X-Message-Number: 23877
Date: Mon, 12 Apr 2004 20:09:33 EDT
Subject: more Goedel

Thomas Donaldson writes in part:
If you dismiss Goedel's theorem as "meaningless 
playing with symbols" you'll get a significant number of mathematicians
disgusted with you. 
That's not quite what I said. I said that the Goedelian undecidable sentence 
has no mathematical significance. He only showed that a particular type of 

sentence is undecidable, and that particular type is an isolated oddity, with no
relevance to the real world that I can see.

Again, although Goedel used a complicated and ingenious labeling method, and 
the ingenuity obscured the issue, the essence of it is just this:

1. I write, "Sentence number one is unprovable."

2. I label that sentence "number one."

In that language, sentence number one is then indeed unprovable, and in fact 
undecidable--and in the metalanguage, or in your head, the sentence is even 
true, despite being undecidable in the primary language. But that tells us 

nothing useful, except that we have to be careful about language, which we 
knew or should have known.

Further, we already knew that formal systems have weaknesses far more serious 
than undecidable sentences. Those weaknesses include the axioms and the 

primitives or undefineds, as well as the rules of logic. Those are much, much 
serious questions than undecidable sentences.

Thomas also writes:

This is not to say, however, as I pointed out in my previous message,
that we should not test the results of our playing with symbols against
our real experiences. Not to do this is REALLY meaningless playing with
symbols. Trying to do such tests is science.

That is on the money. My main efforts in Youniverse relate to real-world 
choices. Here there are no certainties, but plenty of room for improvement.

Robert Ettinger

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