Why Define Cardinality?
zuhair
In previous posts to this Usenet I tried to make a set theoretic
definition of Cardinality of a set? However all trials actually failed
to have a general defined notion of Cardinality! No single one till
now has come with a definition of Cardinality that is general enough.
One may say that:
A Cardinal is a function from classes to classes such that for any
classes A and B:
Cardinality (A) = Cardinality (B) iff A equinumerous to B.
Were A equinumerous to B iff (there exist an injection from A to B and
there exist an injection from B to A)
Also it is desirable to have the following property added to the
above:
Cardinality(A) is a set iff A is a set.
So this is what cardinality is all about!
We can simply do that by adding the one place function symbol
"| |" to the list of primitives of the language of ZF or NBG\MK,
and add the following axioms:
Axiom:|x|=|y| <-> x equinumerous to y
Axiom:|x| is a set <-> x is a set
And after that we can go define relations between cardinalities like
smaller than and greater than in the Cantorian way.
I think that this would be the most general approach to deal with
Cardinality.
The thing that I am wondering about, is:
In what sense the various definitions of cardinality as special
kinds of *sets* is superior to the above rather simple and most
general approach?
What are we going to gain over the above when we *Define*
Cardinality?
Why we need to *Define* cardinality, why not just stipulate it as a
primitive concept and axiomatize as above?
What are the drawbacks of the primitive approach?
Seeing that all trials to have a general *defined* notion of
Cardinality actually failed, especially when trying to define
Cardinality beyond Regularity and Choice, and also when trying to
define Cardinality of proper classes in absence of Choice, then one
might as well stipulate Cardinality as a primitive function as I did
above.
So WHY DEFINE CARDINALITY?
Zuhair
David Libert
> Hi all,
>
> In previous posts to this Usenet I tried to make a set theoretic
> definition of Cardinality of a set? However all trials actually failed
> to have a general defined notion of Cardinality! No single one till
> now has come with a definition of Cardinality that is general enough.
Yes, it seems that way to me also. Your ones with hereditarily subnumerous
looked like they would probably be good to me at first, and it took me a
while to see the problem for them with amorphous sets.
> One may say that:
>
> A Cardinal is a function from classes to classes such that for any
> classes A and B:
>
> Cardinality (A) = Cardinality (B) iff A equinumerous to B.
>
> Were A equinumerous to B iff (there exist an injection from A to B and
> there exist an injection from B to A)
>
> Also it is desirable to have the following property added to the
> above:
>
> Cardinality(A) is a set iff A is a set.
>
>
> So this is what cardinality is all about!
Yes. That is a good summary. Also the < relation defined by injections.
That is I think a matter of taste. I like the usual approach, but
your suggestion is reasonable too.
There are other things like this. In the Foundations of Math mailing
list, Harvey Friedman proposed a foundational system that had sets and
ordered pairs as primitive, making sensible obvious axioms on ordered pairs
similar to what you did above. Instead of mucking around with Kuraotski
ordered pairs.
I remember Bill Taylor commented favourably on this, I can't remember
if that was in FOM or here in sci.math or sci.logic.
Or there is a famous paper. We have the usual interpretation of
natural numbers into ZF as finite von Neumann ordinals.
There is a famous paper by philosphers, What Numbers Cannot Be.
Maybe Benaceraf is one of them. Objecting to this interpretation.
Because it introduces extraneous truths about numbers such as
1 epsilon 2 not fundamental to our intended understanding of numbers.
I had read that some mathematicians reacted against this paper with
our common idea that we are interested really in the abstract structure
and not the implimentation, so this was structuralism in mathematics
provoked as a reaction against that paper, and was an impetus for
category theory. Category theorists make a big deal of the fact their
definitions are only up to isomorphism, and don't introduce extraneous
distinctions beyond the true interest.
I myself am not very sympathetic to these points of view. I think it
is good to have a very simply axiomatized theory as base.
To me the issue of what is our true interest in these notions
we want to interpret is typically so clear that it doesn't create
real problems. So don't redo all the notations just to focus
clarity on an area that we all understood well intuitively.
