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A new Theory of Numbers

  1. Apr 15, 2003 #1
    Hallo Dear people !

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

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

    I'll be glad to get your remarks and insights.

    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:smile:

    Special thanks to Hurkyl:

     
    Last edited by a moderator: May 21, 2003
  2. jcsd
  3. Apr 15, 2003 #2
    Strange ... i can't find the attachment anywhere !
     
  4. Apr 15, 2003 #3
    test
     
    Last edited by a moderator: Apr 18, 2003
  5. Apr 15, 2003 #4
    But I have got it!
     
  6. Apr 15, 2003 #5
    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.
     
  7. Apr 16, 2003 #6

    Hurkyl

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    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
     
    Last edited: Apr 16, 2003
  8. Apr 16, 2003 #7
    Dear Hurkyl,

    First, thank you for your comments, I'll try to answer you.

    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, lets 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
     
    Last edited by a moderator: May 17, 2003
  9. Apr 16, 2003 #8

    Hurkyl

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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 lets 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 existance. This appears to me to be a totally seperate 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.



    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
     
  10. Apr 16, 2003 #9
    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
    http://www.cfh.ufsc.br/~dkrause/DoriaSPqset.pdf

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

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

    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
     

    Attached Files:

    Last edited by a moderator: Apr 16, 2003
  11. Apr 17, 2003 #10

    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.

    Now, please try to follow my ideas, and please tell me what do you think.

    Yours,

    Doron
     
    Last edited by a moderator: May 21, 2003
  12. Apr 17, 2003 #11

    Hi Moni !


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


    Yours,


    Doron
     
  13. Apr 17, 2003 #12

    Hurkyl

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    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 existance 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
     
  14. Apr 18, 2003 #13
    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:

    Beads-only(=Discreteness-only) are uncountable,

    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
     
    Last edited by a moderator: Apr 18, 2003
  15. Apr 18, 2003 #14

    Hurkyl

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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.


    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

     
    Last edited: Apr 18, 2003
  16. Apr 18, 2003 #15

    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 lets 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
     
  17. Apr 19, 2003 #16
    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
     

    Attached Files:

    Last edited by a moderator: Apr 19, 2003
  18. Apr 19, 2003 #17

    Hurkyl

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    I think I'm gonna 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.



    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

     
    Last edited: Apr 19, 2003
  19. Apr 19, 2003 #18

    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

     
  20. Apr 19, 2003 #19
    Dear Hurkyl !

    Before I give you my definition for "well known", let me say
    something about transfinite cardinals.

    You wrote:
    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 lets take another look on Cantor's proof.

    On the liar's paradox
    ---------------------

    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 lets 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 existance 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 lets build finite n x n matrix (end their diagonals)
    which each one of them hes 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
     

    Attached Files:

    Last edited by a moderator: Apr 19, 2003
  21. Apr 19, 2003 #20

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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 existance.



    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.

    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
     
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