Skolem’s paradox
#notes on this
As a consequence of Cantor’s theorem, there exists a set of uncountable cardinality
Löwenheim-Skolem theorem: if a first-order theory has an infinite model, then it has a countable model
Seems to be a paradox!
By Cantor’s theorem we have that and so there is a where
But since is countable then there are only countably many such that !
Contradiction?
Resolution
When evaluating the semantics of a first-order sentence such as , we must recognize that the meaning of first-order language is relative to the model in which we interpret it
For instance, - and - bindings quantify exactly over the domain of rather than any other domain that we had in mind. Likewise, the inclusion relation need not represent “genuine” inclusion but need only be some binary relation between elements of which satisfies the appropriate properties
Though we have some intention for what first-order sentences “ought” to mean, has no mandate to respect that intention.
As humans, we have a certain sense and intuition around what it means for a set to be uncountable. Guided by that sense and intuition, we write the definition of as a first-order expression of uncountability
But this definition was predicated on a certain understanding of first-order logic which need not follow
As such, the fact that may have a meaning far differing from being uncountable.
(For what it’s worth, I think the article does a better job explaining this than me. Go read it! §2.2)
Details on how has “strange interpretations”
Transitive models
A model is called transitive if (1) every member of is a set and any every member of a member of is also a member of ; and (2) the relation in is the same as in the language being used to describe
Conceptually, in a transitive model the interpretations of membership and bijectivity are “correct” (relative to the metalanguage)
An example of how a transitive model can “misinterpret” axioms
Powerset axiom is
But only quantifies over members of
(In §2.4¶4 they say that the source Resnik 1966 follows “this phenomenon through the case of the real numbers”. Seems interesting! #onwards)
Assuming is is transitive, where can it differ from the metalanguage for Skolem’s paradox to arise?
can differ on
The form of is
Since they agree on “bijective”, then they must differ on ""
Skolem’s paradox is intimately tied to first-order logic
Does not arise in second-order settings
Standard proofs fail in constructivist settings
Personally, it seems that this situation suggests that the semantics of first-order logic are not strong enough. If we are confident that we know what uncountability means, and we are confident that is a proper expression of uncountability in first-order logic, then it seems that any appropriately-faithful interpretation of would demand an uncountable model.
The source also includes some discussion of philosophical consequences of Skolem’s paradox (§3) which I chose to skip/skim for now (#onwards?)