# Is CAT the Key to Understanding Numbers?

XOR b XOR c), (a XOR b XOR c, c), (c, a XOR b XOR c), (a XOR b XOR c, c), (c, a XOR b XOR c), ...So, by using the above notation, we can identify each bead, and bythat, we can identify them when they are mixed together, but in thisexample, we need to use the two concepts, and not just one.Now, there is a question:Is this example a proof for the existence of a computable associations between continuum and discreteness ? If the answer is yes,then we can ask ourselves:What is the minimum number of bits that we need for making thiscomputation

Hallo Dear people !

In the attached website there is a short Email exchange, describing the foundations of a new theory of numbers.

http://www.geocities.com/complementarytheory/CATpage.html

A quick reference of my acronyms:
---------------------------------

CAT = Complementary Associations Theory.

CD = Continuum XOR Discreteness (CAT's opposite concepts).

Association = Any possible mutual influence between opposite concepts
(under CAT its between Continuum and Discreteness).

EP = Explorable Product (exists iff it is an Association between CD).

AL = Association Level is an invariant quantity, being kept through
CD Associations.

CR = Computational Root is EP in AL.

RU = Redundancy and Uncertainty concepts, are used as invariant
structural degree of CR, determining its exact position in AL.

FRU = Full RU is the first CR in AL.

~RU = Not RU is the last CR in AL.

PRU = Partial RU is any CR which is not FRU and not ~RU.

FIS = Fractalic Information Structure, used to represent numbers
which are based on CRs.

NAB = Natural Axiom's Base is any property in some theory, that have
elements which are inaccessible to the other proprties of the
theory.

Yours,

Doron

Special thanks to Hurkyl:

Dear Hurkyl !

Thank you very much for being pedantic .

I made very important changes in my article, and I'll be glad

Some Graphics of my number system representation of ALs 1 to 7 ,
you can find here:

http://cyborg2000.xpert.com/ctheory/ [Broken]

It is written in java by a friend of mine, and it is based on
Cartesian Product so, there are some left-right permutations
that can be ignored.

Yours,

Doron.

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Strange ... i can't find the attachment anywhere !

test

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But I have got it!

I've never seen so many acronyms and capitals outside of an IT magazine.

This looks interesting though. Could Doran please clarify all of this. It's kind of a mess right now.

I think you missed a point of Dr. Tom's objections... in the context of replacing ZF, you really need to present your system in explicit formalism. For "low-level" mathematics, there needs to be a clear and precise algorithm for translating any statement into explicit formalism. e.g.

"The union of two sets is a set"

is okay because is clear how to translate this into explicit formalism:

For all x, y: There exists z: z = x U y

For all x, y: There exists z: For all w: w in z <=> w in x or w in y

Anyways, onto the content. I skimmed through and saw the line:

"ZF is ~RU only Theory because {a,a,b}={a,b}..."

You seem to neglect that ZF knows how to simulate the alternative. Let's introduce new symbols '(' ')' with the definition:

(a, b) = {a, {a, b}}
...

And let's call (a, b) an "ordered pair"

By nesting "ordered pairs", we can generate sequences of things. For instance:

((a, a), b) is unequal to (a, b)

Your diagram of a FIS can be modeled in ZF as:

(((., .), (., .)), ((., .), (., .)))

Hurkyl

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Dear Hurkyl,

Lets start from the last part.

You write:

Your diagram of a FIS can be modeled in ZF as:

(((., .), (., .)), ((., .), (., .)))

My question is:

Do you know if this ZF ability have been used as a building block
for representing a number-system, which is not based on:

0 = { }

1 = {{ }} = {0}

2 = {{ },{{ }}} = {0,1}

3 = {{ },{{ }},{{ },{{ }}}} = {0,1,2}

4 = {{ },{{ }},{{ },{{ }}},{{ },{{ }},{{ },{{ }}}}} = {0,1,2,3}

and so on ?

And I do not mean sequences, but something like base value expansion.

I'll really be glad to learn from you.

Another thing which is connected to the above is, that by writing
RU (Redundancy AND Uncertainty) we mean that those concepts
are both used when we define any element under CAT, for example:

((a,a),b) is actually = ((a XOR b,a XOR b),c), and by (a XOR b)
we meam that we can't know if it is a or it is b (uncertainty).

The main idea of this theory is to include a reaserch of our cognition's ability to make Math, as a legal part of the Theory.

I think that without this reaserch, essential things are not included,
and we can miss important points of view about Math.

This is my first aim, before representing its formal side,
to put those ideas "right on the table".

If we are looking at this subject by using a concept like Information, then we can ask ourselves what are the minimal conditions that gives us the ability to identify and count things ?

For example, let's examen this situation:

On the table there are finite unknown quantity of identical beads
and we have to:

A) Find their sum.

B) To be able to identify each bead.

Limitation: we are not allowed to use our memory.

By trying to find the total quantity of the beads (represent the
discreteness concept) without useing our memory (represents the
continuum concept) we find ourselves stuck in 1, so we need
an association between continuum and discreteness if we want to
be able to find the bead's sum.

Lets cancel our limitation, so now we know bead's sum, which is,
for example, value 3.

Now we try to identify each bead, but they are identical, so
we identify each of them by its place on the table.
But this is an unstable solution, because if some one takes the
beads, put them between his hands then shake them, and then put
them back on the table, then we lost their id.

Each identical bead can be the bead that was identified by us
before it was mixed with the other beads.

We shall represent this situation by:

((a XOR b XOR c),(a XOR b XOR c),(a XOR b XOR c))

By notate a bead as 'c' we get:

((a XOR b),(a XOR b),c)

and by notate a bead as 'b' we get:

(a,b,c)

we satisfy condition B but (and this is the important thing)
through this process we define a universe, which exist between
continuum and discreteness concepts, and can be systematically
explored and be used to make Math.

Yours,

Doron

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Let me start off by explaining what set theory is good for.

(a) Set theory provides a common descriptive language across all of mathematics. It gives us the language that let's us talk about "the set of things such that..." and it gives us common manipulations on those objects, such as unions, intersections, cross products, power sets, relations, functions, find and replace operations (requires ZF), choice functions (requires ZFC), et cetera.

(b) Set theory provides a convient axiomatic basis for all of mathematics. This is the interesting part and is related somewhat to your work.

Mathematical theories are generally not specified directly in terms of set theory, even low level mathematics. For instance, number theory comes from the definition:

The ordered pair (N, ++) is called "the natural numbers" if the following is true (these are Peano's axioms):

++ is a unary operator on N

There exists an element of N, call it 0, such that for any element n of N, ++n is not 0.