There have been real problem areas in mathematics. Dave Renfro
talks about that, how 19th century analysts had some controversy
about what constitued a function. And so the set theortic reduction
was a real step of progress.
Or we see all the time confusion about are we using AC or not,
or are we usiing ordinary AC or global choice. So the set theoritc
work does a real job.
But the sort of hypothetical cases from that Numbers paper by
philosophers. A mathematician asks, by ther naturals do you
mean omega as model of the intuitive naturals, or do you mean
some other isomorphic copy.
These issues don't arise in real world mathematics. They arise
with a philospher imagining mathematicians.
So I think when we proceed as usual, we are in practice doing
something close to what you say. We know what we really essentially
want about cardinality or ordered pairs or naturals, and without
spelling it out the interpretations we give do their job.
And if we wanted to be explicit we could make a footnote, that
the cases of real interest are the interpretations of base theories
as you say into our language.
To me the usual, a simply defined axiomatization with small
language, and then simple interpretative definitions over it, is
simple and it is good enough to do the job in the real world.
Think of the problems that actuially arise in mathematics.
Fir students learning it. For us learning new areas. Conceptual
problems like the definition of function.
These are of a different order than being confused about
the Kuratoski definitioon of order pairs.
But all this is more a matter of temperment.
Anyway, regarding cardinals specifically. I like ZFC, ie
AC included specifically. It is my base theory, and in there
I like von Neumann cardinals.
I am fascinated by AC though, and accordingly I also like to
look back at ZF ~AC models. Not as the picture of the true
world, but to look at pathologies and say thank God I
believe AC.
But for those I still like regularity. Regularity gives
something with clear meaning built from below. Without it
all these circles. Where does it all begin.
So accordingly, I don't mind depending on regularity.
So in the moments when I step back from AC, I don't mind
resting on regularity, and accordingly I like Scott's trick.
So those do it for me: von Neumann cardinals and Scott's
trick. They cover the cases of interest.
They don't introduce newe confusions and they are simple.
Look at all the troubles we have been having deciding
if the other cardrinailites are sets. For von Neumann
cardinals and Scott's trick with regualarity used, the
corresponding step is so easy.
And that's good. Don't labour over these little points.
Settle them fast and get on to the real parts that are
genuinely hard and not these manufactured difficulites
like the philophers gave us for numbers.
So these particular usual definitions I like do rest
on AC or regularity.
So is that an issue, should we expect to get by with less?
One thing along these lines is showing how AC gives some
help with defining cardinality. My article citing a Jech
result :
[1] David Libert "'Partial' AC/Well-Ordering?"
sci.math June 20, 2000
http://groups.google.com/group/sci.math/msg/16e537e0dd63ddc1
We need AC if we want cardinalities to have their own cardinality.
Aatu noted in a recent thread that without AC or regularity or
an assumotion such as every set is isomorphic to a well-founded
set, he didn't know of a definition of cardinality.
After reading your article above, it occuured to me to try
for a result along these lines.
So drop regularity. Inside ZFC (usual with regularity)
we will build inner models of ZFC with regularity dropped
similar to what I did in
[2] David Libert "A new definition of Cardinality"
sci.logic, sci.math Nov 23
http://groups.google.com/group/sci.math/msg/721cb8170033cf84
Namely, make atoms by disjointifying from sets, and make these
atoms into fake sets in descending singleton towers.
So these towers can be permuted at will, so we will make FM styke
models as in
[3] David Libert "Cohen symmetric choiceless ZF models"
sci.logic July 6, 2000
http://groups.google.com/group/sci.logic/msg/b4271c2585d2f1e5
I will make a proper class of sets, each being a set of singleton towers.
I will put in bijections between each pair of sets in this proper class
and require all permutations to respect them. I allow any set of towers
from this class to permute to any other set of towers. I permute the tower
along with the set it was in. And induce between sets by the special
bijections.
I end up with a permutatiion model, with a proper class of sets each bijective
to each other, where any can be permuted to another.
Now I make omega copies of the above, and allow permuations among the copies.
I end up with omega many proper classes of sets which can all be permuted
within each of the those proper classes. But it is omage proper classes in
the outer AC model. In the inner permutation model it looks like amorphous
many proper classes, since we can permute among them.
