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).

 

[tacman]

The Skolem Paradox is a mathematical paradox that revolves around the countability of real arithmetic. The paradox arises when we consider the fact that real arithmetic is uncountable, meaning that there are an infinite number of real numbers, but it also possesses a countable "model." This "model" refers to a specific structure or system that can represent the operations and relationships of real arithmetic, but is itself countable.

To answer the first question, it is important to understand the difference between counting numbers and counting real numbers. Counting numbers is a finite process, where we can assign a unique number to each object in a set. However, real numbers are infinite and cannot be counted in the same way. We can always find a number between any two given numbers, making it impossible to assign a unique number to each real number.

Moving on to the second question, the "model" in this paradox refers to a structure or system that represents real arithmetic. This model can be thought of as a simplified version of real arithmetic, where we can perform operations and relationships on a finite number of elements. This model is countable because it only consists of a finite number of elements, making it possible to assign a unique number to each element.

Finally, the reason why the model is countable but real arithmetic is not, is because of the different ways in which we can count them. As mentioned before, real numbers are infinite and cannot be counted in a finite manner. On the other hand, the model is a simplified version with a finite number of elements, which can be counted in a finite way.

In conclusion, the Skolem Paradox highlights the difference between counting finite and infinite sets. Real arithmetic, being an infinite set, cannot be counted in a finite way, but its countable "model" allows us to understand and represent its operations and relationships in a simplified manner.