Cardinality beyond Choice and Regularity
zuhair
We 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:
For all A: A is a set -> Cardinality (A) is a set.
Cardinality might be stipulated as a primitive concept, i.e.
a primitive one place functions symbol "| |" added to the list of
primitives of the language of ZF\NBG\MK, and axiomatized with the
above two assumptions:
1. For all x,y ( |x|=|y| <-> x equinumerous to y )
2. For all x ( x is a set -> |x| is a set )
Of course the second condition is for theories permitting the
existence of "proper" classes in their universe of discourse, like
NBG\MK\Ackermanns'(e,=,| |),etc..,
for ZF(e,=,| |) only 1 is axiomatized.
This primitive Cardinality is the most general approach to
Cardinality.
The second approach is to *DEFINE* cardinality as special kind of sets
in such a manner as to satisfy the above two conditions.
It is well known that Cardinals are defined under the assumptions of
Choice or Regularity.
The question that present itself is:
Can we define Cardinality for every set in ZF minus Regularity?
i.e. beyond Choice and Regularity.
Lets examine, the cardinals that we know
(1) Free-Russell's Cardinals:
Cardinality(A) is the class of all sets equinumerous to A.
Those are incompatible with Z, since
they entail the existence of the set of all sets in Z, or in
NBG\MK they would be proper classes. However in
NF and related systems they are as general as the primitive
concept of Cardinality, but the problem with these theories is
that they are very complex, and difficult to understand, using
concepts of stratification of formulas which is not desirable,
even the finite axiomatization of NFU , though its axioms
do not use stratification, but yet most of its theorems
relies on it.
Those Cardinals were the first defined cardinals
in history of human kind.
(2) Von Neumann's Cardinals.
Cardinality(A) is the least of all Von Neumann
ordinals
equinumerous to A.
Those depend solely on Choice, so they cannot survive beyond it.
(3) Modified Scott Cardinals:
These are defined for every set A as follows:
Cardinality(A) is the set of all well-founded sets equinumerous to
A of the least rank.
Scott Cardinals do not require Choice, but they require
Regularity,however the above amended definition works in
absence of Regularity, but the problem is that the sentence
" every set is equinumerous to a well founded set" (Coret's axiom)is
not a theorem of ZF, so it must be axiomatized.
So these Cardinals work in ZF-Regularity+ Coret
So to some extent they do work beyond Regularity and Choice.
(4)My version of Cardinals:
Those are the Cardinals I defined as:
Cardinality(A) is the class of all sets equinumerous to A having every
member of their transitive closures strictly subnumerous to A.
Or simply:
Card(A) is the class of all sets equinumerous to A that are
hereditarily strictly subnumerous to A.
In ZFC those cardinals are non empty sets. However those Cardinals
unlike Von Neumann's do not necessarily require full choice, they
actually require the following assumption.
(Z)For every set x: Card(x) is a non empty set
or in ZF style
(Z)For every x Exist y ( y=Card(x) & ~y=0 ).
Now that assumption if added to ZF as an axiom.
then ZF+Z is weaker than ZFC, perhaps strictly weaker actually.
Actually we can drop Regularity as well, so we can have
ZF+Z-Regularity.
Now this definition works in ZF+Z-Regularity.
So these cardinals do work beyond Regularity and Choice, provided the
assumption Z above.
The interesting matter is to see if ZF+Z-Reg would be
weaker or stronger than ZF-Reg+Coret, or perhaps
neither weaker nor stronger! they might simply work
under different conditions beyond Regularity and Choice.
The interesting thing in the last definition of mine, is that it
doesn't require Coret's axiom, which is a step downward in diving
beyond Regularity and Choice.
Zuhair
zuhair
Frege-Russell's Cardinals
zuhair
Frege-Russell's
WM
> > We may say that a Cardinal is a function from classes to classes
Cardinality is not a well-defined concept.
>
The cardinality of the set of all finite paths of the binary tree is
aleph_0.
The cardinality of all infinite paths of the binary tree is aleph_1
(given CH).
