[lojban-beginners] Re: logical proofs?

[Andrew Archibald <archibal@math.mcgill.ca>]

On Sun, Aug 22, 2004 at 09:19:58PM -0700, Robin Lee Powell wrote:
> On Mon, Aug 23, 2004 at 03:38:30PM +1200, mikevdg@gulik.co.nz wrote:
> > Is it possible to do logical proofs in lojban? 
> 
> Absolutely.  Give me a short one, and I'll translate it (*much* too
> tired to do this now).

Proof that there are infinitely many prime numbers:

Suppose that we have a finite list of prime numbers.  Then construct
the number that is one greater than the product of all the prime
numbers in the list.  This number is, by a previous result (you could
prove it if you wanted), guaranteed to be divisible by some prime
number - perhaps itself.  But the number is not divisible by any of
the numbers in our finite list of primes, so there must be at least
one prime not on the list (although it may not be the new number).  So
no finite list can contain all the prime numbers. 

If you'd like this a little more self-contained (you'll still need to
depend on some properties of multiplication, addition, and ordering of
the natural numbers) you can prepend the definition of a prime number:
a natural number (i.e., positive integer) such that it can be written
as a product of natural numbers in exactly one way, as one times
itself.  It then follows by induction that every number is divisible
by a prime number. 



It might be interesting to write this as a proof by contradiction.
Such proofs are always potentially confusing, with their temporary
assumption of some "fact" which turns out to be impossible. 


Andrew Archibald