Take an FM style model of this with finite support.
Then I think the permutatioins show there is no defininable cardinality
in your sense in this model. Namely fo outside the support of a purported
definition to get to a proper class in our list with no support. If its
cardinaily evaluation didn't depend on atoms from inside it, you could
permute it to another untouched proper class and make those 2 have
same defined cardinality.
So its cardinality depended on atoms from inside. But permute these
around inside the proper class to get a proper class of members for
the cardinaity.
That's as much as I wil try to write now as "proof".
Anyway this model fails the bijective with well founded as well
as regularity. It fails AC by the permutations.
So this seems if something like above is correct not to have
any cardinality definable in your sense. Similar to Aatu's
observation.
So this seems to show, to define cardinality we need AC or
regularity or that bijective to well-founded axiom.
But if we have those we can go back to the simple definitions
of von Neumann cardinals or Scott's trick.
Anyway, above I argued philosophically but not mathematically
why to define cardinality if we can, then the simplicity issue
for those defintions whent the background (AC, regularity or the
other axiom) is available, and now I have given a mathematical
(well sort of : a vague exposition) argument why to have those
axioms available.
--
David Libert ah...@FreeNet.Carleton.CA
David Libert
[Deletion]
> So these particular usual definitions I like do rest
> on AC or regularity.
>
> So is that an issue, should we expect to get by with less?
>
> One thing along these lines is showing how AC gives some
> help with defining cardinality. My article citing a Jech
> result :
>
> [1] David Libert "'Partial' AC/Well-Ordering?"
> sci.math June 20, 2000
> http://groups.google.com/group/sci.math/msg/16e537e0dd63ddc1
>
>
> We need AC if we want cardinalities to have their own cardinality.
Opps that was the wrong reference.
Instead I meant
[2] David Libert "Clarification of the Question"
sci.logic May 29, 2000
http://groups.google.com/group/sci.logic/msg/59c963e4ef829f50
The part I meant above was near the end of [2] where I wrote
> So ZFC (ZF + Axiom of Choice (AC)) defines cardinal numbers as
>special examples of sets of that many elements. For example, the number
>0 is the empty set. The number 1 is a one element set of a specific
>form. The number 5 is a 5 element of the corresponding form. ZFC
>proves that every set is bijective with exactly one set of the special
>form, so this form is used to define numbers by taking for each size a
>representative set of that size.
>
> ZF (no AC) proves that the above definition works for all finite
>sets. ZFC proves it works for all sets, including infinite ones. In
>fact this last implication reverses over ZF, ZF proves the above
>definition provides a number for every set <-> AC. So this definition
>breaks down without AC (at least for infinite cardinals, for finite its
>ok).
>
> There is another way to define cardinal numbers in ZF. This
>definition associates to each set a cardinal number measuring its size
>(though a different system of numbers than the ZFC version above). This
>is a good definition in that two sets get assigned the same number iff
>they are bijective with each other. On the other hand, these numbers
>don't have their own cardinality: ie now the set representing 5
>does not have 5 members. This method is using Scott's trick.
>
> It would be possible to amalgamate the two methods, for example using
>the ZFC style over finite, and the ZF over infinite, doing this all in
>ZF. So you could arrange that 5 has 5 elements. But you can only do
>this so far over ZF alone. At infinite levels you have to include cases
>where the numbers are not representative sets of size their own number.
>I seem to recall reading in something by Jech that it has been proven
>that if ZF is consistent then there is a model of ZF in which there is
>no definable method of assigning to every set X a set N(X) s.t.
>for all X,Y N(X) = N(Y) <-> X and Y are bijective with each other
>and such that for all X X is bijective with N(X).
--
David Libert ah...@FreeNet.Carleton.CA
Tim Little
> We can simply do that by adding the one place function symbol
> "| |" to the list of primitives of the language of ZF or NBG\MK,
> and add the following axioms:
>
> Axiom:|x|=|y| <-> x equinumerous to y
> Axiom:|x| is a set <-> x is a set
Should this function be idempotent, i.e. for all x, ||x|| = |x|? This
property is sometimes useful in ordinary reasoning with cardinalities.