The union of all finite index sets is
{1} U {1, 2} U {1, 2, 3} U ... = {1, 2, 3, ...}
as well as
{1} U {1, 2} U {1, 2, 3} U ... U {1, 2, 3, ...} = {1, 2, 3, ...}
Hence, the projective union of all finite subtrees is the infinite
tree. The projective union of all finite subtrees contains aleph_0
paths, the infinite tree contains aleph_1 paths. Therefore, we obtain
aleph_0 = aleph_1.
According to other proofs, aleph_0 =/= aleph_1.
Set theory has been contradicted.
Regrads, WM
Virgil
<356e8e33-0c88-4a5d...@p35g2000yqh.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 13 Dez., 21:18, zuhair <zaljo...@yahoo.com> wrote:
>
> > > �We may say that a Cardinal is a function from classes to classes
>
> Cardinality is not a well-defined concept.
depends who defines it. When you define anything we all agree that it
usually fails to be well defined.
> >
> The cardinality of the set of all finite paths of the binary tree is
> aleph_0.
>
> The cardinality of all infinite paths of the binary tree is aleph_1
> (given CH).
>
> The union of all finite index sets is
> {1} U {1, 2} U {1, 2, 3} U ... = {1, 2, 3, ...}
> as well as
> {1} U {1, 2} U {1, 2, 3} U ... U {1, 2, 3, ...} = {1, 2, 3, ...}
>
> Hence, the projective union of all finite subtrees is the infinite
> tree. The projective union of all finite subtrees contains aleph_0
> paths, the infinite tree contains aleph_1 paths. Therefore, we obtain
> aleph_0 = aleph_1.
In order to prove your claim:
First, you must give an clear and unambiguous definition of what you
mean by "the infinite tree", including a definition of what constitutes
a path in such a tree, which you have not yet done, and seem quite
incapable of doing.
Second, you must give an unambiguous definition of a finite subtree of
that tree, including a definition of what constitutes a path in such a
subtree.
Third, you must give an unambiguous definition of "projective union".
Fourth, you must prove that the projective union of all finite subtrees
contains only aleph_0 paths, and also contains all paths of the infinite
tree.
Fifth, you must prove that your "projective union of all finite
subtrees" of that infinite tree is, in fact, in all relevant ways
isomorphic to that infinite tree itself.
And there are undoubtedly a number of essential steps I have overlooked
but which you still must complete in order to have a valid proof of your
claim.
Only then would you be able to claim that aleph_0 = aleph_1.
But since several of those essential steps have proven way too difficult
for you in the past, I do not contemplate your succeeding at any time
within the current century.
George Greene
>
> We 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)
This is arguably too restrictve. Injectivity is not the relevant
criterion.
One could also say that A::B iff there is an injection on A into B
AND a SURjection on A onto B. If you are going to reverse the
orders,
you could also say that A::B iff there is a surjection on A onto B AND
a
surjection on B onto A.
Intuitively, all these are equivalent but the mechanics of proofs
using them
(e.g. how much choice you "need") can differ.
The far greater problem, though, involves the simple invocation of
"exists"
in a first-order context. Standard classical first-order logic in
point of fact
simply Does Not Know Shit about existence. In the standard semantics,
a non-empty-domain of things-that-exist IS TAKEN AS GIVEN *IN
ADVANCE*.
The logic investigates things that must be true about these things,
ASSUMING
that the axioms are true about them and ASSUMING that they ALREADY
exist.
Whether any of these things does or doesn't exist IS NOT a legitimate
question.
In this semantic context, ALL of these things DO exist and NOTHING
ELSE exists!
To the extent that you are trying to use the first-order language to
say anything about
what "really" exists, this is HIGHLY problematic!
The truth of (for example) the downward Lowenheim-Skolem theorem
Depends Crucially on the semantic model's existence that certain
bijections "don't exist" WHEN IN POINT OF FACT THEY OBVIOUSLY DO ...
the model is simply leaving them out of its domain. The fact that
other models
could put these bijections INTO THEIR domains implies that these
bijections
Really Do exist after all; the smaller models ARE LITERALLY LYING
about what
does vs. doesn't "exist".
