The magic of Hereditarily Hereditary Cardinals.

[zuhair]

Hi all,

This topic comes as a continuation to the previous two topics
presented into this usenet:

1. A new definition of cardinality at:

http://groups.google.com/group/sci.logic/browse_thread/thread/3e92b5c2185a1450?hl=en

2. Towards a new definition of cardinality at:

http://groups.google.com/group/sci.logic/browse_thread/thread/a33b3ffbf242c883?hl=en

I'll concentrate here on the later topic.

First of all, I would like to say that the question weather
choice is needed to prove that there exist a cardinal for every set
(cardinal defined in the first topic(see link above)) is still an open
question.

However I do think that the second definition of cardinality do not
require neither choice nor regularity.

Let me present this definition again:

"A cardinal is a class of all hereditarily hereditary sets strictly
subnumerous to some set".

Define(hereditary):
x is hereditary <->
for all y ( y e Tc(x) -> y strictly subnumerous to x )

Were Tc(x) stands for the 'transitive closure of x' defined
in the standard manner.

Tc(x)=U{x,Ux,UUx,UUUx,......}

were "U" stands for "union" as define in Z set theory.

Now we come to define "hereditarily hereditary"

Define(hereditarily hereditary):
x is hereditarily hereditary <->
(x is hereditary & for all y ( y e Tc(x) -> y is hereditary ))

c is said to be the cardinality of x if and only if c is the set of
all hereditarily hereditary sets strictly subnumerous to x.

c=cardinality(x) <->
c={y| y is hereditarily hereditary & y strictly subnumerous to x}

I shall use the symbol "HH" to denote the predicate
"hereditarily hereditary" and the symbol "<" to denote
"strictly subnumerous" , so the above would
be written as:

cardinality(x) = c <-> c={y| HH(y) & y < x }

Define(cardinality(x)):

cardinality(x)=c <-> for all y (y e c <->(HH(y) & y < x)).

So to simplify matters, a hereditarily hereditary set x is a set that
is larger(strictly supernumerous) to every member in it, and also
every member y of x is strictly larger than any of y members, and
those are also larger than their members etc...

So for example the cardinality of the empty set would be the empty set
itself since there are no sets that is strictly subnumerous to the
empty set. And of course the empty set
itself also be a hereditarily hereditary set!

Now the cardinality of any singleton set x would be the set of all
hereditarily hereditary subnumerous to x, and the set fulfilling these
criteria is the empty set, so the cardinality of any singleton set
would be {{}}. which is hereditarily hereditary

Now the cardinality of any pair would be the set of all hereditarily
hereditary sets subnumerous to that ordered pair, and these would be
the empty set and {{}}, so the cardinality of any pair would be
{{},{{}}}, etc...

Now the cardinality of Omega, would be Omega itself, actually the
cardinality of any countable set would be Omega.

Of course Omega is hereditarily hereditary set also.

Now all infinite subsets of Omega are also hereditarily hereditary
also.

Now the cardinality of aleph-1 would be the set of all hereditarily
hereditary sets subnumerous to it, so it would include all finite
ordinals and all infinite subsets of omega as members in it, and this
set is itself also hereditarily hereditary, since it must be larger
than omega since all infinite subsets of omega are members of it,
actually this set would be equinumerous to power(omega).

Observations:

1. Every cardinal is itself a hereditarily hereditary set.

2. These cardinals are too small in size, thus it would be expected
that all of them would be sets and not proper classes.

3. The cardinality of every countable set is the countable Von Neumann
cardinal equinumerous to it.

4. I think it can be proved that for every arbitrary set x , there
exist a cardinality of it that is a set, in ZF minus Regularity.
and of course without choice.

5. Each cardinal contain the smaller cardinals as members in it.

6. These cardinals motive the continuum hypothesis

The cardinality of aleph_1 would be the set of all finite ordinals and
all infinite subsets of omega, which is as we know equinumerous to
power omega.

Now lets define the term "comparable"

x is comparable to y iff
(Exist an injection from x to y or Exist an injection from y to x).

So for example if we stipulate the following axiom:

For any two comparable sets x and y if y is the cardinality of x
then y is equinumerous to x.

Then we'll have the continuum hypothesis restricted to comparable
sets.

The nice result is when we have incomparability.

So without choice suppose x is a set that is infinite but Dedekindian
finite, then its cardinality would be Dedekindian infinite, actually
it would be Omega. but yet its cardinality is incomparable with it.

now regarding these kind of sets, we'll see that all of its infinite
proper subsets would be strictly subnumerous to it, but yet
have the same cardinality namely Omega.

