Re: ka/ni kama

[Edward Cherlin <cherlin@drv.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
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