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A Laver table is an infinite family of magmas that may give rise to a large number function.[1] They were first defined by Richard Laver in 1992. For $$n \geq 0$$, a size-n Laver table is a binary operator $$\star$$ over $$\mathbb{Z}_{2^n}$$, with the following properties:

$a \star 1 = a + 1 \pmod{2^n}$ $a \star (b \star c) = (a \star b) \star (a \star c)$

The period of the function $$a \mapsto 1 \star a$$ is a function of $$n$$, which we will denote as $$p(n)$$. The first few values of $$p(n)$$ are $$1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, \ldots$$ (OEIS A098820), a slow-growing function. $$p$$ is provably divergent in the system ZFC + "there exists a rank-into-rank cardinal." Unfortunately, the latter axiom is so strong that the consistency of the system is seriously doubted. Since the divergence of $$p$$ has not been proven otherwise, it remains an unsolved problem.

Let $$q(n)$$ be minimal so that $$p(q(n)) \geq 2^n$$, the "pseudoinverse" of $$p$$. $$q$$ is a fast-growing function that is total iff $$p$$ is divergent. The first few values of $$q$$ are $$0, 2, 3, 5, 9$$. The existence of $$q(n)$$ for $$n \geq 5$$ has not even been confirmed, but under the assumption of the previously mentioned axiom, Randall Dougherty has shown that $$q(n + 1) > f_{\omega+1}(\lfloor \log_3 n \rfloor - 1)$$ in a slightly modified version of the fast-growing hierarchy,[2] and $$q(5) > f_9(f_8(f_8(254)))$$.[3] Dougherty has expressed pessimism about the complexity of proving better lower bounds, and no upper bounds are known as of yet.

Explanation Edit

For $$\lambda \in \text{Lim}$$, let $$\mathcal{E}_\lambda$$ be the set of elementary embeddings $$V_\lambda \mapsto V_\lambda$$ (explained on the rank-into-rank cardinal page). For $$j,k \in \mathcal{E}_\lambda$$, we define the operator $$j\cdot k$$ (or $$jk$$) as follows:

$j \cdot k = \bigcup_{\alpha < \lambda} j(k \cap V_\alpha)$

That is, we "apply $$j$$ to $$k$$ approaching $$V_\lambda$$." This operator has $$j(kl) = (jk)(jl)$$, a property known as left-distributivity. [incomplete, please expand]

Examples Edit

A size-2 Laver table is shown below:

1 2 3 4
1 2 4 2 4
2 3 4 3 4
3 4 4 4 4
4 1 2 3 4

The entries at the first row repeat with a period of 2, and therefore $$p(2) = 2$$.

A size-3 Laver table is shown below:

1 2 3 4 5 6 7 8
1 2 4 6 8 2 4 6 8
2 3 4 7 8 3 4 7 8
3 4 8 4 8 4 8 4 8
4 5 6 7 8 5 6 7 8
5 6 8 6 8 6 8 6 8
6 7 8 7 8 7 8 7 8
7 8 8 8 8 8 8 8 8
8 1 2 3 4 5 6 7 8

The entries at the first row repeat with a period of 4, and therefore $$p(3) = 4$$.

Sources Edit

1. Laver, Richard. On the Algebra of Elementary Embeddings of a Rank into Inself. Retrieved 2014-08-23.
2. With $$f_{\alpha + 1}(n) = f_\alpha^{n + 1}(1)$$
3. Dougherty, Randall. Critical points in an algebra of elementary embeddings. Retrieved 2014-08-23.