We say that these sets have fixed cardinality, so their cardianlities
not longer reflects relationships of subnuemerousity or
equinuemrousity any more.

while if a set is comparable to its cardinality, then the cardinality
will not be fixed like in the case with the finite Dedekindian
infinite sets, and would reflect the relationships of subnuemrousity
and equinumerousity in the usual manner.

With the above axiom Power(omega) will always equal aleph-1
in cardinality ,since even if we suppose that power omega is
incomparable with aleph-1, then still both of them would have the same
cardinality.

Strange results ha.

Last point that I want to say.

I don't know how to prove that these cardinals are sets. But they must
be, since they are too small to be proper classes.

I mean even without the axiom of comparability that I made above,
still we can define these cardinals, without the need for choice nor
regularity, i.e. we can prove them to exist for every set in
ZF-regularity.

Anybody have a clue to how this can be done. I think it shouldn't be a
difficult task.

anybody help me with that.

Regards

Zuhair

[David Hartley]

<I've changed cardinality to Zcardinality, to avoid confusion with the
usual meaning>

In message
<211a7044-cfac-4233...@s20g2000yqd.googlegroups.com>,
zuhair <zalj...@gmail.com> writes

>So without choice suppose x is a set that is infinite but Dedekindian

>finite, then its Zcardinality would be Dedekindian infinite, actually
>it would be Omega. but yet its Zcardinality is incomparable with it.

>
>now regarding these kind of sets, we'll see that all of its infinite

>proper subsets would be strictly subnumerous to it, but yet have the .
>same Zcardinality namely Omega

If this is so, then you need a different name for your concept. Anything
claiming to be a type of cardinality would need the property that sets
with the same cardinality are equinumerous.

But is it so...? Suppose X is an infinite Dedekind-finite set.
Replace each element a by X-{a}. Then do the same with the elements of
each of these new elements, and so on. In the limit - which needs
justification - you've got a set of the same (conventional) cardinality
as X but which is HH. It is therefore a subset of its Zcardinal, as is
its transitive closure. Every cofinite subset of X is a member of the
Zcardinal, which is definitely not omega. Of course such a set is
incompatible with the Axiom of Regularity.

This X has the property a U {a} = X, for any a in X, as has every member
of its transitive closure. A sort of maximal counter-example to
regularity. Also in some way an opposite of a von-Neumann ordinal: every
member of the transitive closure is the successor of each of its
elements.

I've no idea how to rigorously construct such sets, but can they be
proved not to exist in the absence of regularity and choice?
--
David Hartley

[David Hartley]

In message <lc9JaWGz...@212648.invalid>, David Hartley
<m...@privacy.net> writes

>But is it so...? Suppose X is an infinite Dedekind-finite set.
>Replace each element a by X-{a}. Then do the same with the elements of
>each of these new elements, and so on. In the limit - which needs
>justification - you've got a set of the same (conventional) cardinality
>as X but which is HH. It is therefore a subset of its Zcardinal, as is
>its transitive closure. Every cofinite subset of X is a member of the
>Zcardinal, which is definitely not omega. Of course such a set is
>incompatible with the Axiom of Regularity.
>
>This X has the property a U {a} = X, for any a in X, as has every
>member of its transitive closure. A sort of maximal counter-example to
>regularity. Also in some way an opposite of a von-Neumann ordinal:
>every member of the transitive closure is the successor of each of its
>elements.

... well that was a bit silly. Such a set can't exist. (No a,b in X can
be disjoint, but c in a in X and c in b in X would imply a = b.)

However, it still may be possible to have an infinite Dedekind-finite
set X where every element has the cardinality of X - {a}, each of their
elements that of X - {a,b} and so on throughout the transitive closure.
Such a set will have a Zcardinal bigger than omega.
--
David Hartley

[David Libert]

I am writing this as a followup to Zuhair's base article of this
thread, since I will be directly responding to things there and
want them quoted. But this post also had general inspiration
from David Hartley's posts in this thread.

zuhair (zalj...@gmail.com) writes:
> Hi all,
>
> This topic comes as a continuation to the previous two topics
> presented into this usenet:
>
> 1. A new definition of cardinality at:
>
> http://groups.google.com/group/sci.logic/browse_thread/thread/3e92b5c2185a1450?hl=en
>
> 2. Towards a new definition of cardinality at:
>
> http://groups.google.com/group/sci.logic/browse_thread/thread/a33b3ffbf242c883?hl=en
>
>
> I'll concentrate here on the later topic.
>
> First of all, I would like to say that the question weather
> choice is needed to prove that there exist a cardinal for every set
> (cardinal defined in the first topic(see link above)) is still an open
> question.
>
> However I do think that the second definition of cardinality do not
> require neither choice nor regularity.
>
> Let me present this definition again:
>
> "A cardinal is a class of all hereditarily hereditary sets strictly
> subnumerous to some set".

