1. Can someone help me understand Gödel's incompleteness theorem? Does it really prove that a TOE is impossible? Thanks!

2.

3. I don't think it says anything about the possibility or otherwise of a theory of everything. It only applies to formal systems (i.e. mathematics).

Crudely put, it basically proves that any formal system will always be incomplete; i.e. there will always be true statements that can be written using the system that cannot be proved to be true within the system.

I'm sure someone with a real understanding will be along to correct that...

This book touches on it, as well as computability (the Church-Turing thesis), the nature of consciousness, music, art the structure of language and many other inter-related things:
Gödel,Escher,Bach: An Eternal Golden Braid: Douglas R. Hofstadter: 9780465026562: Amazon.com: Books

4. Originally Posted by Strange
Crudely put, it basically proves that any formal system will always be incomplete; i.e. there will always be true statements that can be written using the system that cannot be proved to be true within the system.
It would be better to say that for any formal system of a certain amount of logical power (i.e., ability to derive statements from other statements), that system will not be able to derive some statements, nor will it be able to derive the negation of these statements.

Truth is really irrelevant to the incompleteness results. They can be presented entirely in terms of what a system can and cannot prove.

5. But, what it means? Well, Coursly (as the theorem is just made for a certain subset) it means that no set of axioms can be made without contradictiong itself. This includes math:
How big is Infinity
As posted before.

This is of course just a very course explanation. Strange has already answered aswell. It does indeed form a relevant part in Quantum Philosophy however.

6. Here is a vague, hand-wavey way to illustrate this - but hey, this is a physics forum!!

The set of symbols - letters - that make up an alphabet is finite, and therefore countable. In our case it has cardinality 26, or if we include punctuation, capitals and spaces we have something in the mid sixties, but countable nonetheless. Suppose we make use of this fact to enumerate these, and, for reasons I will explain shortly, designate as follows: a= 10, b = 20, c = 30 etc.

Then a word will be of the form 2050140 = ben (my real name). This is clearly a number, i.e. an element in . Now I can stick these words together to make a sentence, sentences to make a paragraph, paragraphs to make a book, etc, and I will still have an element in .

OK, I simply used zero as a delimiter to avoid under-counting, that is, I know that 1020 = ab and not l = 12, which could happen if 12 = ab or l. So we see there is no word, sentence, paragraph, library, or even collection of libraries (Congress of Libraries??) that cannot be expressed as an element in the countable set

But, since Cantor showed that the set of real numbers is uncountable, that means there will be uncountably many elements in this set that cannot be given a name using words or sentences or paragraphs, or.... whatever

But it gets worse. is the powerset on the natural numbers. Cantor's diagonal argument can also be used to show this is uncountable. So, let us assume that, in a complete theory of the natural numbers, that is for each subset of i,e, an element of the powerset, there must be at least one true statement (using words, sentences, paragraphs,....) in some theory of natural numbers, so there must be uncountably many true statements of number theory.

Therefore, there must be uncountably many MORE true statements of number theory than there are words, sentences, paragraphs, books or libraries (since these are countable by my construction).

It follows that any system that tries to describe number theory as a complete set of true statements is doomed to failure. This is version of an incompleteness theorem.

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