About: Laver table

Description
Metadata
Settings
- owl:sameAs
- Inference Rule:

About: Laver table Sponge Permalink

An Entity of Type : owl:Thing, within Data Space : 134.155.108.49:8890 associated with source dataset(s)

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

Attributes	Values
rdfs:label	Laver table
rdfs:comment	\[a \star 1 = a + 1 \pmod{2^n}\] \[a \star (b \star c) = (a \star b) \star (a \star c)\] 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.
dcterms:subject	Functions Numbers Unsolved problems
dbkwik:googology/p...iPageUsesTemplate	dbkwik:resource/7cBS1V3uWaVowycFZIqhOg== dbkwik:resource/cig5f-r4Y52f4lCgil0iCg==
abstract	\[a \star 1 = a + 1 \pmod{2^n}\] \[a \star (b \star c) = (a \star b) \star (a \star c)\] 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 \(q(n + 1) > f_{\omega+1}(\lfloor \log_3 n floor - 1)\) in a slightly modified version of the fast-growing hierarchy, and \(q(5) > f_9(f_8(f_8(254)))\). Dougherty has expressed pessimism about the complexity of proving better lower bounds, and no upper bounds are known as of yet.

OpenLink Virtuoso version 07.20.3217, on Linux (x86_64-pc-linux-gnu), Standard Edition
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2012 OpenLink Software