> What are the drawbacks of the primitive approach?
Limitations on what you can prove with it, perhaps?
- Tim
zuhair
> On 2009-12-10, zuhair <zaljo...@gmail.com> wrote:
>
> > We can simply do that by adding the one place function symbol
> > "| |" to the list of primitives of the language of ZF or NBG\MK,
> > and add the following axioms:
>
> > Axiom:|x|=|y| <-> x equinumerous to y
> > Axiom:|x| is a set <-> x is a set
>
> Should this function be idempotent, i.e. for all x, ||x|| = |x|? This
> property is sometimes useful in ordinary reasoning with cardinalities.
I don't see where this property is essential in the definition of
cardinality, neither Scott's cardinals, nor Frege's cardinals have
this property, Only Von Neumman's cardinals possess them, and I don't
think it is of any importance really. The cardinality of the
cardinality of a set is not itself an issue, I guess.
zuhair
OK, give me an example of such limitation.
>
> - Tim
zuhair
> zuhair (zaljo...@gmail.com) writes:
> > Hi all,
>
........snip.......
>
> Anyway this model fails the bijective with well founded as well
> as regularity. It fails AC by the permutations.
>
> So this seems if something like above is correct not to have
> any cardinality definable in your sense. Similar to Aatu's
> observation.
>
> So this seems to show, to define cardinality we need AC or
> regularity or that bijective to well-founded axiom.
>
> But if we have those we can go back to the simple definitions
> of von Neumann cardinals or Scott's trick.
>
> Anyway, above I argued philosophically but not mathematically
> why to define cardinality if we can, then the simplicity issue
> for those defintions whent the background (AC, regularity or the
> other axiom) is available, and now I have given a mathematical
> (well sort of : a vague exposition) argument why to have those
> axioms available.
>
> --
> David Libert ah...@FreeNet.Carleton.CA
The issue is, does the primitive approach I wrote above fail to reveal
the cardinality of these amorphous proper classes?
Second the definitions that I wrote for Cardinality especially the
last one of mine present on this link:
http://groups.google.com.jm/group/sci.logic/browse_thread/thread/d0c31a297367393e?hl=en
under topic: another definition of Cardinality.
Now this definition assumes
(1)Stronge extensionality.
(2) H_(<#x) is a set.
(3) Minimal transitive closures of Equinumerous sets exist.
Non of the above require regulariy, nor the full AC axiom.
ZF+minus Regularity+(1)+(2)+(3)
is weaker than ZFC, and actually even weaker than ZF.
So these cardinals work under condtions weaker than ZF, so they are
expected to be stronger than Scott's cardinals.
Actually I think that ZF+(1)+(2) can prove (3), but I am not sure.
(when I write ZF it means Regularity included).
Stronge Extensionality however shun your permutational models.
Zuhair
Frederick Williams
>
> We can simply do that by adding the one place function symbol
> "| |" to the list of primitives of the language of ZF or NBG\MK,
> and add the following axioms:
>
> Axiom:|x|=|y| <-> x equinumerous to y
> Axiom:|x| is a set <-> x is a set
Tarski just has |x| = |y| iff x equinumerous to y. And Suppes follows
him.
--
Pigeons were widely suspected of secret intercourse with the
enemy; counter-measures included the use of British birds of
prey to intercept suspicious pigeons in mid-air.
Christopher Andrew, 'Defence of the Realm', Allen Lane
Herman Jurjus
> Tarski just has |x| = |y| iff x equinumerous to y. And Suppes follows
> him.
Which reminds me: does anyone here know the correct pronunciation of
that name, 'Suppes'? (I recently heard someone pronounce it a bit like
'shoe-piece'. Someone else pronounced it more or less like 'suppers'
(plural of 'supper').
--
Cheers,
Herman Jurjus
Frederick Williams
I pronounce it the second may only because that's how I heard a teacher
of set theory (Fred Benenson) pronounce it.
