Re: Godel Goofed? Nope.

[Quantum_Cat <quantum-no-spam@cotse.com>]

Hiya, Cooter.

 *>> You are familiar with the title of Godel's paper?  "On Formally
 *>Undecidable
 *>> Propositions Of Principia Mathematica And Related Systems".
 *>>
 *>> Since we are referring to Godel's Theorem, and he limited it's scope to
 *>PM, I

 *>Whoops (you forgot to consider "And Related Systems")

You're still trying to cloud the issue.  Godel built his system from the notation 
and rules of inference of PM.  Hence the "Related".  His proof holds true for PM 
as well.

The power was that his proof works for any axiomatic system.  But his paper was 
limited to his system and PM.  So... do you want to quibble some more? ;)

 *>{Conventionally his theorem is taken to mean unprovable within ANY possible
 *>system, rather than merely PM)...

Parse that better.  His theorem proved that in any axiomatic system of sufficient 
power, there would be statements which remain unprovable.

 *>(Besides if you are now claiming that it ONLY applies to PM, then your
 *>original point (that my system of infallible reasoning can not work) is
 *>lost)...

I doubt your system is an axiomatic system of much power.  ;)

 *>> will do so as well.  Therefore, the accuracy of my translation is adequate
 *>for
 *>> our purposes.  If you want the actually statement in German, I'm sure you
 *>can
 *>> find it.  If you want it in English,  Try page 17 of GEB.  The
 *>mathematical
 *>> symbols involved make it difficult or impossible to post it meaningfully
 *>here.
 *>>
 *>>  *>The scope extends beyond PM...
 *>>
 *>> Red Herring.  Please stick to the topic at hand.
 *>>

 *>Not at all Red Herring, the WHOLE point is the Godel's Theorem specifically
 *>limits the possible chances of my system of Infallible Reasoning,

If your system is an axiomatic system of sufficient power, then his proof will 
work there too.

 *>if it ONLY applies to PM, then this point is entirely lost. We both know
 *>that is applies to more than PM (if it applies at all), so why bother
 *>arguing against this self-evident truth?

Specifically axiomatic systems.  I have no idea what kind of system you think 
you've come up with.  ;)

 *>>  *>The analogy (mathematical mapping)
 *>>
 *>> Analogies and mathematical mappings aren't synonymous.  Please stop trying
 *>to
 *>> cloud the issues by misusing terms.
 *>>

 *>A Mathematical Mapping is the Mathematical form of an Linguistic Analogy...

A linguistic analogy is not a mathematical analogy.  ;)

 *>>  *>that I claim is that both of these sentences are ill formed statements
 *>for
 *>>  *>the same reason of self reference.
 *>>
 *>> You make the claim that statements of self reference are ill formed, and
 *>ill
 *>> formed statements are self referal.  Circular reasoning.  Perhaps now is
 *>the time
 *>> for you to disclose your definition of "ill formed" in order to avoid this
 *>> fallacy in the future.  ;)
 *>>

 *>Sure... In order for a Sentence to form a Valid Statement there must be an
 *>Object that this Statement Specifically refers to, for the Truth Value to be
 *>Attached.

 *>EXAMPLE:
 *>{George Washington was the first president of the United States} In this
 *>case the object of the statement is {first president of the United States}.
 *>If I would have said {George Washington was the}, it would lack the object,
 *>thus fail to form a valid statement.

Oh goodie!  You fell into it good this time!  ;)

Take the statements:

	The following sentence is false
	The previous sentence is true

Each of those statements fits your criteria.  They both have an "Object that this 
Statement Specifically refers to, for the Truth Value to be Attached"

They are therefore well formed according to your rule.

Yet you cannot deny that they are self referenced, although indirectly.

You've attempted to ban direct self referencing statements like "This statement 
is false" by claiming that it is ill-formed.

What are you going to do now that we have statements that fit your definition of 
well-formed, yet retain self referencing?

You know that whatever method you come up with, it simply requires another small 
step of further indirect referencing to invalidate your new method.  Ad 
infinitum.  Therefore, there are statements that are impossible to prove in any 
sufficiently powerful axoimatic system.  

Checkmate.  ;)

Vogons For a Turlette Free Apple II Community!

Quantum_Cat