For all n is in N, ++n is in N.

For all m and n in N, ++m = ++n => m = n

If any subset S of N has the properties that 0 is in S and for any s in S, ++s is in S, then S = N.

All of the properties of the natural numbers are derived from these 4 axioms. For the truly hardcore logician, the natural question is "How can I be sure that such a system can even exist?!" That is where set theory comes into play. The construction

0 = { }
1 = {{ }}
2 = {{ }, {{ }}}
...

++x = x U {x}

is done to prove that there really does exist some (N, ++) that satisfies the axioms of the natural numbers.

Well, technically it doesn't prove that such a thing really exists; it just proves the validity of ZF implies the validity of the natural numbers... but we consider ZF to be "sufficiently obvious" for that detail to be omitted.

So the point of set theory is not to provide interesting descriptions of things, but to prove that interesting descriptions make sense. And even then, pure set theory usually stops after providing the natural numbers and the basic operations, allowing everything else to be built up from there. One proves the integers exist by creating them as equivalence classes of ordered pairs of natural numbers. Rational numbers are created the same way out of integers. The real numbers are created via equivalence classes of sequences of rational numbers.

In that light, I really think you're developing your theory in the wrong way. Set theory is a tool that provides a language and basic manipulations on objects for more advanced mathematics, and provides a way for advanced theories to assert their own existence. This appears to me to be a totally separate goal from what you're trying to accomplish!

Indeed you are trying to describe a new way of looking at things, but the way mathematics is abstracted, that does not mean you need to replace the old way. In all likelyhood, if your new way is sound, the old way will be fully capable of emulating your new way, but that's actually a good thing because that provides you with a way to justify that your new way makes sense, and it doesn't detract in any way from your new way's abilities.

IMHO your focus should be on producing an axiomatic formulation of your theory; i.e. something that has the same form as the axioms of the natural numbers above. You don't even have to include classical sets in your definition; for example the axioms of the natural numbers, or those of euclidean geometry, can be formulated without any use of sets.

Do you know if this ZF ability have been used as a building block
for representing a number-system

I doubt it. Set theory is, as mentioned, used as a tool rather than for development. Axioms are used to describe number systems, and set theory is used to prove that the number system can exist (and even then pure set theory is a last resort).

I have no doubt that such a number system could be developed that way (for instance, I think I could build the natural numbers with that method instead of the traditional method), it's just that building number systems from pure set theory is not a common practice.

Hurkyl

Dear Hurkyl !

My Theory belongs to a universe exists between Quasi-Set Theory
and ZF-Set Theory.

An Information about Quasi-Set Theory you can find here:

http://arxiv.org/PS_cache/math/pdf/0106/0106098.pdf [Broken]
http://www.cfh.ufsc.br/~dkrause/DoriaSPqset.pdf [Broken]

Another interesting article about CH problem, you can find here:

http://arxiv.org/PS_cache/quant-ph/pdf/9902/9902060.pdf [Broken]

I,ll be glad to continue Emeiling with you if you are still find this subject interesting.

The attached pdf file is RU example.

Yours,

Doron

#### Attachments

• ru-example.pdf
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Originally posted by ObsessiveMathsFreak
I've never seen so many acronyms and capitals outside of an IT magazine.

This looks interesting though. Could Doran please clarify all of this. It's kind of a mess right now.

Hi ObsessiveMathsFreak !

A quick reference of my acronyms:
---------------------------------

CAT = Complementary Associations Theory.

CD = Continuum XOR Discreteness (CAT's opposite concepts).

Association = Any possible mutual influence between opposite concepts
(under CAT its between Continuum and Discreteness).

EP = Explorable Product (exists iff it is an Association between CD).

AL = Association Level is an invariant quantity, being kept through
CD Associations.

CR = Computational Root is EP in AL.

RU = Redundancy and Uncertainty concepts, are used as invariant
structural degree of CR, determining its exact position in AL.

FRU = Full RU is the first CR in AL.

~RU = Not RU is the last CR in AL.

PRU = Partial RU is any CR which is not FRU and not ~RU.

FIS = Fractalic Information Structure, used to represent numbers
which are based on CRs.

NAB = Natural Axiom's Base is any property in some theory that have
an elements which are inaccessible to the other proprties of the
theory.

Yours,

Doron

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Originally posted by Moni
But I have got it!

Hi Moni !

I'll be glad to know what do you have to say about CAT.

Yours,

Doron

I'm harping on the idea of "dethroning" ZFC in mathematics because the goal of quasi-set theory is to provide a particular language of description, not to provide axiomatic underpinning to the entirety of mathematics... so I find it a waste of effort to talk about perceived limitations of ZF. Put your effort into axiomizing a QST that does what a good QST should do instead of spending effort in a philosophical discussion about the merits of having a hidden ZF underpinning. In the end, if you base physics on a QST, everything is exactly the same whether or not QST has "more expression-power" than ZFC... and it's all the same if QST is taken as the fundamental or if QST has a hidden ZF underpinning.

Anyways, that out of the way, I'll take a better look at the content of your theory when I have more time to fully read it (unless you want to continue the philosophical discussion).

On a side note, though, the article on the continuum hypothesis is flawed. It rests on the assumption that there does not exist an injection from an "interset" into the real numbers... however it is very easy to demonstrate the existence of such a bijection in ZFC:

Given: S is an interset. i.e. |N| < |S| < |R|

Well order S and R by <.

Define the function &phi as follows:

&phi(min S) = min R
i.e. &phi maps the minimum value of S (with respect to the well ordering) onto the minimum value of R.

Recursively define &phi on the rest of S by:
&phi(s) = min {r | for all t < s, &phi(t) != r}
(< is the well ordering on S)

Because R is well ordered, the above statement is well-defined. Because S is well ordered, this recursive definition is well-defined and complete.

Intuitively, &phi is defined iteratively, at each step we map the smallest unused element of S onto the smallest unused element of R.

Suppose s < t for s and t in S (and < the well-ordering used above)
It's pretty easy to show via transfinite induction that &phi(s) < &phi(t).

So &phi is an injection from S into R (a.k.a. a bijection from S into some subset M of R).

(The well ordering theorem and transfinite induction are my favorite axiom of choice equivalents, so too bad for those of you who would prefer a Zorn's Lemma proof!)

Hurkyl

Hi Hurkyl !

If |R| = Continuum then how you can find ordered elements in it ?