Herman Rubin
As I know him personally, the pronunciation is like soo-peas.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
zuhair
> zuhair (zaljo...@gmail.com) writes:
> > Hi all,
>
generally like the way how you explain things. However day after day
it appears to me as if it is impossible to come with a *defined*
notion of Cardinality of every set in ZF minus Regularity.
I don't know if that can be actually proved
somehow, but it appears so.
Of course assuming additional conditions on ZF minus Regularity like
every set is equinumerous to a well founded set, or H_<#x exist and
a minimal transitive closure of equinumerous sets exist, besides
stronge Extensionality, etc... all such assumtions which are weaker
than ZF, I say assuming those assumptions one can
define cardinality for sets under such assumptions. But to have
a general notion of cardinality in ZF minus Regularity without
additional assumptions, doesn't seem to be workable. One would really
need to try to prove such a thing perhaps by model theory or the
alike.
Anyhow perhaps I am mistaken, but one should take into consideration
that the less assumptions we add to ZF minus Regularity the more it
becomes difficult to define cardinality, and the more the definition
of such cardinals become complex and more complex.
I say seeing all that, why not one simply regard Cardinality as a
primitive concept, and axiomatize it as I did above.
Matters would be much simpler.
However till now, nobody really answered my question of:
What we are gaining by *defining* cardinality, I am not seeing an
obvious example of the benefits of *defining* cardinaly, there is
nothing much to gain over the simple approach of stipulating that
cardinality is a primitive one place function symbol and axiomatize
it
actually even by one axiom.
Axiom of Cardinality: |x|=|y| iff x equinumerous to y
This is more than enough. but preferable we add the axiom
Axiom: x is a set -> |x| is a set.
I thin the opposite direction can be proved in MK (in FOL(e,=,| |)) +
Axiom of Cardinality + Axiom above.
it is a very simple approach, and since cardinality is stipulated as
a primitive function of the language, then every object in the
universe
of discourse of the above theory would have cardinality, so it is the
most general approach.
Can anybody give me a clear example of the benefits of *defining*
cardinality over the simple approach of *primitivizing* cardinality.
Zuhair
zuhair
wrote:
> zuhair wrote:
>
> > We can simply do that by adding the one place function symbol
> > "| |" to the list of primitives of the language of ZF or NBG\MK,
> > and add the following axioms:
>
> > Axiom:|x|=|y| <-> x equinumerous to y
> > Axiom:|x| is a set <-> x is a set
>
> Tarski just has |x| = |y| iff x equinumerous to y. And Suppes follows
> him.
Yea, that is the minimal requirement no doubt, but it is desirable to
have the second condition , actually all what we need is to add
For all x ( x is a set -> |x| is a set )
if x is a proper class then I think it would be provable that
cardinality(x) would be a proper class too.
Zuhair
David Libert
> On Dec 10, 4:15=A0am, ah...@FreeNet.Carleton.CA (David Libert) wrote:
>> zuhair (zaljo...@gmail.com) writes:
>> > Hi all,
>>
> ........snip.......
>>
>> as regularity. =A0It fails AC by the permutations.
>>
>> =A0 So this seems if something like above is correct not to have
>> any cardinality definable in your sense. =A0Similar to Aatu's
>> observation.
>>
>> =A0 So this seems to show, to define cardinality we need AC or
>> regularity or =A0that bijective to well-founded axiom.
>>
>> =A0 But if we have those we can go back to the simple definitions
>> of von Neumann cardinals or Scott's trick.
>>
>> why to define cardinality if we can, =A0then the simplicity issue
>> for those defintions whent the background (AC, regularity or the
>> other axiom) is available, and now I have given a mathematical
>> axioms available.
>>
>> --
>> David Libert =A0 =A0 =A0 =A0 =A0ah...@FreeNet.Carleton.CA
>
> The issue is, does the primitive approach I wrote above fail to reveal
> the cardinality of these amorphous proper classes?
No, your axiomatization is fine, even for such cases. I am giving an
example, without regularity or AC, where cardinality can;t be defined.
If instead you axiomatize it, as you did, then the whole point is you
don't need to define it. Like the subject of this thread. So your
axiomatization works instead.
