A couple of book questions...(one of them about Completeness)

[Tyler Durden]

Well, this is quite a post, and I agree with most of it.

As for the Godel stuff, there's a part of it with which I disagree (or at 
least as far as I take what you said).


>If you want
>to compare something mathematically you -must- use the same axioms and
>rules of derivation. The -only- discussion there is one of two parts:
>Is the sequence of applications/operators valid? (ie Proof)
>Is the sequence terminal, does it leave room for more derivation?
>	(ie Publish or Perish)


Well, not necessarily, unless I misunderstand you. Take the Fermat's last 
theorem example I gave (a^n+b^n=c^n for a,b,c,n integers but n>2).
And let's say I want to "prove" (or disprove) the statement "This has no 
solution for n>2.

There are two 'distinct' methods of determining the validity of the 
statement. One is by what is normally considered a "proof". In other words, 
by building up from axioms using the logical rules of the system.

The other is to actually find a solution for a,b,c and n. In this case the 
statement will have been disproven, but not by a series of logical 
statements and axioms. It is now seen to be "untrue", but not via the 
methods of "proof". Thus, the statement is untrue, and (possibly) unprovably 
untrue (which is the same thing as saying the statement's negation is 
"unprovably true").

Now if subsequent truths need to be made but require the statement above 
(a^n+b^n=c^n has no solution for n>2), even though we know that it is true 
(or untrue, in my example above), to build subsequent truths we need to 
include this statement as an axiom even though we know it's true. It's 
"true", but unprovable.

But perhaps this is what you meant.


>And no, there is zero confusion on what true means under Godel or Cauchy.

Yes, I agree, and the confusion to which I referred had to do with the term 
"true" as it seemed to be used by various parties in the conversation. From 
this alone I think a big "take away" here is that "true" in the Godelian 
sense means something probably quite different from what many believe it to 
be.




>The reality is that most people have problems grasping concepts or
>ideas because there is a conflict with other ideas/concepts they hold dear
>and near. In most cases of mental block it is an emotional issue not an
>intellectual one. People have a hard time learning not because they are
>stupid but because they don't deal with their emotional landscape
>effectively.

Couldn't agree more. "Reason is the whore of desire." Well, not always, but 
its clear to me that most of the time we start with the conclusion we want 
and then work backwards! Most human beings seem to stumble upon some little 
piece of flotsam and then cling onto it for dear life, not knowing they can 
actually swim (or perhaps they don't need to!). I don't consider myself an 
exception, except for the fact that knowing this, I constantly try to expose 
myself to information and experiences that do not correspond to what I 
currently believe. As the spanish mystic St John of the Cross wrote:

"To come to be what you are not, you must go by a way in which you are not. 
To come to know what you know not, you must go by a way in which you know 
not."

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