Don't forget that the Continuum concept is uncounable by definition.

It can be done only with countable sets.

(According CAT, however, Continuum-only power is uncountable
, 0^0 = 1 = exist ,
and Discreteness-only power is uncountable
, (n>0)^0 = 1 = exit ,

so, there is no Math before CD associations for example:

String-only(=Continuum-only) is uncountable,

So, my math begins after we have a chain (string-beads associations) )

If you use expressions like "Intuitively, &phi is ..."
then I don't understand why do you think thet philosophical discussion
about Math is a waste of efforts.

I think that a good philosophical discussion gives us the ability to examen concepts, before we choose to use them in a formal way.

Yours,

Doron

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The Well Ordering Theorem is an equivalent of the Axiom of Choice. It states that for any set S, there exists a relation < such that S is well ordered with respect to <.

The well ordering given by applying well ordering theorem to a set with a natural ordering (like the real numbers) does not necessarily have anything to do with that natural ordering. However for some number systems, such as the natural numbers or the ordinal numbers, the natural ordering is a well ordering.

Incidentally, the ordinal numbers provide a counterexample (in ZF!) to your assertion that you can only well-order countable sets; there are uncountable sets of ordinal numbers, but the natural ordering on the ordinal numbers is a well ordering.

If you use expressions like "Intuitively, &phi is ..."
then I don't understand why do you think thet philosophical discussion
about Math is a waste of efforts.

I'm not sure what you mean by that. I gave a rigorous definition of &phi, and I gave the intuitive description to help the reader understand the definition.

My comments about the philosophy being a waste of time is based on what I perceive is the purpose of developing a quasi-set theory. From what I've seen, the primary motivation is to provide a rich language for describing collections, with the intent to build up analysis on these quasi-sets.

Taking that as the primary motivation, philosophizing about the meaning of set theory is a waste of time for the following reasons:

Quasi-set theory can be created in a way independent of set theory. Set theory isn't a brick wall that needs to be torn down before anyone can do anything different! One of the great benefits of the axiomatic method is that it encapsulates theories, allowing them to be independant from one another.

Quasi-set theory intends to be the basis of a new kind of analysis on these quasi-sets. QST does not want to incoroporate "old" mathematics; instead it wants to be a foundation for "new" mathematics (that may or may not resemble the old stuff). If QST never has to make a reference to real analysis or algebraic geometry, it's a happy theory. However, replacing ZFC entails providing a foundation for the rest of mathematics (in their current incarnations!); this is a task that is disjoint from the goal of developing a QST.

Also, there are all ready some nice ZFC based tools for describing interesting collections. One of the more interseting ones is replacing characteristic functions with arbitrary functions. For a set A, its charactistic function &ChiA is the function with:

&ChiA(x) = 1 if x in A
&ChiA(x) = 0 if x not in A

Intuitively, &ChiA(x) is how many times A contains x. The natural generalization is to "pretend" that arbitrary functions are characteristic functions... we can define a multiset F to be an arbitrary function f, and we call f(x) the number of times F contains x.

Hurkyl

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The well ordering theorem (and all of the axiom of choice equivalents) are strongly nonconstructive. It merely states that well orderings exist; in general there's no hope of explicitly knowing what the well ordering looks like (because the axiom of choice is independant from ZF).

There is debate over whether or not the axiom of choice should be used, but one main point that is often overlooked (much like non-euclidean geometry was overlooked when it was first discovered) is that the only thing that really matters is the properties of your application. If you want to design a physical theory that is incompatable with the axiom of choice, you're free to do so. If your neighbor designs a physical theory that relies on the axiom of choice, he can too. Then you subject both to experimentation and figure out which is better than the other! And if the axiom of choice theory wins out, that doesn't mean that in the future a better theory can't come along that is incompatable with the axiom of choice.

On a side note, it's interesting that few people worry about the axiom of infinity (which essentially says "there is a set of natural numbers"), even though it let's us do similarly pathological things such as invent a number system (the real numbers) for which we cannot explicitly individually identify the vast majority of its elements!

Hurkyl

Hi Hurkyl !

Does Well Ordering Theorem (<) depends on the assumption that
each element's value is well known, before it can be put into order ?

You write:

&ChiA(x) = 1 if x in A
&ChiA(x) = 0 if x not in A

Well, how do you know if x in A OR if x not in A ?
Is it depends on the assumption that each elemnt's value is
well known, before you find if &ChiA(x) = 1 OR &ChiA(x) = 0 ?

ZF's Axiom of extensionality can be written like this:

X and Y are different sets iff there is a set Z in X and not in Y
or in Y and not in X.

I think that this axiom depends on the hidden assumption that
elements are always well known, before any exploration can be
done.

In CAT, however, because Redundancy & Uncertainty are used
as one of the base concepts, this is not an hidden assumption
but useful tools for making Math.

In the attched pdf file there are 3 pages:

Page 1 uses FRU CR and ~RU CR in AL 5, to demonstrate RU concepts.

Page 2 showes that base value expansion is a privet case of a FIS under CAT.

Page 3 demonstrates a FIS built by different CRs from different ALs.

Doron

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• ru-example2.pdf
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I think I'm going to have to ask you for a better definition of "well known". I presume you mean:

x is well known iff with a finite number of steps one can derive a formulae P from the axioms of set theory such that x is the unique set satisfying P.

At least that's what I think about the term... but since I can see how it could mean different things I'd like to get a more precise meaning before replying to some of your post.

&chiA(x) = 1 if x in A
&chiA(x) = 0 if x not in A

Well, how do you know if x in A OR if x not in A ?

I know one of the two statements "x in A" or "x not in A" is true from the definition of not.

Technically, &chiA is "too big" to be a function; in its full glory it can only be described as a proposition:

"&chiA(x) = y" is defined to be the proposition "(y = 1 and x in A) or (y = 0 and x not in A)"

For the programme of generalizing sets, though, we really need to have an honest to goodness function. Fortunately, we usually only care about a particular domain (such as the set of all possible position-momentum pairs), and any proposition can be reduced to a relation when restricted to a domain.

If you'd like a more direct approach...

Suppose you want to prove &chiA exists on the domain D.

D is a set and {0, 1} is a set
Therefore {0, 1} * D is a set. Call it C. (* = cartesian product)
Define &chiA = {(a, b) in C | a = 1 and b in A or a = 0 and b not in A}

This is a rigorous construction of the function &chiA on the domain D.

Of course, in general we aren't able to compute &chiA(d) for any d in D; we just know that &chiA exists, and &chiA(d) = 1 iff d is in A.

