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Skolem paradox

  1. Oct 17, 2003 #1
    mathworld defines the paradox like this:"Even though real arithmetic is uncountable, it possesses a countable "model.""
    now here a few a questions:
    1. why cant you count in real arithmetic, surely you can count numbers (-: ?
    2. what is this "model"?
    3. why the "model" is countable but the arithmetic isnt?
     
  2. jcsd
  3. Oct 17, 2003 #2

    Hurkyl

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    I have to guess at the meaning of some of the terms but...


    Recall that "countable" when applied to a set means that the set can be placed into a 1-1 correspondence with the natural numbers. For example, with the rational numbers, we can write the enumeration

    1 - 1/1
    2 - 2/1
    3 - 1/2
    4 - 3/1
    5 - 1/3 (we already have 2/2)
    6 - 4/1
    7 - 3/2
    8 - 2/3
    9 - 1/4
    10 - 5/1
    11 - 1/5
    ...

    Every rational number will appear in this sequence, so the rational numbers are countable.

    However, the real numbers are uncountable; it is impossible to make such an enumeration (via one of Cantor's diagonal arguments).


    I presume by saying "real arithmetic is uncountable" it means that there are uncountably many real numbers.

    As for the countable model... I presume that they mean when the axioms are weakened to be written in first-order logic. I don't know what one does to the axiom of completeness, but I know there's an important type of field called a real closed field (aka "formally real field"), and I presume that the axioms of a real closed field are what replaces the axiom of completeness. A countable model of a real closed field is the algebraic numbers (the field of all real roots of integer polynomials).
     
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