As is well-known, the Lowenheim-Skolem Theorem (I’m taking it from Schoenfield’s text–my comments will be in square brackets)
If $ T$ is a countable [first-order] theory having a model, then $ T$ has a countable model.
allows for the existence of countable transitive models of $ ZFC$ .
Furthermore, following Skolem, Schoenfield writes:
…We can certainly formalize enough mathematics in a countable theory to prove that the set of real numbers is uncountable. How can such a theory have a countable model? The explanation is this. The set of real numbers in the model is indeed countable. But this mapping is not in the model; so it does not make invalid the theorem of the theory which states that there is no bijective mapping from the set of real numbers [in the model] to the set of natural numbers [in the model].
It is also well-known that for any countable transitive model of $ ZFC$ one can enumerate the dense sets (since there only countably many of them, but not within the model) and then define from this enumeration a generic filter (and in such manner claim it exists–this from Asaf Karagila’s answer to my math stack exchange question, “A question regarding the Continuum Hypothesis (Revised)”). This allows one to create from any countable transitive model $ \mathfrak M$ of $ ZFC$ a forcing extension $ \mathfrak M[G]$ of $ ZFC$ (which is also another countable transitive model).
However, one should note that such models are ‘delusional’ in the following sense–that set-theoretic notions like ‘uncountability’, the generic filter $ G$ , large cardinals, etc., are formed by relations between the countable sets within the model in question and do not involve ‘actual’ uncountability, $ G$ , large cardinals or anything of the sort. The set-theoretic notions in question are simply interpretations of these relations imposed from the outside, nothing more. Therefore, these countable structures, free of any interpretation in the language of set theory, are of inherent mathematical interest.
But are they? One could argue that it is these set-theoretic notions, through ambiguities of definition analogous to what Skolem discovered about ‘uncountability’, allow these relations within the countable structures in question to exist, so in that sense, the interpretation (the ‘delusion’) is necessary for these relations to exist and to be mathematically interesting.
The above prompts the following question:
Are the countable structures underlying the countable transitive models of $ ZFC$ interesting objects of mathematical research in and of themselves, or is some intepretation in the language of set theory necessary in order to make them interesting objects of mathematical research?
[As evidence for the view that the countable structures underlying the countable transitive models of $ ZFC$ are mathematical interesting in their own right, consider the results listed in David A. Madore’s 2017 preprint, “A zoo of ordinals”. Though the results contained therein contain interpretations, the results seem dependent upon the underlying ordinal structure, not on the interpretation.]
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