativity. You directly prove it takling the
problem and viewing it as a class.

Proving it with classes is left as an exercise. Well you
would first prove that the class 0 + z = z where z is the induction
variable, and then prove the class x + y = y + x where x is the

induction variable, and use the previous lemma.

The only reason to do it with sets, is a weak proof tool, which
does not accept axiom schemas that make use of classes.
In DC Proof there is no way to add the class based

Peano induction axiom. So you need ZFC, you will borrow
the Subset axiom to transfer your problem class into a set,
and then use the set based Peano.

Peter schrieb am Dienstag, 2. November 2021 um 20:08:11 UTC+1:
> Dan Christensen wrote:
> > On Monday, November 1, 2021 at 5:13:47 PM UTC-4, Mostowski Collapse wrote:
> >> There is no other choice than use classes. Its a fundamental
> >> principle. More fundamental than sets. The only logic you
> >> need to deal with classes is FOL.
> >>
> >> You find classes everywhere. Its only a paramterized FOL
> >> formula. So whenever somebody writes A(x) or so its classes.
> >> Already the induction axiom in Peano, uses classes:
> >>
> >> P(0) & forall x(P(x) => P(x+1))
> >
> > I prefer:
> >
> > ALL(a):[Set(a) & ALL(b):[b in a => b in N] => [0 in a & ALL(b):[b in a => S(a) in b] => ALL(b):[b in N => b in a]]
> >
> > No classes. Just sets.
> In the best-known theories of sets-and-classes, all sets are classes.
> Though not all classes are sets.
> > The usual setup. See: http://mathworld.wolfram.com/PeanosAxioms.html
> >
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com
> >
> --
> The world will little note, nor long remember what we say here
> Abraham Lincoln at Gettysburg

Maybe Dan-O-Matik should prove this, via DC Proof:

forall x,y (x + y = y + x)

And then compare his proof with text book proofs.

What is different?

Half of the books on the market will not use sets.

Mostowski Collapse schrieb am Dienstag, 2. November 2021 um 21:35:17 UTC+1:
> The problem with this Peano, its not a light weight Peano.
> A mathematician would probably not use it for some parts
> of number theory. What makes it heavy, it imports ZFC
>
> or some set theory. To have reasoning "b in N" etc.. But
> classes are only FOL, they do not introduce a new relation "in".
> For example to prove commutativity:
>
> forall x,y (x + y = y + x)
>
> via induction, it doesn't make sense to first form a set "a"
> for the commutativity. You directly prove it takling the
> problem and viewing it as a class.
>
> Proving it with classes is left as an exercise. Well you
> would first prove that the class 0 + z = z where z is the induction
> variable, and then prove the class x + y = y + x where x is the
>
> induction variable, and use the previous lemma.
>
> The only reason to do it with sets, is a weak proof tool, which
> does not accept axiom schemas that make use of classes.
> In DC Proof there is no way to add the class based
>
> Peano induction axiom. So you need ZFC, you will borrow
> the Subset axiom to transfer your problem class into a set,
> and then use the set based Peano.
> Peter schrieb am Dienstag, 2. November 2021 um 20:08:11 UTC+1:
> > Dan Christensen wrote:
> > > On Monday, November 1, 2021 at 5:13:47 PM UTC-4, Mostowski Collapse wrote:
> > >> There is no other choice than use classes. Its a fundamental
> > >> principle. More fundamental than sets. The only logic you
> > >> need to deal with classes is FOL.
> > >>
> > >> You find classes everywhere. Its only a paramterized FOL
> > >> formula. So whenever somebody writes A(x) or so its classes.
> > >> Already the induction axiom in Peano, uses classes:
> > >>
> > >> P(0) & forall x(P(x) => P(x+1))
> > >
> > > I prefer:
> > >
> > > ALL(a):[Set(a) & ALL(b):[b in a => b in N] => [0 in a & ALL(b):[b in a => S(a) in b] => ALL(b):[b in N => b in a]]
> > >
> > > No classes. Just sets.
> > In the best-known theories of sets-and-classes, all sets are classes.
> > Though not all classes are sets.
> > > The usual setup. See: http://mathworld.wolfram.com/PeanosAxioms.html
> > >
> > > Dan
> > >
> > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > Visit my Math Blog at http://www.dcproof.wordpress.com
> > >
> > --
> > The world will little note, nor long remember what we say here
> > Abraham Lincoln at Gettysburg

Dan Christensen wrote:

> I wrote the attached proof in response to a discussion at Quora. Here I
> show that, in set theory, there exists no set of all those things that
> are NOT PURPLE.
> In the language of set theory: For every set p, there does NOT exist a
> set p' of all those elements NOT in p.

you capitalists are crazy, making it clear so everybody understand:

https://www.bitchute.com/video/qxVpu2QWH1fh/
The Vaccinated are dying - They are telling you to your face

this is what happens bringing governments and businesses together. In
capitalism.

