From: timmaly@is.dal.ca (Snowmit)
Newsgroups: alt.philosophy.debate,sci.math,sci.logic
Subject: FUN WITH PARADOX
Date: 11 Jun 1997 17:11:24 GMT
Summary: Continuing the paradox theme
Keywords: Godel, Paradox

Hello all, I just finished reading the very long (and interesting) thread 
about paradox, so I thought I might share some paradoxes I've run into
(I'm trying to build a collection you see)
 
	First though, some notes about paradoxes:

	For the record, my understanding of aparadox is that it is
statement or a set of statements which are phrased in a such a way that we
are unable to come to a certain conclusiona about whether it is true or
false.  This a working definition and I invite disagreements.

	The paradoxes we've been playing with so far have mostly been of
the "Liar" type (ie variations on "This Sentence is False")  The jey to
this paradow seems to be in that fact that it is self-referential and in
fact, many attempts to eliminate the existence of paradox have dealt with
trying to eliminate self-reference.
	I really liked the idea of the hyper game, it was one I'd never seen
before.

	Here are a bunch more self-referential paradoxes:

(The title of a book by Robert Martin)
	There are two errors in the the title of this book.

Grelling's paradox. 
	If one wished, one could divide all of the English adjectives into
two categories: autological and heterological. An autological adjective is
one which is self-descriptive; "pentasyllabic", "awkwardfulness", "short".
A heterological adjective is not self-descriptive; "edible", "hungry",
"monosyllabic", "long". Now, the question we must ask ourselves is: "Is
'heterological' heterological?" 

Russell's paradox:
	This has to do with early set theory (which is basically the study
of groups of things).
	What you do is similar to Grelling's paradox, in that you devide
all of the sets in to two groups; sets which contain themselves and sets
which do not.  Now create a set called S "the set of all sets which are
not self-containing".  Now, does S belong to this set?

Another variation comes from an explanation of Godels' incompleteness
theorem by Douglas Hofstadter in his book Metamagical Themas:
	"It sounds a bit like a science fiction robot called 'ROBOT R-15'
droning (of course in a telegraphic monotone): 
              ROBOT R-15 UNFORTUNATELY UNABLE TO COMPLETE TASK T-12 -VERY
              SORRY.
       Now, what happens if TASK T-12 happens, by some crazy coincidence,
to not be the assembly of some strange cosmic device but merely the act of
uttering the preceding telegraphic monotone?"

Which brings us to Godel's incompleteness theorem:
	"To every w-consistent recursive class k of formulae there
correspond recursive class signs r, such that neither uGenr nor Neg(uGenr)
belongs to FLG (k) (where u is the free variable of r)"
	In other words, you can always create a mathematical statement
which through tricky isomorphism can be made to say "I cannot be proven in
system X." (where X is the system in which G is written). 

I came up with this one while sitting in Symbolic Logic:
There is nothing to stop me from creating the translation key

        A: There is an ampersand after sentence A
        B: There is an ampersand before sentence B

With this key, A&B is True, while B&A is False!

A&B: A  B   A&B
     T  T    T

B&A: A  B   B&A
     F  F    F

But:
1|A&B   assumption
_________
2|A     1,&E
3|B     1,&E
4|B&A   2,3,&I

In fact, I'm at a loss as to what to do with A&A (I think it's false)

Another one by me:
	P: There is a tilde in front of sentence P.

P has a truth-value of False, but so does ~P.

P: P   ~P: ~P
   F       F

Even more strange is:

        Q: There is nothing in front of sentence Q

   Q   ~Q   Q&Q
   T   T     F (?)

The important thing about the last group of sentences (starting with the 
robot) is that they work not because they are "this sentence is false" but
because at heart they say that "this formal system isn't powerful enough
be both complete and consistent".  As someone (who's name eludes me)
pointed out, Zeno's paradox demonstrated that we didn't have a formal
system which could properly explain motion.
	The amazing thing about Godel's incompleteness theorem is that it
shows that there will always be a way of creating a paradox (and hence
demonstrating either an incompleteness or an inconsistency) in every
formal system.  Thsi poses some problems for the idea that we can know
absolute truth, because we people ar not immune to Godel!
	For example: (feel free to taylor this sentence to whomever you
want) "Abian cannot consistently assert this sentence!"  There is a piece
of the truth from which Abian is forever seperated ;)

	Hmmm, I do go on.  
		I would like to hear what other paradoxes people have run
in to and also if my understanding of Godel is in anyway accurate (and if
any of this post made any sense at all...
--
	  			  Take Air,
   there are them that could        Snowmit of the Loftie Mountain
   and them that couldn't	     Chief Technician, 
   and them that do	    	     Founder of Cootm
   who often shouldn't		     Truffles Like Him (shouldn't you?)
 

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				-The Tick
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