Zuhair's acticle continues to review definitions of some of those terms.

So above is quoted Zuhair's 2nd definition, which was introduced in thread 2.

In thread 1. Zuhair discussed a 1st defintion. I quote that first definition
from the base article of thread 1.

http://groups.google.com/group/sci.logic/msg/77269fe363cf905f

>I would like to suggest the following definition:
>
>4) The cardinality of any set x is: The class of all sets
>that are equinumerous to x were every member of their transitive
>closure is strictly subnumerous to x.

I will be discussing these two definitions and related ones, so I will
introduce a uniform notation for definitions like this.

All these definitions are to define cardinailty of set x, and they define
the set of all sets having certain cardinality restrictions on themselves and
in their transitive closures. Specifically the cardinaily restrictions on
various parts are either <= #x or < #x.

So I will denote these respective restrictions by <= or by < .

These definitions can be rephraseed as placing cardinality restrictions on
various members of Tc({y}) for y a possible member of the set being defined.
I take Tc({y}) instead of Tc(y), so that y will be unriformly included in the list.

The definitions can be phrased on placing cardinailty restrictions on various
depths of membership in Tc({x}).

The first definition from 1. required its member set y to be
have #y , #x, in other words at depth 0 in Tc({y}) to have <=.

In Tc(y), in other words depth >= 1 in Tc({y}), the defintion
required <.

So I represent this first definition by signature <= <. That is matching
restricton to depths starting at depth 0. Let the last member of the signature
apply to the entire infinite tail of remaining depths.

We turn now to the 2nd definition, this from thread 2. , which I repeat:

> "A cardinal is a class of all hereditarily hereditary sets strictly
> subnumerous to some set".

So this defintion has < at depth 0.

The definition uses hereditarily hereditary instead of hereditary as the
first definition. What that amounts to, is putting a cardinality restriction
at depth >= 2 in Tc({y}).

This definition imposes no cardinialty restriction on depth 1. I will denote
the absense of a cardinality restrciton as _ (ie like a blank).

So this 2nd definition has signature < _ < .

So two changes have been made from 1st to 2nd definition: depth 0 has
changed from <= to < , and depth 1 < restriction has been dropped.

I will be discussing such defintions in general out to finite length signature.

A relevant case other than the two Zuhair gave, for background discussion is
length 1 signature. For these I will use outer () to show I am talking about
a signature.

So for example, HC is (<=) for x countable, and H_kappa is (<)
for #x = kappa.

So Jech's paper from thread 1. is about HC. And discussions of Zuhair's
definitions mentioned (<) as a preliminary result.

We are interested in whether these classes defined by signatures are sets
or proper classes.

First note, that thus question for a siganature only depends on the final,
ie tail, restriction in the signature. (... ?) at the tail depth has
copies of (?). If there are only set many ways have (?), then
any (... ?) signature above that level, even with no cardinality restrictions
at all is included in some finite (length original part of signatire)
iteration of power set over (?) .

So it all amount to deciding singleton signatures.

(_) is a proper class: in fact it is V.

Any other singleton signature is imposing a set sized restriction.

My final answer on all these is that they are provably sets in ZF using
regularity and replacement, but no choice.

This was the generalization of Jech's argument, as I claimed in thread 1.

My first thread 1. article gave a countermodel for ZFC but with regularity
dropped, and my 2nd thread 1. aricle claimed a countermodel for ZC,
Zermelo's theory with AC, no replacement.

Ig these are correct, iot settles all cases of signatures, regarding being
sets or proper classes.

In particular, both Zuhair's definitions are sets in ZF using regularity and
replacement but no AC.

But now a second question. David Hartley raised the point, a reasonable
definition of "cardinality" should assign the same "cardinal" to
euqipollent sets (ie having a bijection between them).

ZFC proves both Zuhair's definitions have this property.

But I think I have found ZF ~AC models where this fails for both defintions.
Showing these defintions are not good ones just in ZF.

Before I get to that, I will note a related ZF question which I thought of
before the final models, still interesting.

So this issue of this question makes no problems for 1st defintion <= <.
Because of the <=. #x is apaprently recoverable as the unique largest appearing
at depth 0.

