Nightmare Laver tables

The nightmare Laver tables are certain kinds of generalized Laver tables that appear spontaneously without being constructed from the classical Laver tables and which cannot be obtained from algebras of elementary embeddings. To motivate the notion of a nightmare Laver table, we begin with set theoretic result which is easy to prove but which has profound consequences.

Lemma: Suppose that $j\in\mathcal{E}_{\lambda}$. Then $(j*j)(\alpha)\leq j(\alpha)$ for each $\alpha<\lambda$.

Proof: Suppose that $\alpha<\lambda$. Then let $\beta$ be the least ordinal with $j(\beta)>\alpha$. Then

$$V_{\lambda}\models\forall x<\beta,j(x)\leq\alpha.$$

Therefore, by applying the elementary embedding $j$ to the above formula, we have

$$V_{\lambda}\models\forall x<j(\beta),(j*j)(x)\leq j(\alpha).$$

Therefore, since $\alpha<j(\beta)$, the above formula holds for $x=\alpha$. Therefore, $(j*j)(\alpha)\leq j(\alpha)$. QED

A nightmare Laver table is a generalized Laver table $M$ such that there exists $x\in M$ with and $\alpha\in crit[M]$ such that $(x*x)^{\sharp}(\alpha)>x^{\sharp}(\alpha)$. Equivalently, a nightmare Laver table is a generalized Laver table $M$ such that there are $x,y\in M$ with $(x*x)*y\in Li(M)$ but where $x*y\not\in Li(M)$. If $M$ is a nightmare Laver table and $\simeq$ is a congruence on $M$ such that if $x\simeq y$, then $crit(x)=crit(y)$, then by the above lemma $M/\simeq$ is not isomorphic to a subalgebra of some $\mathcal{E}_{\lambda}/\equiv^{\gamma}$.

The term “nightmare” was chosen because if one were to find a reduced nightmare Laver table as a quotient algebra of elementary embeddings, then the notion of a rank-into-rank cardinal would be inconsistent which would be bad for set-theory.

The following set $x$ is a nightmare Laver table over the alphabet 0,1 of cardinality 64. To see that $x$ is truly a nightmare Laver table, simply observe that $001\in Li(M)$ but $01\not\in Li(M)$. Take note also that there are many instances of $0$ in the strings in $x$ while there are only a few instances of $1$ in the strings in $x$.

{ 0, 1, 00, 10, 01, 000, 100, 010, 011, 1000, 0000, 0100, 0110, 00000, 00001, 01000, 01100, 000000, 000010, 011000, 010000, 0000000, 0000100, 0100000, 0100001, 00001000, 01000000, 01000010, 010000000, 010000100, 0100001000, 11, 001, 101, 0001, 1001, 0101, 0111, 10000, 10001, 01001, 01101, 000001, 000011, 010001, 011001, 0000001, 0000101, 0110000, 0110001, 00000000, 00000001, 00001001, 01000001, 01000011, 000010000, 000010001, 010000001,
010000101, 0100000000, 0100000001, 0100001001, 01000010000, 01000010001 }