# How to show that the application of two elementary embeddings is an elementary embedding?

Let $$\mathcal L$$ be a language of first-order logic. Given two structures $$\mathfrak M$$, $$\mathfrak N$$ for $$\mathcal L$$ with domains $$M$$, $$N$$ respectively, an elementary embedding from $$\mathfrak M$$ to $$\mathfrak N$$ is a function $$j:M\to N$$ such that for any $$\mathcal L$$-formula $$\phi(x_0,\ldots,x_n)$$ with free variables $$x_0,\ldots,x_n$$ and any choice of $$x_0,\ldots,x_n$$ from $$M$$, $$\mathfrak M\vDash\phi(x_0,\ldots,x_n)$$ iff $$\mathfrak N\vDash\phi(j(x_0),\ldots,j(x_n))$$. An elementary embedding is nontrivial if it is not the identity. From here let $$\mathcal L$$ be the first-order language of set theory, i.e. first-order logic with two relation symbols $$\in$$ and $$=$$ (for brevity $$=$$ can be omitted and recovered from extensionality), and let $$M\vDash\phi$$ be short for $$(M,\in)\vDash\phi$$. Define the von Neumann hierarchy as usual, setting $$V_0=\varnothing$$, $$V_{\alpha+1}=\mathcal P(V_\alpha)$$, and for limit ordinal $$\alpha$$, $$V_\alpha=\bigcup_{\beta<\alpha}V_\beta$$.

There is some literature on elementary embeddings from $$V_\lambda$$ to $$V_\lambda$$ in the context of large cardinals. An important operation is application, where given two elementary embeddings $$j:V_\lambda\to V_\lambda$$ and $$k:V_\lambda\to V_\lambda$$, another function $$jk:V_\lambda\to V_\lambda$$ can be obtained, defined as $$\bigcup_{\alpha<\lambda}j(k\cap V_\alpha)$$ and called the application of $$j$$ to $$k$$. It is a common result (e.g. stated in Laver's "On the Algebra of Elementary Embeddings of a Rank into Itself", 1992, arXiv) that if $$j:V_\lambda\to V_\lambda$$ and $$k:V_\lambda\to V_\lambda$$ are elementary, $$jk$$ is elementary. Lemma 1 in the preprint is said to imply elementarity of $$jk$$, however Laver states that it can be "checked directly", and lemma 1 appears more complicated than a direct proof. How can elementarity of $$jk$$ be checked directly?

The best progress which I have so far is to consider the statement that $$V_\lambda\vDash\phi(x_0,\ldots,x_n)$$ for some $$\phi$$ and some $$x_0,\ldots,x_n\in V_\lambda$$. By elementarity of $$k$$ this is equivalent to $$V_\lambda\vDash\phi(k(x_0),\ldots,k(x_n))$$. Choose a limit ordinal $$\alpha<\kappa$$ such that $$k(x_0),\ldots,k(x_n)\in V_\alpha$$. As $$k$$ agrees with $$k\cap V_\alpha$$ on the images of the sets $$x_i$$ (for each $$0\leq i\leq n$$), this is equivalent to $$V_\lambda\vDash\phi((k\cap V_\alpha)(x_0),\ldots,(k\cap V_\alpha)(x_n))$$. Considering this as a formula with $$n+1$$ parameters $$k\cap V_\alpha$$, $$x_0$$, ... $$x_n$$, by applying the elementary embedding $$j$$ we obtain equivalence with $$V_\lambda\vDash\phi(j(k\cap V_\alpha)(j(x_0)), \ldots, j(k\cap V_\alpha)(j(x_n))$$. However to show $$jk$$ is elementary, the goal will be to show equivalence with $$V_\lambda\vDash\phi(j(k\cap V_\alpha)(x_0), \ldots, j(k\cap V_\alpha)(x_n))$$ (for some $$\alpha<\kappa$$, not necessarily the $$\alpha$$ chosen earlier), and I am not sure how to get there.

There is also a MO question "Dehornoy's proof that the application of two elementary embeddings is an elementary embedding", however it seems to use a different definition of application.

• See Lemma 1.6 of Dehornoy's handbook article. Apr 10 at 22:32
• @NoahSchweber Thank you! I will add the content as a community wiki answer.
– C7X
Apr 11 at 1:24

Noah Schweber gave a reference to lemma 1.6 in Dehornoy's article "Elementary Embeddings and Algebra" in volume 2 of The Handbook of Set Theory (2010, ed. Foreman, Kanamori), an open-access copy is available here. For convenience, here is the part of the proof showing that $$jk$$ [denoted $$j[k]$$ in the source] is elementary:

When $$\gamma$$ ranges over $$\lambda$$, the various mappings $$k\upharpoonright V_\gamma$$ are compatible. As $$j$$ is elementary, $$j(k\upharpoonright V_\gamma)$$ is a partial mapping defined on $$V_{j(\gamma)}$$, and the partial mappings $$j(k\upharpoonright V_\gamma)$$ and $$j(k\upharpoonright V_{\gamma'})$$ associated with different ordinals $$\gamma,\gamma'$$ agree on $$V_{j(\gamma)}\cap V_{j(\gamma')}$$. Hence $$j[k]$$ is a mapping of $$V_\lambda$$ into itself.

Let $$\Phi(\vec x)$$ be a first-order formula. For each $$\gamma$$ in $$\lambda$$, we have $$(\forall\vec x\in V_\gamma)(\Phi(\vec x)\iff\Phi((k\upharpoonright V_\gamma)(\vec x))),$$ hence, applying $$j$$, $$(\forall\vec x\in V_{j(\gamma)})(\Phi(\vec x)\iff\Phi(j(k\upharpoonright V_\gamma)(\vec x))),$$ so $$j[k]$$ is an elementary embedding of $$V_\lambda$$ into itself.

I am assuming that each of the two block formulas is preceded by an implicit "$$V_\lambda\vDash$$".