Skolem’s paradox

In: 3

I’m answering this with the understanding that ELI5 doesn’t literally mean you could explain it to a 5 year old. For reasons I hope are obvious, you can’t explain model theory to a 5-year-old. What I instead hope to give is a (very rough) explanation that could be understood by a motivated high school student. I will state the paradox first, then do my best to explain what it means.

So – what exactly is a the paradox? Well, the main idea is that the Löwenheim-Skolem theorem ensures that if a set of axioms has an infinite model, then it has a model that is countable. Cantor’s theorem ensures, for a consistent axiomatization of set theory, the existence of uncountable sets. These two things seem at odds with each other.

So what does any of that mean? Well, a *model* of a set of axioms is, roughly speaking, a structure for which those axioms hold. This structure involves an underlying set, with some functions, predicates, and relations defined on it. What it means for a model to be countable is if the set underlying the model is countable.

The paradox resolves itself when we realize we are dealing with, and erroneously equating, two different notions of a set. The Löwenheim-Skolem theorem refers to the definition of a set given by the ambient language with which we actually formulate model theory, whereas Cantor’s theorem uses the notion of a set that follows the axioms of our model.

By Cantor’s theorem, given an axiomization of set theory there will always exist a set within that theory such that the axioms cannot prove a bijection to the real numbers. This is not related to the cardinality of the underlying set of the model: All we know is there exists an *element of the underlying set* of the model, call it **p**, that satisfies the statement “**p** is uncountable”, with a notion of uncountability that comes from the model rather than the ambient language.

If this feels arcane and very out of the bounds of ELI5, that’s because it is. Model theory is a deep and fascinating rabbit hole in math that often feels like black magic.

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