Hurkyl

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Maybe some examples of more interesting collection types that are definable in ZF might be enlightening.

The basic collection is, of course, the set. A set is an unordered collection of distinct objects.

Then we have the ordered pair. An ordered pair is exactly what it sounds; it contains a pair of objects, and there is an ordering on them allowing us to distinguish one as the "left object" and one as the "right object".

Rigorously, (a, b) is defined as {a, {a, b}} and has the property that (a, b) = (c, d) iff a = c and b = d. In particular, (a, b) != (b, a) iff a != b.

Then there is the &Iota-tuplet. Tuplets describe a collection of things (not necessarily distinct) along with a scheme of identifying them.

Rigorously, the &Iota-tuplet is defined as follows: for a given index set &Iota, an &Iota-tuplet is simply a function whose domain is &Iota.

This definition works by using &Iota as the scheme for identification. Suppose &tau is an &Iota-tuple. Then, for any &iota in &Iota, the &iota-th element of our tuple is simply &tau(&iota).

For an example, take &Iota = {1, 2, 3}. Then, &iota-tuples are simply ordered triples. We often use the shorthand notation:
&tau = (a, b, c) means: &tau = {(1, a), (2, b), (3, c)}
or equivalently, &tau(1) = a, &tau(2) = b, &tau(3) = c

This particular definition is interesting because it doesn't require the index set to be ordered. For instance, my index set could be types of elementary particles; &Iota = {electron, neutron, proton, mu antineutrino} and then I can have a tuple of things with this as the index set. For instance, I could define the particle count of an atom to be an &Iota-tuplet, and the particle count &tau of an ordinary helium atom is:

&tau = {(electron, 2), (proton, 2), (neutron, 2), (mu antineutrino, 0)}

Notice, also, that this definition does not assume distinguishability between individual particles of each type.

How about a really interesting example captured naturally by ZF?

Let's take the classical example of a spinless particle decaying into two photons emitted in opposite directions. Let us define 4 distinct symbols of two types: (we don't care as what they're actually defined, just that they are distinct):

type i:
|east> - abstracts the state of an eastbound photon
|west> - westbound photon

type ii:
|up> - photon with spin up
|down> - photon with spin down

And some appropriate commutative * operator which we use to mean a particule is simultaneously in different states. I.E. |east> * |up> means the particle is both eastbound and has up spin.

And from here we take the real vector space over sets of products. Allow me to single out a single element of that vector space:

&Psi = 0.5 * {|east> * |up>, |west> * |down>} + 0.5 * {|east> * |down>, |west> * |up>}

&Psi, given a suitable interpretation, precisely captures the nature of the uncertainty of our experiment. There's a 0.5 chance that we have an eastbound particle with up spin and a westbound particle with down spin, and there's a 0.5 chance we have an eastbound particle with down spin and an eastbound particle with up spin.

Hurkyl

Dear Hurkyl !

Before I give you my definition for "well known", let me say

You wrote:
there are uncountable sets of ordinal numbers
So you accsept Cantor's theorm and the diagonalization argument.

Please let me show you what I have found:

Another look on Cantor's Theorem
--------------------------------

General:

B > A if A not equal B but there is a bijection between A
and a subset of B.

Cantor's proof, showes that P(X)>X
----------------------------------

Step 1: bijection between A and a subset of B.
-------
Let be function f such as f(x)={x} for example:

1<-->{1} 2<-->{2} 3<-->{3} ...

By this we can say that P(X)>=X .

Step 2:
-------
Lets contradict our assertion and say that P(X)=X so, there must
be a function that can show a 1 to 1 correspondence between
each member in X to each subset in P(X).

Lets define subset S in P(X) which includes ALL
members of X, that are not included in the subsets of P(X),
which they are in 1 to 1 correspondence with them, for example:

X <--> P(X)
-----------
if
a <--> {c,d}
b <--> {a}
c <--> {a,b,c,d}
d <--> {b,e}
e <--> {a,c,e}
.
.
then S includes {a,b,d,...,and so on} .

In set X there exist some member (lets call it t)
and we metch t with subset S (t <--> S).

Now we can ask: is t in S, or t not in S ?

Lets check it.

a) t in S:
----------
But according to the definition of S, t can't be included in S,
otherwise there will be a copy of it in S, which contradicts
the definition of S.

b) t not in S:
--------------
But according to the definition of S, t must be included in S,
but then there will be a copy of it in S, which contradicts
the definition of S.

So, we find that we can't complete the 1 to 1 correspondence between
each member in X with each subset in P(X).

According to step 1 we know that P(X)>=x and because we can't complete
the 1 to 1 correspondence between each member in X, with each subset in P(X), we have no other choice but to conclude that P(X)>X.

This proof, and the diagonalization argument, are (as much as i know)
the basics of the development and research of the transfinite
cardinals in Math.

---------------------------------------------------------------------
Now let's take another look on Cantor's proof.

---------------------

The western logic is a false/true logic, where each statement of it
is examined by those two terms.

Cantor's Theorem uses the logic of the liar's paradox in step 2
of his proof.

Lets examine the liar's paradox from a different point of view.

It goes like this:

The Cretan has said :"All Cretans are liars !"

If a Cretan can be sometimes a liar or sometimes honest so, there is no paradox, but a false statement if he means "all the time",
or a true statement if he means "sometimes".

But if a Cretan is a liar or honest all the time, then he can't say the above statement, because there is something which is common to the liar and to the honest Cretans: They can't say about themselves that they are liars.

Lets examine why.

If a Cretan is a liar or honest then he is in one of those
states before he says such a statement so:

a) A honest Cretan can't say a statement which includes himself
as a liar.

b) A liar Cretan can't say a statement which includes himself
as a liar.

So, the statement: "All Cretans are liars !" does not exist.

As the statement does not exit, the paradox that was build
on top of it, does not exist.
---------------------------------------------------------------------

Now let's examine again the definition of subset S in Cantor's theorem.

Lets define subset S in P(X) which includes ALL members of X
that are not included in the subsets of P(X), which they are in
1 to 1 correspondence with them.

The Defenition that define subset S in P(X) can't exist because its
logic is idendical to the existence of the cretan's statement.

As the statement does not exit, the paradox that was build
on top of it, does not exist.

As subset S does not exit, the paradox in step 2 of cantor's theorem does not exist, and we can't conclude that P(X)>X.

