Z.Set Size Theory
zuhair
The following is a set size theory.
Primitive e.
Axioms:
1) ExAy (x= size y)
2) AxAyAz (y=size x , z= size x -> y=z)
3) AxAyAzAu: z in x, u in y -> size{z}=size{u}
4) AxAyAaAb:a.x={},b.y={},size x=size y,size a = size b -> size aUx =
size bUy.
5) Ay ( y!={} -> size {} < size y )
6) AxAyAaAb: a.x={},b.y={}, size x<size y, size a = size b -> size aUx
< size bUy.
7) AxAy ( size x < size y <-> size y > size x ).
I think that this set size theory is basic to any consistent theorum of
size.
Zuhair
zuhair
Perhaps, this theory only works for finite sets?
Zuhair
MoeBlee
MoeBlee
It was supposed to be this:
zuhair wrote:
> Hi All,
>
> The following is a set size theory.
>
> Primitive e.
>
> Axioms:
> 1) ExAy (x= size y)
> 2) AxAyAz (y=size x , z= size x -> y=z)
> 3) AxAyAzAu: z in x, u in y -> size{z}=size{u}
> 4) AxAyAaAb:a.x={},b.y={},size x=size y,size a = size b -> size aUx =
> size bUy.
> 5) Ay ( y!={} -> size {} < size y )
> 6) AxAyAaAb: a.x={},b.y={}, size x<size y, size a = size b -> size aUx
> < size bUy.
> 7) AxAy ( size x < size y <-> size y > size x ).
You don't know how to state a theory.
'size' is also primitive here (I'll call it 's').
And your first axiom:
ExAy x = s(y)
That says everything has the same size.
Maybe you mean:
AyEx x = s(y).
That says everything has a size.
But you don't need such an axiom (and, if you use Frege's method for
improper descriptions) you never need such an axiom. If 's' is
primitive (and with Frege's method, even if 's' is defined), it is
already a theorem of identity theory that AyEx x=s(y), since (using
Frege's method) it is a theorem schema of identity theory:
For all terms t in which x does not occur free:
Ex x = t.
Then axiom (2) is unnecessary also, since it too is already a theorem
of identity theory.
(Or does your theory not incorporate identity theory? If that is the
case, then '=' is another primitive, and you can't presume anything
about it that isn't entailed by your axioms.)
Then axiom (3) has the symbol '{ }'. Where did that come from? You've
not defined it. So, as you've given it, it's another primitive.
And that's as far as this particular post of mine will bother with
your nonsense.
MoeBlee
Virgil
"zuhair" <zalj...@yahoo.com> wrote:
> zuhair wrote:
> > Hi All,
> >
> > The following is a set size theory.
> >
> > Primitive e.
> >
> > Axioms:
> > 1) ExAy (x= size y)
I could accept 'AyEx (x = size(y))', but not 'ExAy (x= size (y))'
Zuhair's version says every y has the same set size.
My version says every y has a set size, but does not require all sizes
to be the same.
zuhair
right. this is right.
>
> But you don't need such an axiom (and, if you use Frege's method for
> improper descriptions) you never need such an axiom. If 's' is
> primitive (and with Frege's method, even if 's' is defined), it is
> already a theorem of identity theory that AyEx x=s(y), since (using
> Frege's method) it is a theorem schema of identity theory:
>
> For all terms t in which x does not occur free:
>
> Ex x = t.
>
> Then axiom (2) is unnecessary also, since it too is already a theorem
> of identity theory.
>
> (Or does your theory not incorporate identity theory? If that is the
> case, then '=' is another primitive, and you can't presume anything
> about it that isn't entailed by your axioms.)
>
> Then axiom (3) has the symbol '{ }'. Where did that come from? You've
> not defined it. So, as you've given it, it's another primitive.
As if you don't know what { } means. I should define the sun to you.
hun
MoeBlee
> As if you don't know what { } means. I should define the sun to you.
That's not the point! You are introducing a theory. If you have a
symbol, then it is primitive, in which case you give axioms that make
it work the way you want, or it is defined, in which case you define it
in a way that makes it work the way you want. You have no axioms or
definition for { } that make it work as you PRESUPPOSE it does. Your
PRESUPPOSITIONS about { } are taken from ANOTHER THEORY, not YOUR
theory. So if you want all that stuff from the other theory that makes
{ } work the way you want, then you have to ADD those axioms and
definitions from that other theory.
> I should define the sun to you.
Sarcasm would suit your style better if you weren't such an ignorant
twerp.
MoeBlee
Virgil
"zuhair" <zalj...@yahoo.com> wrote:
> MoeBlee wrote:
> > zuhair wrote:
> > > Hi All,
> > >
> > > The following is a set size theory.
> > >
> > > Primitive e.
> > >
> > > Axioms:
> > > 1) ExAy (x= size y)
> > > 2) AxAyAz (y=size x , z= size x -> y=z)
> > > 3) AxAyAzAu: z in x, u in y -> size{z}=size{u}
> > > 4) AxAyAaAb:a.x={},b.y={},size x=size y,size a = size b -> size aUx =
> > > size bUy.
> > > 5) Ay ( y!={} -> size {} < size y )
> > > 6) AxAyAaAb: a.x={},b.y={}, size x<size y, size a = size b -> size aUx
> > > < size bUy.
> > > 7) AxAy ( size x < size y <-> size y > size x ).
> >
> > You don't know how to state a theory.
> >
> > Then axiom (3) has the symbol '{ }'. Where did that come from? You've
> > not defined it. So, as you've given it, it's another primitive.
>
> As if you don't know what { } means.
