Are counting numbers or natural numbers a set within real numbers?

Discussion in 'Numerical Analysis' started by Martin, Jun 3, 2011.

  1. Martin

    Martin Guest

    The definition of a real number is a quantity along a continuum. For
    example -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. Counting numbers (or
    natural numbers as they are called) are basically how we count out
    loud and contain 1, 2, 3, 4, 5. Since 1, 2, 3, 4, 5 is a subset of -5,
    -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 counting numbers are a subset within
    real numbers. Agree or disagree? Martin
     
    Martin, Jun 3, 2011
    #1
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  2. Martin

    calvin Guest

    You're confusing integers and real numbers.
     
    calvin, Jun 4, 2011
    #2
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  3. No, this is NOT the definition. I suggest that you should
    actually learn this subject before making further ridiculous
    claims. Take a course in real analysis. Look up "Cauchy sequence"
    or "Dedekind cut".
    This set does NOT form a "continuum". It is discrete.
    Mathematics is not done by popular poll. However, in this
    case, this is the first statement you have made that is correct.
     
    Pubkeybreaker, Jun 4, 2011
    #3
  4. Martin

    Martin Guest

    I understand the entire set of reals are not limited to integers or
    whole numbers and did not confuse the two though I understand how it
    may have appeared so.
     
    Martin, Jun 4, 2011
    #4
  5. Martin

    Martin Guest

    This is precisely the definition: In mathematics, a real number is a
    value that represents a quantity along a continuum http://en.wikipedia.org/wiki/Real_number
    I suggest that you should
    I would like to.
     Look up "Cauchy sequence"
    Cauchy sequence convergant for R or converge in the reals. A sequence
    a_1, a_2, ... metric d(a_m,a_n) satisfies lim_(min(m,n)-
    >infty)d(a_m,a_n)=0.
    set partition of the rational numbers into two nonempty subsets S_1
    and S_2 all members of S_1 are less than those of S_2 and S_1 has no
    greatest member.
    Agreed.
    However, in this
    How can a question of agree or disagree be correct or incorrect or
    rather how can a question be correct or incorrect?
     
    Martin, Jun 4, 2011
    #5
  6. The very next paragraph in that page reads:

    "These descriptions of the real numbers are not sufficiently
    rigorous by the modern standards of pure mathematics. The discovery of
    a suitably rigorous definition of the real numbers — indeed, the
    realization that a better definition was needed — was one of the most
    important developments of 19th century mathematics."

    So, no, that is *not* "precisely the definition". It's not even a
    definition, it's a nonrigorous description.

    See also http://math.stackexchange.com/questions/14828/set-theoretic-definition-of-numbers/14842#14842
     
    Arturo Magidin, Jun 4, 2011
    #6
  7. Martin

    nmm1 Guest

    Which is itself flawed! "Certainly, N and Z are entirely different
    animals; set-theoretically, you can even show that they are disjoint."

    The concept of disjointness applies only with two subsets of the
    same superset. You can define sets where Z is a subset of R,
    ones where they are disjoint, ones where they overlap, and even
    more arcane possibilities.


    Regards,
    Nick Maclaren.
     
    nmm1, Jun 4, 2011
    #7
  8. Martin

    nmm1 Guest

    I suggest that you look further into it.

    Something that real mathematics (though not some of the computer
    science that claims to be mathematics) is rather keen on is absolute
    consistency, in that the statements A and not A should not both be
    true. If you start applying the concept of disjointness to two
    sets except in the context of being subsets of a superset, they
    can be both disjoint and not disjoint. As that very reference
    indicates!
    And, without some superset to define the rules of doing that, the
    union operator is undefined!


    Regards,
    Nick Maclaren.
     
    nmm1, Jun 4, 2011
    #8
  9. There is no such requirement in ordinary set theoretic
    mathematics. Even if we for some reason thought it prudent to talk of
    two sets as disjoint only when they both are subsets of some set, this
    would make no difference: given sets A and B they are both subsets of A
    union B.
    We can define all sorts of sets. What do you take to be the relevance
    of this trivial observation?
     
    Aatu Koskensilta, Jun 4, 2011
    #9
  10. There's nothing further to look into. If we inspect the definition of
    disjointness in any standard set theory text we find no mention of any
    supersets. We might find talk about some fixed superset U, and
    operations on subsets of U, in introductory discrete mathematics texts,
    of course.
    Something real mathematicians are usually not particularly keen on is
    contentless blather about "absolute consistency" and so on.
     
    Aatu Koskensilta, Jun 4, 2011
    #10
  11. The answer to

    Are counting numbers or natural numbers a set within real numbers?

    is "yes and no". One can construct the real numbers from the natural
    numbers in various ways, and according to the ways that I know of the
    answer is "no", but one identifies the natural number 6 (say) with the
    real number 6. Or one can start with the real numbers which, being a
    field, have 0 and 1 among them, and then define the natural numbers to
    be

    0, 0+1, 0+1+1, etc.

    So the answer is yes.
     
    Frederick Williams, Jun 4, 2011
    #11
  12. That's not a problem. The usual notions of true and not are such that
    if A is true, not A is false; and if A is false, not A is true.
     
    Frederick Williams, Jun 4, 2011
    #12
  13. Martin

    Ronald Bruck Guest

    There's nothing wrong with this statement, since disjointness depends
    only on the resolution of the statement "x = y", which is a primitive.
    Reading the full paragraph, the authors define Z as something like N x
    {0,1}, where a "0" corresponds to a "-" and a "1" to a "+" (removing
    the extra representation of 0). Nothing of this form is in N, so
    they're disjoint, AS SETS.

    Of course, in the usual development this is followed by a redefinition
    of N as N x {1}, and only then is N \subset Z.

    -- Ron Bruck
     
    Ronald Bruck, Jun 4, 2011
    #13
  14. Okay, so what we actually have here is that you have absolutely no
    idea whatsoever about what you are talking about.

    That is complete, absolute, abject nonsense, without any basis
    whatsoever in set theory. You have no idea what you are talking about.

    Take a sentence out of context; add a dash of ignorance, and what you
    get is Nick Maclaren's pronouncements on mathematics.
     
    Arturo Magidin, Jun 4, 2011
    #14
  15. The "very reference" indicates no such thing. Do stop projecting your
    ignorance on others.

    Two sets are disjoint if and only if their intersection is empty.
    Period.

    Sets cannot be both "disjoint" and "nondisjoint". Don't confuse the
    *name* *you* give a set with the set itself.
    The existence of unions is part of Set Theory. It's an axiom, in
    fact.

    In ZF set theory, the Axiom of Pairing asserts that for every sets A
    and B, there exists a set Z such that A and B are elements of Z.
    Applying then the Axiom of Separation, we conclude that for every sets
    A and B, there is a set Z whose elements are *precisely* A and B.

    The Axiom of Unions states that if W is any set, then there is a set Y
    such that x is an element of Y if and only if there exists an element
    w of W such that x is an element of W; that is, Y is the union of the
    elements of W.

    Combine the two, and you get that for *any* two sets A and B (with no
    "superset to define the rules of doing" anything) there is a set which
    is equal to the union of
    A and B.

    If you don't know the subject, perhaps you might want to ask instead
    of issue nonsensical assertions?
     
    Arturo Magidin, Jun 4, 2011
    #15
  16. Martin

    Guest Guest

    Thank you. Martin
     
    Guest, Jun 6, 2011
    #16
  17. Hi. I like your question and prefer to use physical examples and
    graphical analysis to consider it. Traditionally, real analysis does
    build its continuum out of the natural number by moving along to
    develop more types that fill in the line. When people bothered to name
    a number 'real' shouldn't we admit that they took this name seriously?
    So I justify my physical instances which will expose the arbitrary
    choice of unity along the real line, whereas within the counting
    numbers no such arbitrary choice is possible.

    Simply compare one centimeter against one inch and we see that the
    encoding of the continuum's address requires a definition of unity. We
    name this thing the unit and denote it by the symbol
    1
    or
    1.00000...
    Likewise when we draw a line upon a piece of paper to address the
    continuum we are forced to plop down two arbitrary positions; zero and
    unity, at which point we feel confident that we can address the rest
    of the representation.

    Relativity theory further blurs the representation by allowing for
    numerous references, the offsets from one reference to another being a
    means of address which invokes relative offsets, and in the
    multidimensional form taking on even more complexity.

    I criticize the mathematician's real number firstly for its name. The
    real world seems symbolically quantifiable, but only to some finite
    precision. We might pull out a caliper and witness an instrument
    capable of unencrypted continuous transcription abilities, whereas the
    tape measure as it is used usually to produce an intermediate number
    cannot provide this level of integrity. Still, as a matter of design
    the numerically quantified piece is the most reproducible and
    transferable design, presuming that the unit of the work was
    accurately transferred in the first place.

    I believe that the continuum can and should stand freely aside from
    the discrete value. I believe that magnitude, as it lacks any
    signature is more fundamental than the real number and so the problem
    can be dissected slightly more. Under this dissection the ray becomes
    more fundamental than the line, and in this way perhaps we can come to
    view the vector as a more primitive format than it is built in today's
    math. Let's not forget also that the term 'dimension' springs from the
    real line as fundamental; the number of dimensions of a system being a
    matter of the quantity of real lines(typically orthogonal) required to
    represent that system. By splitting that real line in half what
    becomes of these dimensional analyses? They do survive, though off by
    one. Still, that one matters. The ray... the continuum... the discrete
    dimension that none of us can deny... time as a mystery still belies
    the fallacy of modern mathematics. That unidirectional ray mimics the
    unidirectional time that we witness, while we stumble to quantify it
    with a real number. Why? Because this is an old presumption, and the
    human mind operates via mimicry. False belief propagates as readily
    and as unchallenged as truth within the mimicry paradigm.

    - Tim http://bandtech.com/polysigned
     
    Tim Golden BandTech.com, Jun 6, 2011
    #17
  18.  
    Michael Stemper, Jun 6, 2011
    #18
  19. Martin

    Ronald Bruck Guest

    For what? His answer is WRONG.

    To say that N is NOT a subset of R is to argue that the technicalities
    of the proof are to build, from N, a set Z (which is set-theoretically
    disjoint from N, but which contains a "copy" of N), then to build a set
    Q which contains a "copy" of Z (but which is disjoint from Z), etc. To
    leave it at that--where N, Z, Q and R are all disjoint--is a little
    like hiring a contractor to build you a house, and when you arrive to
    take possession, you find he's left all kinds of forms and wood scraps
    and unpainted walls. He hasn't finished the job.

    The final step is to REDEFINE Q, Z, and N, so that they ARE subsets of
    R. OK, the original contractor argues that he PUT the walls in, but
    just didn't paint them; but that's not what most people would call
    finishing the job.

    -- Ron Bruck
     
    Ronald Bruck, Jun 8, 2011
    #19
  20. Martin

    fishfry Guest

    Bad analogy. The construction you outlined is the standard way N, Z, Q,
    and R are defined. Each set injects into the next one in a natural way.

    The reason this is not the same as building a house, is that once the
    construction of N, Z, Q, and R is accomplished, we can then freely use
    these sets with their usual properties. Nobody ever cares or mentions
    that Z is not really a subset of Q. It's irrelevant once we've shown
    that we could build Q out of Z if we wanted to.

    If you want an analogy from the field of buildings. it's like the sign
    in an elevator that the inspection certificate is on file at the
    superintendent's office. Nobody cares that the certificate might be
    torn, taped up, old, yellowed, and hard to find. We don't care about it
    except to know that it exists; and even then, we really don't care about
    it one way or another.

    I believe xkcd made this point recently ...

    http://xkcd.com/897/
     
    fishfry, Jun 8, 2011
    #20
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