The backside of a baboon is purple...

On Tuesday, November 2, 2021 at 4:35:17 PM UTC-4, Mostowski Collapse wrote:
> The problem with this Peano, its not a light weight Peano.
> A mathematician would probably not use it for some parts
> of number theory. What makes it heavy, it imports ZFC
>
> or some set theory. To have reasoning "b in N" etc.. But
> classes are only FOL, they do not introduce a new relation "in".
> For example to prove commutativity:
>
> forall x,y (x + y = y + x)
>
> via induction, it doesn't make sense to first form a set "a"
> for the commutativity. You directly prove it takling the
> problem and viewing it as a class.
>

It is not necessary to postulate the existence of a universal set to prove commutativity of + on N.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

So your idea is:

Proof:
If V \ p were a set, then (V \ p) u p = V were also set,
since the union of two sets is a set. But we already
know that V is a proper class, contradiction.

Q.E.D.

Thanks for the proof idea.

Now I have the feeling there is a shorter proof.
You use the universal class V = { x | x = x } to
provoke a contradiction. But the Russell class

R = { x | ~(x e x) } comes also into play. You
use it when you show that V is a proper class.
Basically you derive a contradiction as follows:

1) You assume V is a set,

2) Then by the separation axiom you get:

R = { x | ~(x e x) } = { x | x e V & ~(x e x) } is also a set.

3) Then you make the usual Russell contradiction argument.

You need 17 lines of proof for that. Thats huge.
I have seen much shorter Russell contradiction
arguments. Most likely there is a universal purple

proof, and maybe also a short shorter proof?

Mostowski Collapse schrieb am Mittwoch, 10. November 2021 um 17:20:38 UTC+1:
> So your idea is:
>
> Proof:
> If V \ p were a set, then (V \ p) u p = V were also set,
> since the union of two sets is a set. But we already
> know that V is a proper class, contradiction.
>
> Q.E.D.
>
> Thanks for the proof idea.

The nice thing about this kind of proof of non-existence
of the universal set, you don't need the assumption V = R.
The resulting set R in step 2) can be a subset of V.

So basically you don't need the axiom of regularity. And the
same proof would also work in some of the Non-well-founded
set theories, that do not have the axiom of regularity.

So the stipulation in Wikipedia:

"Many set theories do not allow for the existence of a universal set.
For example, it is directly contradicted by the axioms such as the
axiom of regularity and its existence would imply inconsistencies."
https://en.wikipedia.org/wiki/Universal_set

Is utter nonsense. The axiom of regularity was never needed
to show that the universal class is a proper class.

Mostowski Collapse schrieb am Mittwoch, 10. November 2021 um 17:31:16 UTC+1:
> Now I have the feeling there is a shorter proof.
> You use the universal class V = { x | x = x } to
> provoke a contradiction. But the Russell class
>
> R = { x | ~(x e x) } comes also into play. You
> use it when you show that V is a proper class.
> Basically you derive a contradiction as follows:
>
> 1) You assume V is a set,
>
> 2) Then by the separation axiom you get:
>
> R = { x | ~(x e x) } = { x | x e V & ~(x e x) } is also a set.
>
> 3) Then you make the usual Russell contradiction argument.
>
> You need 17 lines of proof for that. Thats huge.
> I have seen much shorter Russell contradiction
> arguments. Most likely there is a universal purple
>
> proof, and maybe also a short shorter proof?
> Mostowski Collapse schrieb am Mittwoch, 10. November 2021 um 17:20:38 UTC+1:
> > So your idea is:
> >
> > Proof:
> > If V \ p were a set, then (V \ p) u p = V were also set,
> > since the union of two sets is a set. But we already
> > know that V is a proper class, contradiction.
> >
> > Q.E.D.
> >
> > Thanks for the proof idea.