----------------------------------------------------------------------

Someone can say that in finite sets we can see clearly that there are
more subsets in P(X) than members in X so it must be true for infinite
sets, but a mathematical examination of this intuition shows that it
is wrong.

On diagonales and n X n matrix
--------------------------------

^ = power of

B = a base value which is > 1 (for the example we shell use
base 2 or {0,1})

v = the number of cells in a diagonal

v^2 = tha number of cells in n x n matrix

B^(v^2) = the number of the different matrix that we can get
after we put B members in their matrix's cells.

B^v = the number of the different diagonals that we can get
after we put B members in their diagonal's cells.

A = the number of matrix which include a diagonal with specific contents
in aspcific order.

Now let's build finite n x n matrix (end their diagonals)
which each one of them he's its own spcific contents and order.

Lets build them according to v = 3:

_1___2___3_
000 000 000
000 000 001
000 000 010
000 000 011
... ... ...

_1___2__3__
000 000 000|

1|000
2|000
3|000

_1___2__3__
000 000 001|

1|000
2|000
3|001

_1___2__3__
000 000 010|

1|000
2|000
3|010

_1___2__3__
000 000 011|

1|000
2|000
3|011

_1___2___3_
... ... ... ------> 512 different matrix

The number of different diagonals (when v=3):

000
001
010
011
100
101
110
111 -------> 8 different diagonals

Finding A value
---------------

The formula is: B^(v^2) / B^v = A

so, if v=3 and B=2 then we get:

2^(3^2) / 2^3 = 64

When v is a finite value then each time v becomes
bigger, so A becomes bigger.

Theorem: B^aleph0 = aleph0

Proof:

If v=aleph0 then B^(v^2) / B^v = 1

(B^(v^2) = B^v = 1 to 1 correspondence
between B^(v^2) and B^v)

When v=aleph0 and A=1 (and it means that the number of
the matrix = the number of their different diagonals)
each unique diagonal(=aleph0 cells) belonges to a
different matrix which its size = aleph0^2(=aleph0 cells)
and we can't use the diagonalization argument to find if
there is or there is not a bijection between N and R .

If v=0 then B^(v^2) / B^v = 1

The number of cells included in 0 size matrix is:

B^0 * 0 = 0

The number of cells included in aleph0 size matrix is:

B^aleph0 * aleph0 = aleph0

So, B^aleph0 = aleph0

QED

-------------------------------------------------------

We have to pay attention to the fact that there is
a direct proportion between aleph0 and 0 .

Some graphic example you can find in the attached pdf file.

Yours,

Doron

#### Attachments

• para-graph.pdf
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I agree mostly with your analysis of the liar's paradox... rather than say the paradox does not exists, though, I would say that the statement:

"Either all cretans are honest or all cretans are liars"

is a false statement.

The distinction is important, because it's the basis of proof by contradiction. Formally, proof by contradiction is:

P => Q
P => ~Q
------------
~P

Recall that the paradox part of the Liar's paradox comes from assuming "Either all cretans are honest or all cretans are liars" is true, not its mere existence.

I imagine you had in mind a more sophisticated version of the liar's paradox that takes the form:

"This sentence is false"

This one really is a problem, and the resolution is an integral part of formal logic. The resoltion is that statements are not variables. If we try to encode "This sentence is false" directly into formal logic, it would look like:

P := (P = false)

However, P is not a varaible (because it's a proposition!), therefore the above sentence is not a legal definition of a proposition.

There is a similar paradox in set theory:

S := {x | x is not in x}

S is the set of all sets that don't contain themselves. If we assume S contains itself, then it doesn't contain itself, but if we assume S doesn't contain itself, then it must contain itself.

This (and other more sophisticated paradoxes) led to the downfall of Naive Set Theory.

In ZFC, however, the above definition is again illegal. In ZFC, building up a set through a proposition is not allowed. However, ZFC allows you are allowed to restrict a set via propositions, which is instrumental in Cantor's proof.

Anyways, the formal structure of Cantor's proof is as follows:

Define the three propositions:

A(X, &phi) := &phi is a bijection from X to P(X)
B(X, &phi, S) := for all a in X: a in S iff a not in &phi(a)
C(X, S) := &phi-1(S) in S

For all X, &phi: there exists S: A(X, &phi) => B(X, &phi, S)
For all X, &phi, S: A(X, &phi) and B(X, &phi, S) => C(X, S)
--------------------------------------------------------------------
For all X, &phi: there exists S: A(X, &phi) => C(X, S)

For all X, &phi: there exists S: A(X, &phi) => B(X, &phi, S)
For all X, &phi, S: A(X, &phi) and B(X, &phi, S) => ~C(X, S)
--------------------------------------------------------------------
For all X, &phi: there exists S: A(X, &phi) => ~C(X, S)

For all X, &phi: there exists S: A(X, &phi) => C(X, S)
For all X, &phi: there exists S: A(X, &phi) => ~C(X, S)
--------------------------------------------------------------------
For all X, &phi: ~A(X, &phi)

For all X, &phi: ~A(X, &phi)
-----------------------------------------------------------
For all X: ~(there exists &phi: A(X, &phi))

Turning back into words:
For any set X, there does not exist a bijection &phi from X onto P(X)

All of the implications I used above in the proof were ones to which you didn't object, so I omitted the details for the sake of exposing the overall structure of the proof. This proof puts into formal logic what I was saying previously. When you derive a contradiction, that means one of your hypotheses was false. In the liar's paradox, in addition to the hypothesis that formal logic is correct, we have the hypothesis that "Either all cretans are honest or all cretans are liars". Because these hypotheses lead to a contradiction, one of them must be false. Because there is that additional hypothesis that goes beyond formal logic, the liar's paradox is not sufficient to prove that formal logic is wrong.

Note that the statement is an acceptable proposition formal logic. Formally, it is:

"for all x: x is a cretan => x is always honest" or "for all x: x is a cretan => x always lies"

Anyways, I would be incomplete if I didn't try to find a flaw in your proof.

Theorem: B^aleph0 = aleph0
If v=aleph0 then B^(v^2) / B^v = 1

Division isn't even defined for cardinal numbers! And even if it was, your proof rests heavily on the fact that v is finite, it does not extend to the infinite case.

Hurkyl

Hi Hurkyl !

Lets start form the end.

I think you missed the point of my proof about Cantor's diagonal
because I wrote it in a wrong way.

Lets write again my diagonal's proof, and this time, without any division.

On diagonales and n X n matrix
--------------------------------

^ = power of

B = a base value which is > 1 (for the example we shell use
base 2 or {0,1})

v = the number of cells in a diagonal

v^2 = tha number of cells in n x n matrix

B^(v^2) = the number of the different matrix that we can get
after we put B members in their matrix's cells.

