[lojban-beginners] Re: logical proofs?

[Robin Lee Powell <rlpowell@digitalkingdom.org>]

On Mon, Aug 23, 2004 at 03:18:49AM -0400, Andrew Archibald wrote:
> 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:

ni'o jarco lo du'u ci'i mulna'usle cei broda cu zasti

(Had to figure out "prime number"; started with

http://www.lojban.org/tiki/tiki-index.php?page=Number+theory&highlight=prime

(which as you can see I made several changes to) and

http://www.lojban.org/tiki/tiki-index.php?page=Abstract+Algebra&highlight=integer)

> Suppose that we have a finite list of prime numbers.  

.i sruma lo du'u mi'o ponse lo liste be me'i ci'i broda

(This relies on a certain amount of listener generosity for "me'i ci'i";
"ci'i nai" is another possibility.)

> Then construct the number that is one greater than the product of all
> the prime numbers in the list.  

.i ba bo finti ko'a goi lo namcu poi ve jmina li pa lo simxu pilji be ro
le se liste

> This number is, by a previous result (you could prove it if you
> wanted), guaranteed to be divisible by some prime number - perhaps
> itself.  

.i ko'a to se nibli lo pu nu jarco toi cu kakne lo nu fendi fi pa
broda goi ko'e to cumki lo nu du ko'a toi

> 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).  

.i ku'i ko'e na se fendi fi lo se liste broda .i se ni'i bo su'o broda
poi na se list cu zasti to cumki lo nu na du ko'a toi

> So no finite list can contain all the prime numbers. 

.i se ni'i bo lo liste be me'i ci'i broda cu na kakne lo nu vasru ro
broda

.i fe'o

(fe'o seemed about right for the arrogance of QED :-)

> 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.  

da poi mulna'usle cei broda cu ma'u zei mulna'u gi'e mulna'u pilji po'o
pa da

> It then follows by induction that every number is divisible
> by a prime number. 

No, thanks.

> 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. 

Feel free to present a proof by contradiction for me to translate; this
or another.

I think a term for "list" in the mathematical / CS sense might be nice.

-Robin, who is enjoying the heck out of this.

-- 
http://www.digitalkingdom.org/~rlpowell/ *** http://www.lojban.org/
Reason #237 To Learn Lojban: "Homonyms: Their Grate!"