From: "Clive Tooth" <clive@pisquaredoversix.force9.co.uk>
Subject: Re: Goedel question
Date: Wed, 13 Oct 1999 11:39:25 +0100
Newsgroups: sci.math
Keywords: multiplication and successor define addition

Bill Taylor wrote in message <7tmlse$agp$5@cantuc.canterbury.ac.nz>...

>fc3a501@AMRISC01.math.uni-hamburg.de (Hauke Reddmann) writes:
>
>|> In short, arithmetic with + and * is complex enough
>|> to allow the Goedel effect, but without + or * it isn't.
>
>Arithmetic with * but without + is a bit artificial seeming, but I know
>it's been shown possible - can anyone post the details please?
>
>IIRC, you still have to have a predicate for successor, as well as
>one for multiplication.  You can hardly expect to do without it, surely?

Nope. You cannot have successor. If you do, then addition becomes definable:

i+j=k  iff  (i'*k'')'*(j'*k'')'=((i'*j')'*(k''*k''))'

~ ~ ~ ~

i+j=k  iff  (i*k)'*(j*k)'=((i*j)'*(k*k))' almost works as a definition, but
not quite.

--
Clive Tooth
http://www.pisquaredoversix.force9.co.uk/
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