B^v = the number of the different diagonals that we can get
after we put B members in their diagonal's cells.

Now let's build finite n x n matrix (end their diagonals)
which each one of them he's its own spcific contents and order.

Lets build them according to v = 3:

_1___2___3_
000 000 000
000 000 001
000 000 010
000 000 011
... ... ...

_1___2__3__
000 000 000|

1|000
2|000
3|000

_1___2__3__
000 000 001|

1|000
2|000
3|001

_1___2__3__
000 000 010|

1|000
2|000
3|010

_1___2__3__
000 000 011|

1|000
2|000
3|011

_1___2___3_
... ... ... ------> 512 different matrix

The number of different diagonals (when v=3):

000
001
010
011
100
101
110
111 -------> 8 different diagonals

Checking matrix diagonal equality
---------------------------------

The formula is: B^(v^2) > B^v when v has a finite value.

Theorem: B^aleph0 = aleph0

Proof:

If v=aleph0 then B^(v^2) = B^v

When v=aleph0, each unique diagonal(=aleph0 cells) belonges to a
different matrix which its size = aleph0^2(=aleph0 cells).

There are B^(aleph0^2) different matrix (exist along Z axis in a 3D universe) so, there is a room for any unique infi diagonal, and we
can't find if there is or there is not a bijection between
N and R according to cantor's diagonalization argument
(built on top of the assumption that there exist only 1 matrix
with aleph0 size).

(By the way, if we write B^(v^d) when d is any natural number > 1 ,
then the formula B^(v^d) = B^v is good for a matrix, a cube or any
other dimension > 3).

If v=0 then B^(v^2) = B^v

The number of cells included in 0 size matrix is:

B^0 * 0 = 0

The number of cells included in aleph0 size matrix is:

B^aleph0 * aleph0 = aleph0

So, B^aleph0 = aleph0

QED

-------------------------------------------------------

We have to pay attention to the fact that there is
a direct proportion between aleph0 and 0 .

Please look at the attached pdf file, shows this.

Yours,

Doron

#### Attachments

• para-graph.pdf
8.2 KB · Views: 282
Last edited by a moderator:
If v=aleph0 then B^(v^2) = B^v

When v=aleph0, each unique diagonal(=aleph0 cells) belonges to a
different matrix which its size = aleph0^2(=aleph0 cells).

I agree this far.

and we
can't find if there is or there is not a bijection between
N and R according to cantor's diagonalization argument

Okay... so we can't conclude anything from it.

(built on top of the assumption that there exist only 1 matrix
with aleph0 size).

I can easily exhibit 2 infinite matrices; one of all 0's and one of all 1's. Why would you make that assumption? I don't see how it fits into your argument anyways.

If v=0 then B^(v^2) = B^v

The number of cells included in 0 size matrix is:

B^0 * 0 = 0

The number of cells included in aleph0 size matrix is:

B^aleph0 * aleph0 = aleph0

So, B^aleph0 = aleph0

QED

And I have no clue how you could even think to go from the top to the bottom.

Anyways, the other diagonalization argument used in set theory can provide a bijection between the set of countably infinite vectors (B^aleph0) and doubly countably infinite matrices (B^(aleph0^2)). The proof is virtually identical to the proof that aleph0 and aleph0^2 have the same cardinality:

The matrix Aij gets mapped to the vector:

(A00, A10, A01, A20, A11, A02, A30, A21, A12, A03, ...)

It is easy to exhibit the inverse function, making this mapping a bijection:

The vector Bi is mapped to the matrix:

B0 B2 B5 B9 ...
B1 B4 B8 ...
B3 B7 ...
B6 ...
...

And Cantor's diagonalization argument can be used to prove B^aleph0 = |R| when B <= aleph0

Hurkyl

P.S. there was an omission in my previous post; I didn't notate that 'S' was the same in all cases. I can rewrite it, if you like.

Last edited:
Dear Hurkyl !

My English is poor, so you missed the point again.

I'll explain each setp of my proof.

Cantor's diagonalization argument is a tool to proove that
there is another power of infinity which is stronger than the
countable one (the countable one is aleph0=|N|=|W|=|Z|=|Q|).

According to Cantor 2^aleph0=|R| > aleph0=|N|=|W|=|Z|=|Q| and
this is the gate to "Cantor's paradize" which is infinite power
levels of infinite ...2^(2^(2^aleph0)) .

An example of the diagonalization argument in base 2
(we can use any other base) looks like this:

1 <--> .01101...-->> infi
2 <--> .00101...-->> infi
3 <--> .00111...-->> infi
4 <--> .10101...-->> infi
5 <--> .10111...-->> infi
..<--> ...-->> infi
|
v
infi

So, it looks like there is a bijection from N to R,
but then Cantor used a function which created a new number of R
that can not be put in 1 to 1 correspondence with any number of N,
for example:

1 <--> .11101...-->> infi
2 <--> .01101...-->> infi
3 <--> .00011...-->> infi
4 <--> .10111...-->> infi
5 <--> .10110...-->> infi
..<--> ...-->> infi
|
v
infi

The new number in the example (.11010...) can not be put in
1 to 1 correspondence with any number of N, because the function
that created it, made it to be different from each number of R
that exist in the bijection list so, even if we add the new number
to the bijection list, than we can create another new number of R
wich is not in a 1 to 1 correspondence with any number of N
and so on, and so on.

So there exist at least 1 number of R that can not be put in
1 to 1 correspondence with any number of N, and we can conclude
that the infi power of |R| > infi power of |N|.

The above also can be written as 2^aleph0 > aleph0 .

Now let's write again my proof, shows that 2^aleph0=aleph0

On diagonals and n X n matrices
---------------------------------

^ = power of

B = a base value which is > 1 (for the example we shell use
base 2 or {0,1})

v = the number of cells in a diagonal

v^2 = tha number of cells in n x n matrix

B^(v^2) = the different matrices that we can get
after we put B members in each matrix's cells.

B^v = the different diagonals that we can get
after we put B members in each diagonal's cells.

Now let's build finite n x n matrix (end their diagonals)
which each one of them he's its own spcific contents and order.

Lets build them according to v = 3:

_1___2___3_
000 000 000
000 000 001
000 000 010
000 000 011
... ... ...

