The large cardinals above hugeness are a very powerful tool for proving results about finite self-distributive algebras, but mathematicians so far have neglected to exploit the tremendous power of these very large cardinals to prove results about finite objects. Initially around the late 80’s and early 90’s, Laver and other mathematicians have established some very good classical results about the Laver tables. On the other hand, from the later 1990’s to the 2010’s, no mathematician has published any original research that directly relates to what I call Laver-like LD-systems. Furthermore, before my work on the algebras of elementary embeddings, nobody has investigated what will happen in the algebras of elementary embeddings generated by more than one element.

There are two results about the classical Laver tables which presumably require large cardinals to establish. One of these results states that the inverse limit of the classical Laver tables contains free left-distributive algebras on one generator while the other result states that in the classical Laver table $A_{n}$, we have $2*_{n}x=2^{n}\Rightarrow 1*_{n}x=2^{n}$. All of the other results about the classical Laver tables are known to hold in ZFC.

In this post, we shall use large cardinals and forcing to prove the existence of certain classes of finite self-distributive algebras with a compatible linear ordering. The results contained in this note shall be included in my (hopefully soon to be on Arxiv) 100+ page paper *Generalizations of Laver tables*. In this post, I have made no attempt to optimize the large cardinal hypotheses.

For background information, see this post or see Chapter 11 in the Handbook of Set Theory.

We shall let $\mathcal{E}_{\alpha}$ denote the set of all elementary embeddings $j:V_{\alpha}\rightarrow V_{\alpha}.$

By this answer, I have outlined a proof that the algebra $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is locally finite. We therefore have established a deep connection between the top of the large cardinal hierarchy and finite algebras.

In this note, we shall use two important ideas to construct finite self-distributive algebras. The main idea is to generalize the square root lemma for elementary embeddings so that one obtains elementary embeddings with the desired properties.

$\textbf{Theorem: (Square Root Lemma)}$ Let $j\in\mathcal{E}_{\lambda+1}$. Then there is some $k\in\mathcal{E}_{\lambda}$ where $k*k=j|_{V_{\lambda}}$.

$\mathbf{Proof}:$ By elementarity

$$V_{\lambda+1}\models\exists k\in\mathcal{E}_{\lambda}:k*k=j|_{V_{\lambda}}$$

if and only if

$$V_{\lambda+1}\models\exists k\in\mathcal{E}_{\lambda}:k*k=j(j|_{V_{\lambda}})$$

which is true. Therefore, there is some $k\in\mathcal{E}_{\lambda}$ with $k*k=j|_{V_{\lambda}}$. $\mathbf{QED}$

The other idea is to work in a model such that there is a cardinal $\lambda$ where there are plenty of rank-into-rank embeddings from $V_{\lambda}$ to $V_{\lambda}$ but where $V_{\lambda}\models\text{V=HOD}$. If $V_{\lambda}\models\text{V=HOD}$, then $V_{\lambda}$ has a definable linear ordering which induces a desirable linear ordering on rank-into-rank embeddings and hence linear orderings on finite algebras. The following result can be found in this paper.

$\mathbf{Theorem}$ Suppose that there exists a non-trivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$. Then in some forcing extension $V[G]$ there is some elementary embedding $k:V[G]_{\lambda+1}\rightarrow V[G]_{\lambda+1}$ where

$V[G]_{\lambda}\models\text{V=HOD}$.

Therefore it is consistent relative to large cardinals that there exists a non-trivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ such that $V_{\lambda}\models\text{V=HOD}$.

