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We consider algebras with one binary operation~$\cdot $ and one generator, satisfying the left distributive law $a\cdot (b\cdot c)=(a\cdot b)\cdot (a\cdot c)$; such algebras have been shown to have surprising connections with set-theoretic large cardinals and with braid groups. One can construct a sequence of finite left-distributive algebras~$A_{n}$, and then take a limit to get an infinite left-distributive algebra~$A_{\infty }$ on one generator. Results of Laver and Steel assuming a strong large cardinal axiom imply that $A_{\infty }$~is free; it is open whether the freeness of~$A_{\infty }$ can be proved without the large cardinal assumption, or even in Peano arithmetic. The main result of this paper is the equivalence of this problem with the existence of a certain left-distributive algebra of increasing functions on natural numbers, called an {\em embedding algebra}, which emulates some properties of functions on the large cardinal. Using this and results of the first author, we conclude that the freeness of~$A_{\infty }$ is unprovable in primitive recursive arithmetic.

Copyright 1997 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 03 (1997), pp. 28-37
• Publisher Identifier: S 1079-6762(97)00020-6
• 1991 Mathematics Subject Classification. Primary 20N02; Secondary 03E55, 08B20
• Key words and phrases. Left-distributive algebras, elementary embeddings, critical points, large cardinals, primitive recursive arithmetic
• Received by the editors December 16, 1996
• Posted on April 9, 1997
• Communicated by Alexander Kechris
• Comments (When Available)

Randall Dougherty

Department of Mathematics, Ohio State University, Columbus, OH 43210

E-mail address: rld@math.ohio-state.edu

Thomas Jech

Pennsylvania State University, 215 McAllister Building, University Park, PA 16802

E-mail address: jech@math.psu.edu

The first author was supported by NSF grant number DMS-9158092 and by a grant from the Sloan Foundation.
The second author was supported by NSF grant number DMS-9401275.