_1___2__3__
000 000 000|

1|000
2|000
3|000

_1___2__3__
000 000 001|

1|000
2|000
3|001

_1___2__3__
000 000 010|

1|000
2|000
3|010

_1___2__3__
000 000 011|

1|000
2|000
3|011

_1___2___3_
... ... ... ------> 512 different matrices

The number of different diagonals (when v=3):

000
001
010
011
100
101
110
111 -------> 8 different diagonals

Checking matrix - diagonal equality
-----------------------------------

The formula is: B^(v^2) > B^v when v has a finite value.

Theorem: B^aleph0 = aleph0

Proof:

If v=aleph0 then B^(v^2) = B^v

B^(v^2) = the different matrices that we can get
after we put B members in each matrix's cells.

B^v = the different diagonals that we can get
after we put B members in each diagonal's cells.

When v=aleph0, each unique diagonal(=aleph0 cells) belonges to a
different matrix which its size = aleph0^2(=aleph0 cells).

There are B^(aleph0^2) different matrices (exist along Z axis in a 3D universe) and each one of those B^(aleph^2) matrices,already has the "new" R number ,"created" by Cantor's function.

Shortly speaking, all what Cantor's function did is, to jump from
one aleph^2(=aleph0) matrix, to another aleph0^2(=alehp0) matrix
that exist along the Z axis of the 3D universe so, he did not show any new R number that can not be put in 1 to 1 correspondence with any number of N.

So, now we have B^(aleph0^2) matrices and each one of them has
aleph^2(=aleph0) CELLS contains all R (real numbers) digits.

so, let's check again the 2^aleph0 infi power.

If v=0 then B^(v^2) = B^v

The number of CELLS included in 0 size matrices is:

B^0 * 0 = 0

The number of CELLS included in aleph0 size matrices is:

B^aleph0 * aleph0 = aleph0

So, B^aleph0 = aleph0

QED

-------------------------------------------------------

Yours,

Doron

Last edited by a moderator:
Shortly speaking, all what Cantor's function did is, to jump from
one aleph^2(=aleph0) matrix, to another aleph0^2(=alehp0) matrix that exist along the Z axis of the 3D universe so, he did not show any new R number that can not be put in 1 to 1 correspondence with any number of N.

Let me demonstrate the flaw in this reasoning with a trivial example.

Let A = {1, 2} and let B = {3, 4, 5}

Fred claims "B is bigger than A".
George responds "No it's not, they're the same size."
Fred answers "But there can't be any bijection between A and B"
George responds "Sure there is"
Fred challenges "Ok, try to pick a bijection between A and B and I'll find an element of B that isn't placed in 1-1 correspondence with an element in A"
Fred retorts "Aha! notice that 5 doesn't correspond to any element of A!"
George replies "Yah, but if I do 1<->3 and 2<->5, then 5 is in 1-1 correspondence with an element of A. Your argument is flawed!"

You're doing the same thing George did; if you selected any particular mapping from N into R, Cantor's argument finds a real number that does not correspond to any element in N... but then you switch mappings (i.e. change matrices) so that the element found by Cantor is accommodated by your new mapping.

Well, George is wrong and so are you. A bijection is a single mapping, not a collection of them. If the only way you can exhibit a correspondance is to change maps (i.e. matrices) mid-stream, then there cannot be a bijection.

If v=0 then B^(v^2) = B^v

The number of CELLS included in 0 size matrix is:

B^0 * 0 = 0

The number of CELLS included in aleph0 size matrix is:

B^aleph0 * aleph0 = aleph0

So, B^aleph0 = aleph0

QED

And I still have no clue what the heck this is supposed to mean.

Anyways, let's put the shoe on the other foot. I challenge you to do the same thing to my proof that I'm doing to your proof. I'll give my own proof (based, of course, on Cantor's) that 2^|N| > |N|, and I want you to pick out one or more steps and tell me they're wrong. And by that I mean that you actually explain what logical fallicy I made, as opposed to trying to construct a proof that my conclusion is wrong.

By definition, 2^|N| = |S| where S is the set of functions from N into {0, 1}. (a.k.a. the set of binary sequences)

Lemma: For any &phi from N to S: there exists &theta in S: for all m in N: ~(&theta = &phi(m))
-------------------------------------------------------
Let &phi be any function from N to S

define &theta in S as: &theta(n) = 1 - (&phi(n))(n)
(to help decipher this, recall that S is a set of functions, so &phi(n) is a function. (&phi(n))(n) means I evaluate &phi at n, and then evaluate the resulting function at n)

Let m be any element of N:
&theta(m) = 1 - (&phi(m))(m)

Since &theta and &phi(m) differ in the m-th position, &theta and &phi(m) must be unequal.
--------------------------------------------------------

Now, apply some logical rules to the above lemma:

For any &phi from N to S: there exists &theta in S: for all m in N: ~(&theta = &phi(m))

is equivalent to

For any &phi from N to S: there exists &theta in S: ~(there exists m in N: &theta = &phi(m))

is equivalent to

For any &phi from N to S: ~(for all &theta in S: there exists m in N: &theta = &phi(m))

is equivalent to

~(there exists &phi from N to S: for all &theta in S: there exists m in N: &theta = &phi(m))

Now, by definition of bijection, this is equivalent to:

~(there exists &phi from N to S: &phi is a bijection)

And via a theorem on set cardinalities, this is equivalent to

~(|N| = |S|)

Since we know |N| <= |S|, we thus conclude |N| < |S|, and thus:

|N| < 2^|N|

Last edited:
Is each number in the full infi list of the real numbers (R),
is a unique number (I mean no doubles, triples and so on) ?

It's unclear what you mean. Two distinct elements of the set R are, of course, distinct.

But, you said "the ... list". That could mean a great many things...

My best guess is that you mean the following, to which I agree:

There exists a sequence S (over some index set &Iota) such that every real number appears exactly once in the sequence.

In fact, this statement is trivial; for instance, if I choose the real numbers as my index set, then the sequence &tau defined as &taur = r is a trivial "full infinite list" of the real numbers.

Dear Hurkyl !

About the diagonalization argument's function and R set
-------------------------------------------------------

If cantor's function has changed each real number, we can ask:

What happened to the real numbers that were in R set
before cantor's function has changed each one of them in 1 digit ?

Answer: now there exist two R sets, but because
A bijection is a single mapping, not a collection of them,
then we marge the sets but now we have to use Cantor's function again on the merged set, and so on, and so on, ad infinituum...
(never-start bijection)

If cantor's function has not changed each real number ,but gave us
the information to build another R set, than the question is:

What we are going to do with the new R set ?