These facts also jointly imply that different models are going to have
different
opinions about questions of cardinality, which means that THE THEORY
(since
its theorems have to be true IN ALL models of the axioms) CANNOT SAY
shit
about a whole host of questions regarding cardinality. In particular,
ZFC, even if
you expand C to GLOBAL choice, STILL HAS NOT THE FIRST FUCKING CLUE
what the cardinality of the power-set of the naturals is -- or rather,
it has ONLY the
first clue -- it knows that this cardinality is bigger than the
cardinality of the naturals,
and in general it knows that |beth_n| >= |aleph_n|. But it does NOT
know whether
this greater-than-or-equal-to means "greater than" or "equal to" (it
must in fact be
one or the other, NOT "either"), and when it's greater, it does not
know (OR CARE,
for that matter!) how MUCH greater.
All of these questions are just a lot more important than any kind of
robustness
you might be trying to add by doing without regularity or choice. If
you can't
resolve some of THIS then you really are just wasting everybody's
time.
> Also it is desirable to have the following property added to the
> above:
>
> For all A: A is a set -> Cardinality (A) is a set.
>
> Cardinality might be stipulated as a primitive concept, i.e.
> a primitive one place functions symbol "| |" added to the list of
> primitives of the language of ZF\NBG\MK, and axiomatized with the
> above two assumptions:
>
> 1. For all x,y ( |x|=|y| <-> x equinumerous to y )
> 2. For all x ( x is a set -> |x| is a set )
>
> Of course the second condition is for theories permitting the
> existence of "proper" classes in their universe of discourse, like
> NBG\MK\Ackermanns'(e,=,| |),etc..,
> for ZF(e,=,| |) only 1 is axiomatized.
>
> This primitive Cardinality is the most general approach to
> Cardinality.
No, it isn't. "General" means applying basic concepts in as broad
a way as possible. What makes ZFC "general" and makes it viable
both as a foundation for math (generally) and for the model theory of
FOL is that *EVERYTHING* is a set, or is legitimately re-encodable as
a set. Adding new primitives to the language is LESS general.
The whole point about ZFC's relevance as a foundation is that ONE
PREDICATE IS ALL you need! That's WHY we use it!
> The second approach is to *DEFINE* cardinality as special kind of sets
> in such a manner as to satisfy the above two conditions.
Well, yes, if you wanted a "most general" approach, this one would be
more
appropriate. I know, cardinalities don't have to be sets, and neither
do ordered
pairs, for that matter, but That's Not The Point -- the point IS that
sets are
flexible AND GENERAL enough to ACCOMMODATE both of these concepts.
> The question that present itself is:
>
> Can we define Cardinality for every set in ZF minus Regularity?
I'm going to say this again: EVEN WITH choice and regularity,
ZFC *STILL CANNOT* define a cardinality for p(w).
Different models of ZFC HAVE DIFFERENT cardinalities for p(w).
George Greene
> Cardinality is not a well-defined concept.
Idiot, PLEASE. SHUT UP.
You personally are THE PAST MASTER of the non-well-
defined concept. You just MAKE UP terms ALL THE TIME!!
> The cardinality of the set of all finite paths of the binary tree is
> aleph_0.
Well, a stopped clock is right twice a day.
> The cardinality of all infinite paths of the binary tree is aleph_1
> (given CH).
OK, that's your 2nd time, but why should anybody be "given CH"?
There are perfectly good models of ZFC in which CH is false!
There are plenty of models of ZFC in which the cardinality of this
path-class is MUCH BIGGER than aleph_1 !
> The union of all finite index sets is
> {1} U {1, 2} U {1, 2, 3} U ... = {1, 2, 3, ...}
You canNOT legitimately use the word "union" in THIS way in THIS
context BECAUSE "union" ALREADY HAS A MEANING under the
axioms of ZFC. More to the point, IF you actually HAD a well-defined
version of YOUR term "union", then YOU COULD STATE SOME AXIOMS
DEFINING "union" in the way in which you intend to use it. This, of
course,
you cannot do, because you're REALLY stupid.
zuhair
I know all of what you wrote here, you are wright, but these matters
are the non escapable ill background of FOL, we cannot do much about
that.