But it is a question for < _ < the 2nd defintion

Namely because this cardinality(x) only depends on which cardinals, in the
conventional sense, are < #x. (Do conventional cardinatity in ~AC, for
example Scott-Potter).

So the question: is it possible to have distinct cardinals k1 k2 having
exaclty the same predecessors?

(Predecessor as opposed to immediate predecessor: predecessor is just <
top card by an injection).

ZFC answers that no.

What about ZF? I don't know.

But its not obvious to me that ZF answers no. Maybe in some ZF model there could
be such k1, k2.

If there were such, then for #x1 = k1, #x2 = k2,
cardinality(x1) = cardinality(x2) for the < _ < 2nd definition.

And x1 x2 are not equipollent since we assumed k1, k2 distinct.

I haven't yet been able to answer that question in ZF, but here is a partial answer
I found thinking about it.

In ZFC there are no such k1, k2 with either of k1, k2 Dedekind (Dedekind finite
and infinite).

To prove this, without loss of generality for contradiction assume k1 is Dedekind,
and let x1, x2 be sets with #x1 = k1, #x2 = k2.

Then for a member x1, x1 - {a} had cardinaly < #x. (since k1 Dedekind).

So by the k1, k2 assumed prpeerrty #(x - {a}) < k2 (k1, k2 same redecessors).

So x - {a} injects to x2.

This is not a surjection onto x2, since #(x - {a}) < k2.

So extend that injection to x by sending a to another x2 element.

So x1 injects to x2, so k1 <= k2.

We assumed k1, k2 distinct so k1 is a predecessor of k2, but k1 is not
a predecessor of itself and k1 and k2 were supposed to have the same predecessors.

Contradiction from assuming k1, k2 example with k1 Dedekind.

Apart from Zuhair's definitions, that k1, k2 question is interesting.

Anyway that finishes k1, k2.

I turn now to another problem for these defintions. This one I think is complete,
unlike the k1, k2 question. It is a problem as David Hartley raised, can equipollent
sets have same cardinality.

It seems to be a problem for both <= < and < _ < defintions.

A subset B of A is defined to be cofinite in A (briefly: cofinite when context is
understood) iff A - B is finite.

A set A is defined to be amorphous iff
A is infinite and every subset of A is either finite or cofinite in A.

ZFC proves there are no amorphous sets.

Paul Cohen's methods show if ZF is consistent then there are ZF models with amorphous
sets.

Let x = A an amorphous set, for the definitions above. I will conclude from this in ZF.

Every subset B of an amorphous set A is either finite or amorphous.

To see this, if B were a counterecample, it would have B1 subset neither finite nor
cofinite in B. Then B1 would be a subset of A, neither finite nor cofinite in A, contra
A amorphous.

Suppose y member of (<) for x = A.

If y is infinite then it is anorphous. Namely #y < #A by y in (<), so y injects to
A, so y is bijective with an infinite subset of A and hence an amorphous set by the last
property. Being bijective to amorphous, y is amorphous, else inject a non-finite
non-cofinite subset to the range.

So look at the tree Tc({y}) ordered by epsilon, with y at top.

All down that tree everything injects to A by y in (<), so all these have the propery
as y that they are finite or amorphous.

In ZF with regularity that epsilon tree is well-founded.

So if y were infinite, showing some members of the tree are infinite, find y1 in the tree
minimal for being infinite: y1 is infinite but everything below it in the tree is finite.

So everything in the Tc tree below y1 is finite, ie iterated members below y1 are finite.

So y1 is a subset of HF, the heritarily finite sets.

But in ZF, omega bijects to HF, an explicitly definable bijection.

So y1, an infinite sybset of HF, bijects with an infinite subset of omega.

But no infinite subset of omega is amorphous, by taking the evens in an omega listing of
the infinite subset, getting non-finite, non-cofinite subset.

So y1 is not amorphous, contrary to y1 being infinite and in Tc({y}) as discussed above.

From this contradiction we conclude y in (<) for amorphous A is finite.

Simimilarly, any infinite memeber in Tc({y}) gets y1 as above and a contradiction
as above.

We conclude every Tc({y}) member is finite, in other words, y member HF
the hereditarily finite sets.

Hence for x = A amorphous (<) = HF.

Now consider the two cardinality defintions for x = A amorphous.

Regarding first definition <= < .

We require its top members y to be equipollent with A. Sp repeat the above argumwnt
tro contradiction in Tc({y}) on any such member y, this time having from <= <
definition that starting y infinite.