Answer: Well, now there exist two R sets, but because
A bijection is a single mapping, not a collection of them,
then we marge the sets but now we have to use Cantor's function again on the merged set, and so on, and so on, ad infinituum...
(never-start bijection)

Please tell me what do you think ?

Yours,

Doron.

If cantor's function has changed each real number

That's not the case because well-defined mathematical objects cannot change.

So...

If cantor's function has not changed each real number ,but gave us
the information to build another R set

That's not the case either. As previously mentioned, well-defined mathematical objects cannot change.

(R^N is the set of all functions from the natural numbers to the real numbers, a.k.a. the set of all ordinary sequences of real numbers)

Why do you think "Cantor's function" tries to change or create anything?

Last edited:
Maybe this analogy'll help:

Cantor's Proof
--------------
You try and count the real numbers.
Any counting you try, I can find one you missed.
So there is no counting

There is no greatest integer
--------------
You try and pick a greatest integer
No matter which one you pick, I can find one bigger
So there's no biggest one,.

Hi Hurkyl !

An example:

Every R member is a unique member in the aleph0^2 matrix so,
if we take this alph0^2 R set, we can build another aleph0^2 R' set with unique R' members by using the data of Cantor's diagonal, instead of the data of the original diagonal that belongs to aleph0^2 R set, and then
we can do this bijection:

1 <--> R1 unique member
2 <--> R'1 unique member
3 <--> R2 unique member
4 <--> R'2 unique member
5 <--> R3 unique member
6 <--> R'3 unique member
7 <--> ...

Yours,

Doron

#### Attachments

• para-graph.pdf
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Every R member is a unique member in the aleph0^2 matrix so

I presume you really meant to say:

Every real number in the aleph0^2 "matrix" is unique.

Is this correct? Anyways, I guess this is a side issue because this isn't where the main fault in your proof lies.

(Just for the sake of clarity, by "member" I presume you mean that R's binary representation is one of the rows of the matrix. Also, for the sake of completeness, one also needs to specify where the binary point should go for each row and the nonuniqueness of binary representations, but these are minor details that can be considered independantly from the main issue)

if we take this alph0^2 R set, we can build another aleph0^2 R' set with unique R' members by using the data of Cantor's diagonal, instead of the data of the original diagonal that belongs to aleph0^2 R set, and then
we can do this bijection:

1 <--> R1 unique member
2 <--> R'1 unique member
3 <--> R2 unique member
4 <--> R'2 unique member
5 <--> R3 unique member
6 <--> R'3 unique member
7 <--> ...

Ok, so you've created a new matrix that corresponds to a new function from N into R.

I presume by the pre-edited version of your post that your scheme is that R'n is the same as Rn but with the n-th digit flipped.

My question is "Why?"

What good is your new mapping?
Can you prove your new mapping is a bijection? (according to Cantor's diagonal method, it is not)
Can you prove that your new mapping even includes the element that Cantor's diagonal method says wasn't covered by the original mapping?

For an explicit example, suppose the original matrix was:
0 0 0 0 0 0 ...
1 0 0 0 0 0 ...
1 1 0 0 0 0 ...
1 1 1 0 0 0 ...
1 1 1 1 0 0 ...
...

Cantor's diagonal method applied to this matrix selects the sequence:

&alpha = 1 1 1 1 1 1 ...

which does not appear as any row of the matrix.

If one flips the n-th bit of the n-th row you get:

1 0 0 0 0 0 ...
1 1 0 0 0 0 ...
1 1 1 0 0 0 ...
1 1 1 1 0 0 ...
1 1 1 1 1 0 ...
...

And if we execute your splicing scheme we get:

0 0 0 0 0 0 ...
1 0 0 0 0 0 ...
1 0 0 0 0 0 ...
1 1 0 0 0 0 ...
1 1 0 0 0 0 ...
1 1 1 0 0 0 ...
1 1 1 0 0 0 ...
1 1 1 1 0 0 ...
1 1 1 1 0 0 ...
1 1 1 1 1 0 ...
...

So not only does the new matrix still not include the sequence &alpha, but the new matrix doesn't even cover any new sequences; it just duplicates all but one of the rows of the original matrix!

Last edited:
Hi Hurkyl !

Can you prove that your new mapping even includes the element that Cantor's diagonal method says wasn't covered by the original mapping?

First, let's write again my example:

Every R member is a unique member in the aleph0^2 matrix so,
if we take this alph0^2 R set, we can build another aleph0^2 R' set with R' members by using the data of Cantor's diagonal, instead of the data of the original diagonal that belongs to aleph0^2 R set,
and then we can do this bijection:

1 <--> R1
2 <--> R'1
3 <--> R2
4 <--> R'2
5 <--> R3
6 <--> R'3
7 <--> ...

Now, here is my proof:

Cantor's diagonal is not in R set.

R'1 can not be equal to R1.

Let be R'1 = Cantor's diagonal.

R'1 is in the bijection list.

QED

Yours,

Doron

Last edited by a moderator:
You've proven that you can create a new matrix that has all the same rows as the old matrix, but with a single new row that contains one number that the old matrix didn't have. How do you prove that all real numbers are in the new matrix?

How do you prove that all real numbers are in the new matrix?

Well, I have all the infomation that I need from both R and R'
sets.

Yours,

Doron.

You've called the set covered by the original matrix R, but there's no reason to think what you've called R is the entire real numbers. (Of course, it can't be, because Cantor's method shows a real number that is missing)

You mix in R', but there's no reason to think R' even contains a single element not in the set you called R...

Hurkyl

Dear Hurkyl !

Before your last message you wrote:
You've proven that you can create a new matrix that has all the same rows as the old matrix, but with a single new row that contains one number that the old matrix didn't have
You mix in R', but there's no reason to think R' even contains a single element not in the set you called R...

An example:

R set (before Cantor's function is active)

1 <--> .111... = R1
2 <--> .100... = R2
3 <--> .110... = R3
4 <--> ... = R4

R' set (after Cantor's function is active)

1 <--> .011... = R'1 = Cantor's diagonal
2 <--> .110... = R'2
3 <--> .111... = R'3
4 <--> ... = R'4

and then merging between R'1 and R:

1 <--> .011... = R'1 = Cantor's diagonal
2 <--> .111... = R1
3 <--> .100... = R2
4 <--> .110... = R3
5 <--> ... = R4

We are in an endless game, because after each Cantor's function operation,
I can return its results to the bijection list.

So, Cantor's diagonalization argument can't work.

Yours,

Doron

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