Now suppose that $V_{\lambda}\models\text{V=HOD}$. Then there exists a linear ordering $\ll$ of $V_{\lambda}$ which is definable in $V_{\lambda}$. If $j\in\mathcal{E}_{\lambda}$ and $\gamma$ is a limit ordinal with $\gamma<\lambda$, then define $j\upharpoonright_{\gamma}:V_{\gamma}\rightarrow V_{\gamma+1}$ by $j\upharpoonright_{\gamma}(x)=x\cap V_{\gamma}$ for each $x\in V_{\gamma}.$ Take note that $j\upharpoonright_{\gamma}=k\upharpoonright_{\gamma}$ if and only if $j\equiv^{\gamma}k$. Define a linear ordering $\trianglelefteq$ on $\mathcal{E}_{\lambda}$ where $j\trianglelefteq k$ if and only if $j=k$ or there is a limit ordinal $\alpha$ where $j\upharpoonright_{\alpha}\ll k\upharpoonright_{\alpha}$ but where $j\upharpoonright_{\beta}=k\upharpoonright_{\beta}$ whenever $\beta<\alpha$. Define a linear ordering $\trianglelefteq$ on $\{j\upharpoonright_{\gamma}\mid j\in\mathcal{E}_{\lambda}\}$ by letting $j\upharpoonright_{\gamma}\triangleleft k\upharpoonright_{\gamma}$ if and only if there is some limit ordinal $\beta\leq\gamma$ where $j\upharpoonright_{\beta}\ll k\upharpoonright_{\beta}$ but where $j\upharpoonright_{\alpha}=k\upharpoonright_{\alpha}$ whenever $\alpha$ is a limit ordinal with $\alpha<\beta$. By elementarity, the linear ordering $\trianglelefteq$ satisfies the following compatibility property: if $k\upharpoonright_{\gamma}\trianglelefteq l\upharpoonright_{\gamma}$, then $(j*k)\upharpoonright_{\gamma}\trianglelefteq(j*l)\upharpoonright_{\gamma}$. We say that a linear ordering $\leq$ on a Laver-like LD-system $(X,*)$ is a compatible linear ordering if $y\leq z\Rightarrow x*y\leq x*z$. If $V_{\lambda}\models\text{V=HOD}$, then $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ has a compatible linear ordering defined by $[j]_{\gamma}\leq[k]_{\gamma}$ if and only if $j\upharpoonright_{V_{\gamma}}\trianglelefteq k\upharpoonright_{V_{\gamma}}$.

Using generalized Laver tables, we know that the set $\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$ has order-type $\omega$. Let $\text{crit}_{r}(j_{1},…,j_{n})$ be the $r$-th element of the set $$\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$$ ($\text{crit}_{0}(j_{1},…,j_{n})$ is the least element of $\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$). Let $T:\bigcup_{n\in\omega}\mathcal{E}_{\lambda}^{n}\rightarrow V_{\omega\cdot 2}$ be a mapping definable in $(V_{\lambda+1},\in)$ where $T(j_{1},…,j_{m})=T(k_{1},…,k_{n})$ if and only if $m=n$ and if $\gamma=\text{crit}_{r} (j_{1},…,j_{m})$ and $\delta=\text{crit}_{r}(k_{1},…,k_{n})$, then there is some isomorphism $\phi:\langle j_{1},…,j_{m}\rangle/\equiv^{\gamma}\rightarrow\langle k_{1},…,k_{n}\rangle/\equiv^{\delta}$ where $\phi([j_{i}]_{\gamma})=[k_{i}]_{\delta}$. We remark that if $T(j_{1},…,j_{m})=T(k_{1},…,k_{n})$, then the subspaces $\overline{\langle j_{1},…,j_{m}\rangle}$ and $\overline{\langle k_{1},…,k_{n}\rangle}$ of $\mathcal{E}_{\lambda}$ are homeomorphic by an isomorphism of algebras preserving $*,\circ$ ($\mathcal{E}_{\lambda}$ can be given a complete metric that induces a canonical uniformity on $\mathcal{E}_{\lambda}$).

The following technical result is a generalization of the Square-Root Lemma, and a simplified special case of the following results can be found in this answer that I gave.

- $\ell_{1},…,\ell_{p},j_{1},…,j_{m}\in\mathcal{E}_{\lambda}$ and $(k_{r,s})_{1\leq r\leq n,1\leq s\leq p}\in(\mathcal{E}_{\lambda})^{n\cdot p}$.
- $\ell_{1},…,\ell_{p}$ are I1 embeddings.
- $T(\ell_{i}*j_{1},…,\ell_{i}*j_{m},k_{1,i},…,k_{n,i})=x_{i}$ whenever $1\leq i\leq p$.
- $v$ is a natural number.
- there is some $\mu<\lambda$ where $\mu=\text{crit}_{v}(k_{1,1},...,k_{n,1})=\ldots=\text{crit}_{v}(k_{1,p},...,k_{n,p})$.
- $k_{r,1}\equiv^{\mu}…\equiv^{\mu}k_{r,p}$ for $1\leq r\leq n$.
- $\ell_{1}\equiv^{\mu+\omega}…\equiv^{\mu+\omega}\ell_{p}$.

Then there are $(w_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ in $\mathcal{E}_{\lambda}$ where

- $T(j_{1},…,j_{m},w_{1,i},…,w_{n,i})=x_{i}$ for $1\leq i\leq p$,
- there is some $\alpha<\lambda$ where $\text{crit}_{v}(w_{1,i},...,w_{n,i})=\alpha$ for $1\leq i\leq p$, and
- $w_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}w_{r,p}$ for $1\leq r\leq n$.