We conclude for 1st def <= < cardinality(A) = empty.

But this is the cardinality a 0.

Or if you don't like that, by same Cohen methods we can make a ZF model with two
non-ispmorphic amporphouse sets, so these both have forst def cardinality = empty.

Regarding 2nd def < _ < :

The (<) depth 2 down is HF as above. Depth 1, no card restriction is arbitrary subsets
of HF.

Depth 0 is subsets of depth 1, size < #A.

Ok, so at depth 0 we have subsets of P(HF).

HF is bijective in ZF with omga, so P(HF) is bijective with P(omega) the reals.

So any member of < _ < is bijective with a set of reals, hence can be linearly
ordered.

No amorphous set can be linearly ordered. I proved this in

http://groups.google.com/group/sci.logic/msg/f64bbd23fdbcb0aa

So any < _ < member is not amorphous.

But these are subnumerous to A, so if infinite are bijective with an infinite
subset of an anorphous set A, so as above bijective with an amorphous set.

From all this we comclude all < _ < members of cardinality(A) are finite.

So 2nd defintion < _ < of cardinality(A) is all finite subsets of
P(HF). All of these meet the definition so its at least that big.

But this is same as cardinality (omega).

As above, if there are any problems with the recent, the other wasy way
is to take 2 non-isomprphic amporphous sets. Actually that still depends
on last arguemnt that top level finite.

That closing section ended up being more complictated when I wrote it out than
running it in my head.

Well this is at least worth considering anyway, as questions about all this.

--
David Libert ah...@FreeNet.Carleton.CA

[David Libert]

Already a couple of small corrections to my last post to make, of
exposition rather than content.

I messed up the expression of one signature.

Here are Zuhair's other threads related to this:

David Libert (ah...@FreeNet.Carleton.CA) writes:
> I am writing this as a followup to Zuhair's base article of this
> thread, since I will be directly responding to things there and
> want them quoted. But this post also had general inspiration
> from David Hartley's posts in this thread.
>
>
> zuhair (zalj...@gmail.com) writes:
>> Hi all,
>>
>> This topic comes as a continuation to the previous two topics
>> presented into this usenet:
>>
>> 1. A new definition of cardinality at:
>>
>> http://groups.google.com/group/sci.logic/browse_thread/thread/3e92b5c2185a1450?hl=en
>>
>> 2. Towards a new definition of cardinality at:
>>
>> http://groups.google.com/group/sci.logic/browse_thread/thread/a33b3ffbf242c883?hl=en

Here is Zuhair's first definition, from thread 1. :

> In thread 1. Zuhair discussed a 1st defintion. I quote that first definition
> from the base article of thread 1.
>
> http://groups.google.com/group/sci.logic/msg/77269fe363cf905f
>
>>I would like to suggest the following definition:
>>
>>4) The cardinality of any set x is: The class of all sets
>>that are equinumerous to x were every member of their transitive
>>closure is strictly subnumerous to x.

Zuhair's 2nd definition, from thread 2. and this thread:

>> Let me present this definition again:
>>
>> "A cardinal is a class of all hereditarily hereditary sets strictly
>> subnumerous to some set".

So Zuhair's 1st definition was set equinumerous with x, where every
member y has a condition about deeper levels of Tc({y}).

So Zuhair's first definition required y to be equinumerous with x,
not cardinality <= x.

So I should have used = to denote the depth 0 conditition #y = #x.

So I should have given signature = < .

Instead in my last article I have erroneous signature <= < for
the first definition.

Working with it, I used Zuhair's definition properly. I just gave
erroneneius notation for the signature.

For the second definition, I have signature < _ < which I still
think is correct.

Now the second correction. Late in my last article I discussed issues
in ZF, about both definitions having problems there with the possibility
of non-isomorphic sets nonetheless being assigned the same
cardinailty by these definitions. That is what I was thinking about and
what the discussion did. Both raising a question about a difficult
conceivable case and later with a different case finding a relative
independence result.

My error of exposition was to state this backwards in introducing
it:

> But now a second question. David Hartley raised the point, a reasonable
> definition of "cardinality" should assign the same "cardinal" to
> euqipollent sets (ie having a bijection between them).

The topics here was really the reverse direction. If the definition
assigns the same cardinal are the sets really equipollent.

I think the discussion after stated the details correctly but this
introduction stated it backwards.

By the way, the direction I did state in that introduction, is
ok for both Zuhair's definitions.

--
David Libert ah...@FreeNet.Carleton.CA