> Second the definitions that I wrote for Cardinality especially the
> last one of mine present on this link:
>
> http://groups.google.com.jm/group/sci.logic/browse_thread/thread/d0c31a2973=
> 67393e?hl=3Den
>
> under topic: another definition of Cardinality.
>
> Now this definition assumes
>
> (1)Stronge extensionality.
> (2) H_(<#x) is a set.
> (3) Minimal transitive closures of Equinumerous sets exist.
I wrote about these axioms in relation to that definiion, into that thread:
[1] David Libert "Another definition of Cardinality:"
sci.logic, sci.math Dec 10 my 2nd post, 5th in thread
http://groups.google.com/group/sci.math/msg/19734edc6faddb45
I noted there I don't see how axiom 2 relates to this definition.
Also these axioms don't seem to resolve the potential issue of
cardinalities not being sets.
> Non of the above require regulariy, nor the full AC axiom.
>
> ZF+minus Regularity+(1)+(2)+(3)
>
> is weaker than ZFC, and actually even weaker than ZF.
Weaker than ZFC, I agree, we have that as known. Strictly weaker than
ZFC (if that is what you wrre thinking of above) seems a reasonable
conjecture. Though it would take some work to prove that. I don't
think we have done that yet.
About relations to ZF. ZF proving 2 is that ongoing question we have
not settled. And ZF proving 3, I outlined a possible countertermodel
previously, as noted in
[2] David Libert "Another definition of Cardinality:"
sci.logic, sci.math Dec 10 my 1st post, 4th in thread
http://groups.google.com/group/sci.math/msg/ae3e8f93dc2de550
ZF does prove 1, and 1 is indeed strictly weaker, as discussed
in [2].
> So these cardinals work under condtions weaker than ZF, so they are
> expected to be stronger than Scott's cardinals.
I noted above I don't see how 1-3 gets these cardinals to be sets.
So its not clear that 1-3 are good enough to make this definition
work.
2 was not really needed for this definition, as above and from [1].
3 is used for the definition, and I don't see how to get 3 from ZF,
I only see it wioth AC.
So two problems from ZF. To get from ZF to 1-3 (ie to get 3),
then to get from 1-3 to the definition (cardinalities being
sets).
One issue where you have apparently weakened from ZF. Scott
cardinals use regularity. For your new definition, you can drop
regularity, and replace it by axiom 3.
Though as I noted the usual proof of axiom 3 also uses AC.
So its both weaker and stronger.
> Actually I think that ZF+(1)+(2) can prove (3), but I am not sure.
>
> (when I write ZF it means Regularity included).
I don't know. It would be surprising to me if 2 helps to prove 3.
If we drop 2, there is my possible couterexample from [2].
I haven't checked if this satisfied 2, so I am not sure about your
claim here.
I don't see how 2 closely relates to the definition or to 3.
2 is about sets with extra cardinality restrictions at depth in
their transitive closure, 3 is about sets with no such restriction.
I don't see how information about the one kind gives you information
about the other.
> Stronge Extensionality however shun your permutational models.
[2] discussed relations of strong extensionality to usual
extensionality. In particular, they are equivalent when no sets are
members of themselves.
My permutation model earlier in this thread has no set a member of
itself, and it satisfies usual extensionality. So it satisfies
strong extensionality.
> Zuhair
--
David Libert ah...@FreeNet.Carleton.CA
zuhair
> > 67393e?hl=3Den
>
> > under topic: another definition of Cardinality.
>
> > Now this definition assumes
>
> > (1)Stronge extensionality.
> > (2) H_(<#x) is a set.
> > (3) Minimal transitive closures of Equinumerous sets exist.
>
> I wrote about these axioms in relation to that definiion, into that thread:
>
> [1] David Libert "Another definition of Cardinality:"
> sci.logic, sci.math Dec 10 my 2nd post, 5th in thread
> http://groups.google.com/group/sci.math/msg/19734edc6faddb45
>
> I noted there I don't see how axiom 2 relates to this definition.
I posted a proof of that in that subject, I will reiterate it here.
Since for any set x, any member of TC(x) is a subset of TC(x), then
every set x here hereditarily subnumerous to TC(x).
