Existence of a real number that cannot be encoded by the truth values in the “worldly-successive” levels of $V$

Question

In this question, the term “real” implies an infinite binary sequence (a binary sequence of length $\omega$).

Assume that the theory “ZFC + there is a proper class of worldly cardinals” is consistent and use this theory as the background set theory. Additionally, assume that $V \ne L$.

Let $f(\alpha)$ denote the $\alpha$-th worldy cardinal. Let $t(\phi, \alpha)$ denote the truth value (either $0$ or $1$) of a statement $\phi$ (a finite formula of arbitrary complexity) in $V_{f(\alpha)}$.

Assuming that $\beta$ is either $0$ or a limit ordinal, let $r(\phi, \beta)$ denote a real encoded by the following sequence of bits: $$r(\phi, \beta) = t(\phi, \beta), t(\phi, \beta+1), t(\phi, \beta+2), \ldots$$

Let $S$ denote the set of reals such that for any $x \in S$ there exists a statement $\phi$ and an ordinal $\beta$ such that $r(\phi, \beta) = x.$

Question: does there exist a real $y$ which is not an element of $S$? If no, why? If yes, is it possible to prove it and maybe describe a particular example of such real?

Answer 1

Let’s suppose $V = L$, which is relatively consistent with a proper class of worldly cardinals. We know that $L$ is definably well-ordered, so take the smallest bijection $f : 2^{\aleph_0} \to \{f : \omega \to 2\}$ according to this well-ordering.

Then consider the following statement:

There is a set of worldly cardinals. Let $\beta$ be the ordinality of the set of worldly cardinals. Then we can write $\beta$ uniquely as $\beta = 2^{\aleph_0} \cdot \gamma + \delta$ for some $\delta < 2^{\aleph_0}$. Furthermore, we have $\delta < \omega$ and $\gamma < 2^{\aleph_0}$. And the $\delta$th element of $f(\gamma)$ is $1$.

We denote the above statement as $\phi$. Note that $r(\phi, 2^{\aleph_0} \cdot \alpha) = f(\alpha)$. Therefore, all reals can be written as $f(\phi, \beta)$ for some $\beta$. $\square$

The above demonstrates that we cannot prove the existence of a real which is not of the form $f(\phi, \beta)$.

Now suppose we could show that all reals are written in the form $f(\phi, \beta)$. Then we could define a well-order on the reals. But I suspect that even with a proper class of worldly cardinals, we could not produce a definable well-ordering of the reals.

So it appears this question is independent of your axioms.