What Are the Philosophical Implications of Godel's Incompleteness Theorems?

[stevendaryl]

Similarly, the statement "this statement is not provable" is by its form a second-order statement, in that it refers to a statement, and not directly to a logical or empirical fact. The statement is improper, because its referent is a second-order statement. The statement "the statement '2 + 2 = 4' is provable" is a valid and true second-order statement; not a first-order statement -- the first-order statement "2+ 2 = 4" is a legitimate first-order statement within the second-order statement as its first-order referent.

Yes, "this sentence is not provable" seems like a second-order statement, but it's not. That's the beauty of Godel's proof.

What Godel did was to come up with a way to map the "second-order" concept of proof to first-order concepts. The idea is roughly this:

1. Come up with a numerical "code" for each statement of arithmetic. For example, you can write down the statement using ASCII characters. Each ASCII character can be represented as a base-8 numeral. Concatenate all the base-8 numerals in a statement to get a base-8 number. Given a code, we can talk about $S_n$, the nth statement of arithmetic. Every statement of arithmetic will be $S_n$ for some value of $n$
   
2. Similarly come up with a code for each proof in the theory of Peano arithmetic.
   
3. Then corresponding to "the set of provable statements of arithmetic" is a set of numbers, "the set of codes of provable statements of arithmetic".
4. Show that this set of numbers can be defined by a formula of arithmetic. So there is some formula $P(x)$ with one free variable, x, such that for every natural number $n$, we have that $P(n)$ is true (and in fact, provable in Peano arithmetic) if and only if $S_n$ is a provable statement of arithmetic. What's important to note is the formula $P(x)$ is just an ordinary arithmetic formula, involving zero, plus, times, equality, etc.
   
5. Come up with a statement $G$ with corresponding code $g$ (that is, $G = S_g$) such that it is provable in Peano Arithmetic that $G \Leftrightarrow \neg P(g)$

So [/itex]G[/itex] is some statement of arithmetic, and $g$ is some number, and $\neg P(g)$ is another statement of arithmetic. It's all perfectly first-order. However, by the way that $P(g)$ is defined, we can see that $P(g)$ is true if and only statement $S_g$ is provable. So the negation, $\neg P(g)$, is true if and only if the statement $S_g$ is not provable. But $S_g = G$. So we have:

• $G$ is true if and only if $G$ is not provable.

G doesn't literally say "I am not provable", but we can see that it is true if and only if it is not provable.

 

[stevendaryl]

For me a reasonable definition of soundness of a belief is that a belief is sound if and only if and it is true, and validly justified, and some set or sets of true facts and valid precepts sufficient to constitute such valid justification is or are solely what is relied upon for the belief.

It's just a matter of definition. There is a distinction between something being true and there being a convincing argument that it is true. Godel's theorem actually shows that there is such a distinction, in the sense that no notion of "convincing argument" can ever completely capture all true statements.

 

[RockyMarciano]

I thought both the OP riddles and the limitations of distinguishing between n-orders to solve them were suficiently clarified at least from the early thirties and Godel's theorems and actually even before in a more heuristic way. There will always be assertions whose "truth" is unprovable if one insists on remaining within the original system of axioms. There simply are no universal axiomatic systems as Hilbert dreamt, this is formally admited but many seem not to have internalized it (the OP seems to be an example).

 

[stevendaryl]

I thought both the OP riddles and the limitations of distinguishing between n-orders to solve them were suficiently clarified at least from the early thirties and Godel's theorems and actually even before in a more heuristic way. There will always be assertions whose "truth" is unprovable if one insists on remaining within the original system of axioms. There simply are no universal axiomatic systems as Hilbert dreamt, this is formally admited but many seem not to have internalized it (the OP seems to be an example).

The OP wasn't talking about a fixed axiom system.

 

[RockyMarciano]

The OP wasn't talking about a fixed axiom system.

Ah, ok, I just noticed #9.