So the Cardinality(x) would be a subclass of the set of all
sets hereditarily sunbumerous to the minimal transitive
closure of sets equinumerous to x.
Since the later is a set according to axiom 1, thus Cardinality(x) is
a set
(Separation).
>
> Also these axioms don't seem to resolve the potential issue of
> cardinalities not being sets.
>
> > Non of the above require regulariy, nor the full AC axiom.
>
> > ZF+minus Regularity+(1)+(2)+(3)
>
> > is weaker than ZFC, and actually even weaker than ZF.
>
> Weaker than ZFC, I agree, we have that as known. Strictly weaker than
> ZFC (if that is what you wrre thinking of above) seems a reasonable
> conjecture. Though it would take some work to prove that. I don't
> think we have done that yet.
>
> About relations to ZF. ZF proving 2 is that ongoing question we have
> not settled. And ZF proving 3, I outlined a possible countertermodel
> previously, as noted in
>
> [2] David Libert "Another definition of Cardinality:"
> sci.logic, sci.math Dec 10 my 1st post, 4th in thread
> http://groups.google.com/group/sci.math/msg/ae3e8f93dc2de550
>
> ZF does prove 1, and 1 is indeed strictly weaker, as discussed
> in [2].
>
> > So these cardinals work under condtions weaker than ZF, so they are
> > expected to be stronger than Scott's cardinals.
>
> I noted above I don't see how 1-3 gets these cardinals to be sets.
> So its not clear that 1-3 are good enough to make this definition
> work.
>
> 2 was not really needed for this definition, as above and from [1].
It is, see above.
>
> 3 is used for the definition, and I don't see how to get 3 from ZF,
> I only see it wioth AC.
>
> So two problems from ZF. To get from ZF to 1-3 (ie to get 3),
> then to get from 1-3 to the definition (cardinalities being
> sets).
>
> One issue where you have apparently weakened from ZF. Scott
> cardinals use regularity. For your new definition, you can drop
> regularity, and replace it by axiom 3.
Still Scott cardinals can be modified to work out of Regularity.
We can drop Regularity and replace by Coret axiom
every set is equinumerous to a well founded sets.
and redefine Scott cardinals as Aatu did in one thread, stipulating
that every member of a Scott cardinal is well founded.
This will release these cardinals from the constrains of Regularity.
Of course this is provided that we shun the existence of Ur-elements
and quine atoms and the alike sets.
>
> Though as I noted the usual proof of axiom 3 also uses AC.
>
> So its both weaker and stronger.
>
> > Actually I think that ZF+(1)+(2) can prove (3), but I am not sure.
>
> > (when I write ZF it means Regularity included).
>
> I don't know. It would be surprising to me if 2 helps to prove 3.
Yea, Regularity can do the job I think.
>
> If we drop 2, there is my possible couterexample from [2].
> I haven't checked if this satisfied 2, so I am not sure about your
> claim here.
>
> I don't see how 2 closely relates to the definition or to 3.
> 2 is about sets with extra cardinality restrictions at depth in
> their transitive closure, 3 is about sets with no such restriction.
> I don't see how information about the one kind gives you information
> about the other.
>
> > Stronge Extensionality however shun your permutational models.
>
> [2] discussed relations of strong extensionality to usual
> extensionality. In particular, they are equivalent when no sets are
> members of themselves.
>
> My permutation model earlier in this thread has no set a member of
> itself, and it satisfies usual extensionality. So it satisfies
> strong extensionality.
ah I see. But you cannot have a proper class of towers of singletons,
the lemma that for every set y in TC(x) there must exist a finite
sequence x1,x2,x3,....,xn
were each xi+1 e xi , x1 e x and y=xn would prevent us from having a
proper class of these towers, so we can have at most a countable set
of them .
I spoke one day about these sets, these violate Regularity, but still
one cannot have a proper class of these sets, you can have a proper
class of sets having those in their transitive closures yes, but that
should not be a problem.
>
> > Zuhair
>
> --
> David Libert ah...@FreeNet.Carleton.CA- Hide quoted text -
>
> - Show quoted text -