$\mathbf{Proof:}$ For $1\leq i\leq p$, let $A_{i}$

$$=\{(w_{1}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})},…,w_{n}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})}):

T(j_{1},…,j_{m},w_{1},…,w_{n})=x_{i}\}.$$

Then $\ell_{i}(A_{i})$

$$=\{(w_{1}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})},…,w_{n}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})}):

T(\ell_{i}*j_{1},…,\ell_{i}*j_{m},w_{1},…,w_{n})=x_{i}\}.$$

Therefore,

$$(k_{1,i}\upharpoonright_{\mu},…,k_{n,i}\upharpoonright_{\mu})\in\ell_{i}(A_{i})$$ for $1\leq i\leq p$. Since

$k_{r,1}\upharpoonright_{\mu}=…=k_{r,p}\upharpoonright_{\mu}$, we have

$$(k_{1,1}\upharpoonright_{\mu},…,k_{n,1}\upharpoonright_{\mu})=…=(k_{1,p}\upharpoonright_{\mu},…,k_{n,p}\upharpoonright_{\mu}).$$

Therefore, let $$(\mathfrak{k}_{1},…,\mathfrak{k}_{n})=(k_{1,1}\upharpoonright_{\mu},…,k_{n,1}\upharpoonright_{\mu}).$$

Then

$$(\mathfrak{k}_{1},…,\mathfrak{k}_{n})\in\ell_{1}(A_{1})\cap…\ell_{p}(A_{p})\cap V_{\mu+\omega}$$

$$=\ell_{1}(A_{1})\cap…\cap\ell_{1}(A_{p})\cap V_{\mu+\omega}$$

$$\subseteq\ell_{1}(A_{1}\cap…\cap A_{p}).$$

Therefore, $A_{1}\cap…\cap A_{p}\neq\emptyset.$

Let $(\mathfrak{w}_{1},…,\mathfrak{w}_{n})\in A_{1}\cap…\cap A_{p}$. Then there are $(w_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ in $\mathcal{E}_{\lambda}$ where

$$(\mathfrak{w}_{1},…,\mathfrak{w}_{n})=(w_{1,i}\upharpoonright_{\text{crit}_{v}(w_{1,i},…,w_{n,i})},…,w_{n,i}\upharpoonright_{\text{crit}_{v}(w_{1,i},…,w_{n,i})})$$

and

$$T(j_{1},…,j_{m},w_{1,i},…,w_{n,i})=x_{i}$$

for $1\leq i\leq p.$ Therefore, there is some $\alpha<\lambda$ with $\text{crit}_{v}(w_{1,s},...,w_{n,s})=\alpha$ for $1\leq s\leq p$ and where $w_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}w_{r,p}$ for $1\leq r\leq n$. $\mathbf{QED}$

$\mathbf{Remark:}$ The above theorem can be generalized further by considering the classes of rank-into-rank embeddings

described in this paper.

- $U_{1},…,U_{p},V_{1},…,V_{m}$ and $(W_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ are unary terms in the language with function symbols $*,\circ$,
- $L$ is an $np+1$-ary term in the language with function symbols $*,\circ$,
- $v$ is a natural number,
- There is some classical Laver table $A_{N}$ where in $A_{N}$, we have

\[\mu=\text{crit}_{v}(W_{1,1}(1),…,W_{n,1}(1))=…=\text{crit}_{v}(W_{1,p}(1),…,W_{n,p}(1))<\text{crit}(2^{n}).\] - For all $N$, we have $W_{r,1}\equiv^{\mu}\ldots\equiv^{\mu}W_{r,p}(1)$ for $1\leq r\leq n$,
- $U_{1}(1)\equiv^{\mu^{+}}\ldots\equiv^{\mu^{+}}U_{p}(1)$.

If $Y$ is a finite reduced Laver-like LD-system, then let $\approx$ be the relation on $Y^{<\omega}$ where $(x_{1},...,x_{m})\approx(y_{1},...,y_{n})$ if and only if $m=n$ and whenever $\langle x_{1},...,x_{m}\rangle$ and $\langle y_{1},...,y_{n}\rangle$ both have more than $v+1$ critical points, then there is an isomorphism
\[\iota:\langle x_{1},...,x_{m}\rangle/\equiv^{\text{crit}_{v}(x_{1},...,x_{m})}\rightarrow
\langle y_{1},...,y_{n}\rangle/\equiv^{\text{crit}_{v}(y_{1},...,y_{n})}\]
where $\iota([x_{i}])=[y_{i}]$ for $1\leq i\leq n$.

