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Let $V$ denote the Von Neumann universe and $L$ denote the constructible universe. Let $(X_n)_{n\in\mathbb{N}}$ be an i.i.d. sequence of random variables such that $P(X_1=0) = P(X_1 = 1) = 1/2$, and construct a set $y$ as follows. If $X_k=0$, then include $k$ in $y$, otherwise, do not put $k$ in $y$. Since $y\subseteq\omega$, it is clear that $y\in V_{\omega+1}$.

I know it is true that, almost surely, $y\not\in L_{\omega+1}$ (cf. Discovering Modern Set Theory, Chapter 7 Exercise 20). However, I am not sure how to prove this claim. Does anyone have any suggestions or references?

Charles Swaney
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As Asaf Karagila observes, $L_{\omega+1}$ is countable; nullness is a consequence of that. In fact, every level of $L$ up to $L_{\omega_1}$ is countable, and so - even if $V=L$ - for each fixed countable ordinal $\alpha$ a randomly-generated real $y$ is almost-surely not in $L_\alpha$. (This "local countability" incidentally is the easy half of proving that $L$ satisfies the continuum hypothesis - the hard half is provided by the condensation lemma.)

One important aspect of this is a notation mismatch: even if $V=L$, the set $L_{\omega+1}$ is vastly smaller than $(V_{\omega+1})^L$. To be fair, this isn't really a mismatch - what's really going on is that the $L$-hierarchy provably grows much more "narrowly" than the $V$-hierarchy. Of course every so often the two will catch up to each other since $\alpha\mapsto L_\alpha$ and $\alpha\mapsto (V_\alpha)^L$ are each continuous, but this takes a while (in particular, the smallest infinite ordinal where they coincide is much bigger than $\omega_1$). See here for more detail.

Noah Schweber
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