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What is Skolem's paradox?

According to Skolem’s paradox, any statement that has a non-denumerable model has a denumerable model.

A class has ‘denumerably’ many members if it can be put into a one-one correspondence with the class of natural numbers (0, 1, 2, 3,….) A class has ‘non-denumerably’ many members if it cannot be bijected with the class of natural numbers or with any subset thereof. If a class has non-denumerably many members, it is larger than the class of naturals.

An example of a non-denumerable class is the class of real numbers.

It turns out that any statement about non-denumerable classes, including the statement that they exist, can be interpreted as statements about denumerable classes. And this is Skolem’s paradox.

This ‘paradox’ is a theorem—a proven mathematical truth—-not the by-product of a mistake.

Skolem’s paradox is not so much about mathematical objects as it is about the shortcomings of any attempt to understand such things in a purely formal way. A ‘formalization’ or ‘model’ of a given class of mathematical truths is nothing other than a recursive definition of such a class, meaning that that it is a class that has a first element and any given one of whose remaining elements can be reached from that element in a finite number of steps. So Skolem’s ‘paradox’ is really just an artifact of the concept of what a formalization is than it is a deep mystery about the nature of mathematics proper.

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