Then there is some finite reduced Laver-like LD-system $X$ along with

\[x,(y_{r,s})_{1\leq r\leq n,1\leq s\leq p}\in X\]

such that

- $X$ has a compatible linear ordering,
- \[(U_{s}(x)*V_{1}(x),…,U_{s}(x)*V_{m}(x),W_{1,s}(x),…,W_{n,s}(x))\]

\[\approx(V_{1}(x),…,V_{m}(x),y_{1,s},…,y_{n,s}),\] - there is some critical point $\alpha$ where $\text{crit}_{v}(y_{1,s},…,y_{n,s})=\alpha$ for all $s$, and
- $y_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}y_{r,p}$.
- $L(x,(y_{r,s})_{1\leq r\leq n,1\leq s\leq p})\neq 1$.

ind

I challenge the readers of this post to remove the large cardinal hypotheses from the above theorem (one may still use the freeness of subalgebras $\varprojlim_{n}A_{n}$ and the fact that $2*_{n}x=2^{n}\Rightarrow 1*_{n}x=2^{n}$ though).

So it turns out that by taking stronger large cardinal axioms, one can induce a linear ordering on the algebras of elementary embeddings without having to resort to working in forcing extensions. We say that a cardinal $\delta$ is an I1-tower cardinal if for all $A\subseteq V_{\delta}$ there is some $\kappa<\delta$ such that whenever $\gamma<\delta$ there is some cardinal $\lambda<\delta$ and non-trivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ such that $\text{crit}(j)=\kappa$ and where $j(\kappa)>\delta$ and where $j(A)=A$. If $A$ is a good enough linear ordering on $V_{\delta}$, then $A\cap V_{\lambda}$ induces a compatible linear ordering the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ such that $j(A\cap V_{\gamma})=A\cap V_{j(\gamma)}$ for all $\gamma<\lambda$. It is unclear where the I1-tower cardinals stand on the large cardinal hierarchy or whether they are even consistent.

It turns out that we can directly show that if $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ is a non-trivial elementary embedding, then there is a linear ordering $B$ of $V_{\lambda}$ where $j(B)=B$. In fact, if $j:V_{\lambda}\rightarrow V_{\lambda}$ is a non-trivial elementary embedding, $\mathrm{crit}(j)=\kappa$, and $A$ is a linear ordering of $V_{\lambda}$, then if we let $B=\bigcup_{n}j^{n}(A)$, then $B$ is a linear ordering of $V_{\lambda}$ and $j(B\cap V_{\gamma})=B\cap V_{j(\gamma)}$ whenever $\gamma<\lambda$. In particular, if $j$ extends to an elementary embedding $j^{+}:V_{\lambda+1}\rightarrow V_{\lambda+1}$, then $j^{+}(B)=B$. One can therefore prove the results about finite permutative LD-systems by working with the linear ordering that comes from $B$ instead of the linear ordering that comes from the fact that $V_{\lambda}[G]\models V=HOD$ in some forcing extension. One thing to be cautious of when one announces results before publication is that perhaps the proofs are not optimal and that one can get away with a simpler construction.

Philosophy and research project proposals

In the above results, we have worked in a model $V$ where there are non-trivial maps $j:V_{\lambda}\rightarrow V_{\lambda}$ and where $V_{\lambda}\models\text{V=HOD}$ in order to obtain compatible linear orderings on finite algebras. However, if we work in different forcing extensions with rank-into-rank embeddings instead, then I predict that one may obtain from large cardinals different results about finite algebras.

I predict that in the near future, mathematicians will produce many results about finite or countable self-distributive algebras using forcing and large cardinals where the large cardinal hypotheses cannot be removed. I also predict that rank-into-rank cardinals will soon prove results about structures that at first glance have little to do with self-distributivity.

I must admit that I am not 100 percent convinced of the consistency of the large cardinals around the rank-into-rank level. My doubt is mainly due to the existence of finite reduced Laver-like LD-systems which cannot be subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$. However, if no inconsistency is found, then the results about finite or countable structures that arise from very large cardinals would convince me not only of the consistency of very large cardinals but also the existence of these very large cardinals. Therefore, people should investigate the finite algebras which arise from very large cardinals in order to quell all doubts about the consistency or the existence of these very large cardinals.

Since it is much more likely that the Reinhardt cardinals are inconsistent than say the I1 cardinals are inconsistent, I also propose that we attempt to use the algebras of elementary embeddings to show that Reinhardt cardinals are inconsistent. I have not seen anyone investigate the self-distributive algebras of elementary embeddings at the Reinhardt level. However, I think that investigating the self-distributive algebras of elementary embeddings would be our best hope in proving that the Reinhardt cardinals are inconsistent.