h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Dear Sir Fragalot,

I was happy to find this entry, but perhaps due to my lacking language-knowledge, I was not really able to follow the
arguements of the non-intuitive proof of incompleteness.
Especially it is not plain for me:

-what a goedel number is (is it a one-to-one mapping from the collection of all statements to the natural numbers ?)
-what a truth-finding function is (is it a mapping from the natural numbers to {0,1] ?
-what is a collection-calculating function and what is the link between it and a statement ?
or: how can a coll.cal.function be true ? From where to where does it map ?
- is the assertion that "if the coll-cal. function is true, its Gödel-number belongs to its collection" THE Gödel-sentence or only A Gödel-sentence ?

-how can an even number be a true statement ?

I hope these questions contribute to an easier entry.

I think the posting preceeding this one says all that I could on the subject.

Heaven help the poor sub that gets saddled with that one.

May I suggest putting spaces between paragraphs? It won't make me any more intelligent but it might break the subject matter down into chunks that I can deal with one at a time.

see my article "talking of fish" written rather hastily, i'm afraid. what i'm proposing is that we expand logic in the same spirit that we now consider non euclidian geometries. if we admit statements that are outside the set of aristotelian logic, we can propose a universal set of statements that says everything, and (paradoxically in normal senses, but not in the context of a universal language) says that what it says is true. this statement is u. obviously there exists the inverse of u, but you would expect that in a complete language. instead of always saying "no" we can now say "yes" and "no".

I think the easiest way to explain what Godel was on about is like this:

'This statement cannot be proven.'

Now, that's a true statement. But, if you try and prove it, it ceases to be true. Therefore it is both true and unprovable.

Write the above statement in strict mathematical form, using Godel numbers, and you've got a proof that you can't prove everything...

Godel-Escher-Bach is a great book, by the way, 's definitely recommended.

26199