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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.

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

1 Answer 1

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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 "$V_\lambda\vDash$".

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