From: Stephen Turner (sret1_at_ntlworld.com)
Date: Tue Feb 05 2002 - 03:51:10 PST
176:7 INVALID -2.0
James Willson 2002-02-05 08:51:33
>>>>>
Perhaps the most famous fantasy maths problem is the proof that 1 = 0.
It was commonly believed for centuries, but until the groundbreaking work
of Set, all "proofs" that 1 = 0 were unsound.
Set, of course, is the discoverer of k, a fantasy number.
k is defined as 0^0. Set used this to construct the first
sound proof that 1 = 0.
k = k (all fantasy numbers equal themselves)
0^0 = 0^0 (definition of k)
1 = 0^0 (for all n, n^0 = 1)
1 = 0 (for all n, 0^n = 0) QED
This has some interesting implications.
Since 1 = k = 0, both 1 and 0 must be fantasy numbers.
Set extended this result to show that all numbers are fantasy numbers.
(This is called Set's Fantastic Completeness Theorem)
We'll use this to prove the following trivially true:
fc + rc = (f+r)c unless c is a fantasy number
Since all numbers c are fantasy numbers, the proposition reduces to
fc + rc = (f+r)c unless true
and true -> b is true for arbitrary b. QED
Set made many important contributions to fantasy maths.
All future rules will describe some such contribution.
<<<<<
Judgement: Invalid. If true => b is true for all b then all statements are
true. But there are false statements, e.g., multiplication is associative.
Style: I was glad to find this blunder. The idea that 1 does in fact equal
0 is superficially amusing, but it implies that all numbers -- or at least
all of our numbers -- are equal, which would cause the whole number system
to collapse. (Of course the bigger leap in your proof is not the existence
of 0^0, but the axioms that n^0 = 1 and 0^n = 0 for all n.)
There are some positive aspects to this rule. First of all, it exploited
the fact that fantasy numbers haven't yet been defined. (This was also the
reason that I knocked off 0.5 points at 176:6 -- this loophole makes those
theorems much easier to prove or disprove.) Secondly it introduces us to a
mathematician. And finally the notion that all numbers are fantasy numbers
is amusing, and doesn't seem to be problematic. Nevertheless, I think that
this would have caused a lot of trouble, and I'm going to give it -2.0.
--------------------------------------------------------------------------
--
Stephen Turner, Cambridge, UK http://homepage.ntlworld.com/adelie/stephen/
"This is Henman's 8th Wimbledon, and he's only lost 7 matches." BBC, 2/Jul/01
--
Rule Date: 2002-02-05 11:51:29 GMT
This archive was generated by hypermail 2.1.5 : Thu Nov 24 2011 - 10:48 PST