Unraveling the Skolem Paradox: A Look into the Countability of Real Arithmetic

[MathematicalPhysicist]

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 can't 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?

 

[Hurkyl]

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).