The point of an axiom system is that it should be entirely independent
of outside special knowledge (other than formal logic). So what someone
may know about "{}" outside the system is irrelevant within the system.
zuhair
but all variable mentioned in this system are sets , sets according to
ZFC-I (ZRC without axiom of infinity). and {} means an empty set, and
{u} mean a singlton set.
Zuhair
Virgil
"zuhair" <zalj...@yahoo.com> wrote:
Where in YOUR list of axioms does it say that?
zuhair
True. I should have added that, anyhow, that was what I meant.
Zuhair
Virgil
"zuhair" <zalj...@yahoo.com> wrote:
In mathematics, what you meant, but did not say, doesn't count.
zuhair
Ok, no problem I can add, the following phrase "were every variable in
the above axiomatic system is a set based on ZFC-I axiomatic set
theory". I think this is enough. there is not need to go write all the
axoims of ZFC except infinity, beside the axioms of this set size
theory.
Zuhair
MoeBlee
> but all variable mentioned in this system are sets , sets according to
> ZFC-I (ZRC without axiom of infinity). and {} means an empty set, and
> {u} mean a singlton set.
So are you saying now that the axioms and definitions of ZFC-I are to
be adjoined to your axioms and definitions? And if so, then how were we
supposed to know that without you saying it?
MoeBlee
MoeBlee
> Ok, no problem I can add, the following phrase "were every variable in
> the above axiomatic system is a set based on ZFC-I axiomatic set
> theory".
It's not even clear how that would be formalized. If you want ZFC-I to
apply to your theory, then say that by saying that your theory is
axiomatized by ZFC-I plus your added axioms. That is precise.
> I think this is enough. there is not need to go write all the
> axoims of ZFC except infinity, beside the axioms of this set size
> theory.
No, just say that your axioms are those of ZFC-I plus your added
axioms.
MoeBlee
MoeBlee
MoeBlee wrote:
> how were we
> supposed to know that without you saying it?
Okay, you addressed that question in your post to Virgil, so nevermind,
we can move on.
MoeBlee
zuhair
If the whole idea is silly, then why should we move on.
We'll certainly have more interesting discussions after one year
when I will be able to read this subject systematically and
thoroughlly. Reading the book you told me to read, Techniques of Formal
Reasoning: Kalish,Montague and Mar. also Haloms. Do you have other
books you would recommend, since you know very well how far I am
ignorant of logics. After I read these books, should I read Jecks book
on set theory. He have two expensive books on Amazon.com.
Zuhair
zuhair
If the whole idea is silly, then why should we move on.
zuhair
If the whole idea is silly, then why should we move on.
MoeBlee
> MoeBlee wrote:
> > MoeBlee wrote:
> > > how were we
> > > supposed to know that without you saying it?
> >
> > Okay, you addressed that question in your post to Virgil, so nevermind,
> > we can move on.
> >
> > MoeBlee
>
> If the whole idea is silly, then why should we move on.
I mean we can move on from the matter of your not having said that ZFC
is part of your theory; so we can move on to the main subject of this
thread, which is your latest proposal or onto whatever we want to talk
about.
> We'll certainly have more interesting discussions after one year
> when I will be able to read this subject systematically and
> thoroughlly. Reading the book you told me to read, Techniques of Formal
> Reasoning: Kalish,Montague and Mar. also Haloms.
I don't recommend Halmos for starting. I think it's better as an
overview, a review, after having read Enderton's 'Elements Of Set
Theory' and Suppes's 'Axiomatic Set Theory'. I find each of Enderton
and Suppes has certain advantages, so, personally, what I did was blend
the two books into my own list of theorems and proofs taken from the
two books, choosing between their different formlations as I went
along. But I did read each of them alone first. I recommend starting
with Enderton's.
> Do you have other
> books you would recommend, since you know very well how far I am
> ignorant of logics. After I read these books, should I read Jecks book
> on set theory.
Jech's book is considered a classic and comes very highly recommended.
It goes into some more advanced material. I think by the time you're
ready for Jech's book, you'll already have been exposed to enough in
the literature that you'll be able to see for yourself whether Jech is
what you want to study next.
Kunen's book is another more advanced book. But I think the first
chapters would be good for you to read anytime.
There are so many books to recommend, but for starting, I recommend
this course, in this order:
Kalish, Montague, and Mar
Enderton's 'Elements Of Set Theory'.
Enderton's 'A Mathematical Introduction To Logic'
Suppes's 'Axiomatic Set Theory'.
Then blend Enderton's 'Elements Of Set Theory' with Suppes's 'Axiomatic
Set Theory'.
Then you can start branching off from there.
MoeBlee
Virgil
"zuhair" <zalj...@yahoo.com> wrote:
Is there some reason for posting this same post three times?
zuhair
Yes, when I post, "Server Error" is shown. so I post again, thinking
that it wasn't posted.
so it was a techniqual problem.
Sorry for the inconvenience.
Zuhair
zuhair
Thank you very much for this precious information.
Zuhair
Aatu Koskensilta
> Axioms:
> 1) ExAy (x= size y)
> 2) AxAyAz (y=size x , z= size x -> y=z)
> 3) AxAyAzAu: z in x, u in y -> size{z}=size{u}
> 4) AxAyAaAb:a.x={},b.y={},size x=size y,size a = size b -> size aUx =
> size bUy.
> 5) Ay ( y!={} -> size {} < size y )
> 6) AxAyAaAb: a.x={},b.y={}, size x<size y, size a = size b -> size aUx
> < size bUy.
> 7) AxAy ( size x < size y <-> size y > size x ).
I see you're still in the business of producing random formulas. What is
it you wish to accomplish with this seemingly pointless activity?
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus