How large is the supremum of minimal $V$-heights of all first-order set theories formulated in a particular language of FOST?

Question

Fix a language $\mathcal{L}$ of first-order set theory. For this question, we can assume that $\mathcal{L}$ is the language described in Chapter 1 of “An introduction to set theory” [William A. R. Weiss | October 2, 2008].

Assuming that the complexity of a formula is not restricted and every formula has a finite length, construct an infinite list $Z_{\mathcal{L}}$ of all syntactically valid statements, so that every such statement is assigned an unique natural number (i.e. enumerate all syntactically valid statements).

Consider an infinite binary sequence $s$. We can assume that any such sequence corresponds to a set of axioms of a particular set theory $T$: an $i$-th bit of $s$ is non-zero if and only if an $i$-th statement of $Z_{\mathcal{L}}$ is an axiom of $T$. Let $t(s)$ denote a theory encoded by $s$. (For example, there will be some $s$ such that $t(s) = \text{ZFC}$.)

Then $f(s) = \alpha+1$ if and only if there exists the smallest ordinal $\alpha$ such that $V_{\alpha} \models t(s)$; otherwise, $f(s) = 0$. For example, if $t(s)=\text{ZFC},$ then $f(s)$ is equal to the successor of the initial ordinal of the smallest worldly cardinal.

Note that the phrase “any binary sequence corresponds to a set of axioms of a particular set theory $T$” does not make any assumptions about $T$: the theory may be empty (if all bits of $s$ are zero) or inconsistent. That is, almost all sequences will not correspond to a consistent theory, but we are not interested in such sequences: for any such sequence $s$ we will have $f(s) = 0$.

The ordinal $\beta_{\mathcal{L}}$ is defined as follows: $$\beta_{\mathcal{L}} = \sup \{ f(s) : s \in {2^\omega }\}.$$

Here “$s \in {2^\omega}$” means that we take into account all infinite binary sequences (subsets of $\omega$).

Question: is $\beta_{\mathcal{L}}$ a well-defined ordinal? If no, why? If yes, how large is $\beta_{\mathcal{L}}$ in the hierarchy of large cardinals?

Answer 1

Your ordinal $\beta_\mathcal{L}$ is perfectly well-defined: in my opinion it's more easily thought of as $$\sup\{\alpha: \forall \beta<\alpha(V_\beta\not\equiv V_\alpha)\},$$ and this definition should be clearly unproblematic (note that Tarski notwithstanding there is no problem in talking about truth relative to a set-sized structure like the $V_\alpha$s). Moreover, this definition avoids any reference to coding of theories by reals, which adds a lot of unnecessary length to the question.

As to how big $\beta_\mathcal{L}$ is, the key observation is that levels of the cumulative hierarchy are correct about "local" phenomena. For example, $\beta_\mathcal{L}$ is greater than the least measurable cardinal $\mu$ if the latter exists, since the measurability of $\mu$ is visible in $V_{\mu+2}$. To get past $\beta_\mathcal{L}$, you need to look at large cardinal properties which more significantly reach up the cumulative hierarchy - e.g. we trivially have that $\beta_\mathcal{L}$ is less than the least supercompact if the latter exists.