Date: Sat, 18 Oct 1997 18:27:44 -0500 (EST) Message-Id: In-Reply-To: <199710181844.LAA00641@unagi.cybernothing.org> Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Sat, 18 Oct 1997 11:05:28 -0700 To: John Cowan From: Edward Cherlin Subject: Re: ka/ni kama X-Mozilla-Status: 0011 Content-Length: 1901 X-From-Space-Date: Sat Oct 18 18:27:47 1997 X-From-Space-Address: drv.cbc.com!cherlin@cbgate.cbc.com At 7:28 AM -0700 10/17/97, John Cowan wrote: [snip] >There are many interpretations of numbers as sets: >in Cantor's interpretation (which is hardcoded into the >Loglan offshoot -gua!spi), *n* is the set of all >sets of cardinality *n*. Oh, dear. That makes *n* a class, and then we can't have sets of numbers. >In von Neumann's interpretation, >0 is the null set and *n* is the set whose members >are the integers smaller than *n*. Much better. Then we can extend the theory to infinite cardinals and ordinals in a consistent and constructive manner. >And there are others. In Peano's interpretation, any set with a Successor function which satisfies his 5 axioms is a model of the natural numbers, and we can use any object whatsoever as a number if we have the right relations with the other numbers. For example, we could count "partridge, turtle dove, French hen, calling bird, gold ring" rather than "1 2 3 4 5" as long as we provided a rule for continuing without limit. Peano *proved* that all models of the natural numbers are isomorphic, but his proof was wrong (using second-order logic methods not available in first-order theories), so now there are non-standard natural numbers, and of course that leads to non-standard calculus with actual infinitesimals (Abraham Robinson), surreal numbers (John Horton Conway, Donald Knuth), and games as extensions of numbers, with, of course, infinitesimal games (Conway, Berlekamp). Using this theory, Berlekamp has created go positions in which he, a mid-level amateur, can beat the top professionals with either color. Detailed references available on request. Hard-coding math into your ontology is always a mistake. [snip] >-- >John Cowan http://www.ccil.org/~cowan cowan@ccil.org > e'osai ko sarji la lojban co'o mi'e ed. .i e'osai la lojban pluka ko