You are right about Cardinality of P(w). We know for sure that there
is
a Von Neumann ordinal that is bijective to P(w) (in ZFC), but we don't
know which ordinal is that. You are right, we know there exist a Von
Neumann cardinal for P(w)
but we don't know which cardinal is that?
Even my cardinals, don't solve that.
However interestingly one of my approaches to define Cardinality do
really find a specific cardinality for P(w) see that approach:
Axiom 1: For all A Exist x for all y
(y e x <-> (y is hereditarily hereditary & y strictly subnumerous to
A))
y is hereditary iff y strictly supernumerous to every member of its
transitive closure.
y is hereditarily hereditary iff
(y is hereditary & every member of the transitive closure of y is
hereditary)
Define:
x=Card(A) <->
for all y (y e x <-> (y is hereditarily hereditary & y strictly
subnumerous to A))
So Cardinality of a set A is the set of all hereditarily hereditary
sets strictly subnumerous to A.
Axiom 2: For all A (Card(A) equinumerous to A).
So ZF+ 1+2 can specify the cardinality of P(w), which is equal to the
Cardinality of Aleph_0.
This actually motivates Cantor's GCH.
However it is too strong.
I don't know if you find this interesting, but it do solve the last
problem you've mentioned.
Zuhair
zuhair
Another point regarding the subject of weather primitive cardinality
is more general than the defined one. This is something sure up till
now, nobody brought a definition of Cardinality that match the
generality of primitive Cardinality, one example I mentioned already:
can you tell me of a defined Cardinality that can successfully
differentiate between the size of proper classes that are not
equinumerous to each other? to cut matters short for you, there is non
up till now, perhaps this is only possible in Ackermanns' class theory
with Regularity or Choice, but Ackermanns' is not the classical class
theory, NBG\MK is\are, and in these non of the current definitions
work to differentiate the size of proper classes, While primitive
Cardinality does the job swiftly!
Of course no doubt that finding a general definition of Cardinality
that matches that of the primitive Cardinals would be something good
indeed, but I greatly doubt that this is achievable. I don't know if
one can prove that impossibility or possibility either, but for now it
looks at least something that is very difficult to achieve.
Zuhair
Nam Nguyen
> I'm going to say this again: EVEN WITH choice and regularity,
> ZFC *STILL CANNOT* define a cardinality for p(w).
> Different models of ZFC HAVE DIFFERENT cardinalities for p(w).
If I might add to what you've already expressed, cardinality should
be a binary-predicate concept instead of a unary-function one, even
when the sets involved are finite.
zuhair
Why it should be a binary-predicate concept?
What is the intuitive account for this idea?
Zuhair
Nam Nguyen
There are a few reasons but I think the primary one is that we have
to be able _to compare 2 sets_ which is what a binary predicate operation
is for. For example, if we come up with a set aleph_alpha as the cardinality
for the set of naturals in ZF [out of a unary function say card(N) =
aleph_alpha] and a set aleph_beta for the set of reals, we still have
to _compare the 2 sets_ aleph_alpha and aleph_beta; hence we can't avoid a
kind of binary predicate, which is the why for using a binary-predicate.
As for the why-not of using unary function to define cardinality, it's
possible that there are sets that are well-defined but yet whose cardinalities
might be impossible to know (even in principle). For instance, in ZF, let's
define the set S = {x | x is a counter example of GC}. Currently it looks
almost certain that it's impossible to know the "size" (i.e. "cardinality") of
S: it could be infinite, or non-zero finite, or zero. And I don't think it'd
be a good idea to define cardinality in general with unary function, given
this kind of unknownability about some particular sets. The binary-predicate
way of defining cardinality is a "soft"/"relative" comparison way and would
enable us to avoid the "hard"/"absolute" comparison in such cases of impossibility.
Imho.