Paradox with elementary submodels of the constructible tower

[Amir Livne]

This is an argument I thought up after a class on combinatrical properties of the model $\textbf{L}$. Our course is about set theory, not logic, so this paradox desn't seem relevant in its context. Can you help me figure out where I got it wrong?

The constructible heirarchy of sets is a series $L_{\alpha}$ that is defined for all ordinal numbers $\alpha$. The important properties for my argument are:

1. $L_{\alpha}$ is transitive for every $\alpha$
2. If $\alpha < \beta$, then $L_{\alpha}\subset L_{\beta}$
3. The transitive collapse (aka Montowski collapse) of every elementary submodel $M \prec L_{\alpha}$ is $L_{\beta}$ for some $\beta$
4. $L_{\omega_{1}}$ satisfies "every set is countable" and $L_{\omega_{2}}$ does not
5. $L_{\alpha}$ is coutable iff $\alpha$ is countable

So, we take an countable elementary submodel (CESM) $M_{1} \prec L_{\omega_{1}}$, and look at its transitive collapse, $L_{\alpha_{1}}$ for some countable $\alpha_{1}$. We then take an CESM $M_{2} \prec L_{\omega_{2}}$ that contains $L_{\alpha_{2}}$, and collapse it to get $L_{\alpha_{2}}$ with countable $\alpha_{2}$. Then the same procedure yields a model $L_{\alpha_{3}} \supset L_{\alpha_{2}}$ that has an elementary embedding into $L_{\omega_{1}}$. We generate an infinite series, switching between modelling $L_{\omega_{1}}$ and $L_{\omega_{2}}$.

The limit $$L_{\alpha}=L_{\lim_{n<\omega}\alpha_{n}}=\bigcup_{n<\omega}L_{\alpha_{n}}$$ is then the union of both subseries $\{L_{\alpha_{n}}\}_{n=1,3,\ldots}$ and $\{L_{\alpha_{n}}\}_{n=2,4,\ldots}$. But a union of a series of elementary submodels is itself an elementary submodel, since it is a direct limit. In particular $L_{\alpha}$ should be elementary equivalent to both $L_{\omega_{1}}$ and $L_{\omega_{2}}$. This is impossible because of property (4), namely there is a statement true in one and not in another.

Where did I go wrong in my reasoning? All kinds of tips are appreciated...

 

[AKG]

"The union of elementary submodels is itself an elementary submodels" is only true when those submodels are elementary substructures of one another. This means more than just the fact that $L_{\alpha_n} \subset L_{\alpha_{n+2}}$ and $L_{\alpha_n} \equiv L_{\alpha_{n+2}}$, it requires that the inclusion map is the elementary embedding. This would be the case if the $L_{\alpha_n}$ were elementary substructures of their respective $L_{\omega_i}$, but they're generally not. You do get that they're elementarily equivalent subsets of their respective $L_{\omega_i}$, but the inclusion maps are not typically the elementary embeddings. The embeddings here are the inverses of the Mostowski collapses.

Nothing I said above proves that the inclusion maps aren't elementary embeddings, although nothing you said proves that they are. But here's why they're definitely not. Let's take for example $L_{\omega_2}$. For the same that GCH holds in L, we know that $L_{\omega_2}$ thinks there is exactly one cardinal after $\omega$, and that this cardinal is $\omega_1$. If $L_{\alpha} \equiv L_{\omega_2}$ with $\alpha$ countable, then this structure will also think there's a unique cardinal after $\omega$, but it won't think $\omega_1$ is it.

 

[Amir Livne]

I see what you mean...

So, it seems that for every ordinal $\alpha$, the set $\{\delta < \omega_{1} \mid L_{\delta} \prec L_{\alpha}\}$ is closed w.r.t taking limits. I thought about it some more and it's not hard to see this set is unbounded for $\alpha = \omega_{1}$, since for each $\beta < \omega_{1}$ you can find find some $L_{\beta'}$ with $\beta<\beta'<\omega_{1}$ that is closed w.r.t Skolem functions of $L_{\omega_{1}}$. This natually means that the set of δs can't be unbounded for $|\alpha|>\aleph_{1}$.

 

[AKG]

I see what you mean...

So, it seems that for every ordinal $\alpha$, the set $\{\delta < \omega_{1} \mid L_{\delta} \prec L_{\alpha}\}$ is closed w.r.t taking limits. I thought about it some more and it's not hard to see this set is unbounded for $\alpha = \omega_{1}$, since for each $\beta < \omega_{1}$ you can find find some $L_{\beta'}$ with $\beta<\beta'<\omega_{1}$ that is closed w.r.t Skolem functions of $L_{\omega_{1}}$.

Correct. Another simple argument is to do the following construction:

• $X_0 = Hull(\{\beta\},L_{\alpha})$
• $\beta_0 = \min\{\gamma : X_0 \subset L_\gamma\}$
• $X_{n+1} = Hull(L_{\beta_n},L_\alpha)$
• $\beta_{n+1} = \min\{\gamma : X_n \subset L_\gamma\}$
• $L_{\beta'} = \bigcup X_n$

Here $Hull(X,M)$ denotes some Skolem Hull of the set X in the structure M. Since we're working with L, there's a canonical choice for such hull.

This natually means that the set of δs can't be unbounded for $|\alpha|>\aleph_{1}$.

Indeed, the set of such $\delta$'s is empty!