Gödel's Incompleteness Theorem involves a system's, such as mathematics, inability to prove itself by utilizing only itself.
For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.
The issue is that any logical proof of an axiom will necessarily lead to another which will always merely lead to another endlessly, and thus never complete any absolute proof.
The resolve to this, which was either ignored or merely not realized by those involved in its promotion, is the concern of definitions. Within mathematics, are statements such as "1+1=2". But how does one prove that 1+1 really does equal 2? Using only mathematics leads to more complex equalities that eventually lead to the proof that 1+1=2. But every one of those depends upon an axiom of some kind presumed to be true. Logic always begins with something presumed to be true such as to deduce what else must be true.
But there is a difference between a common axiom and a definitional axiom. A typical axiom is merely something that most, if not all, people would accept as true. But a definitional axiom is something that is declared to be true throughout the system involved, such as, "2 ≡ 1+1".
A declared definition for a system cannot be contended with and is not susceptible to truth statement doubts. The definition is a conditional agreement for utilizing the system at all. If the definition is not accepted then the entire system is not accepted. Every system of any nature involved in thought depends on defined concepts that might or might not have been explicated. Any logical proof that leads back to a defined conceptual truth is necessarily true, without exception.
The issue then becomes one of arbitrary definitions and the rationality of the system being proposed. In the case of mathematics, the definitions are hardly arbitrary (being based on simple quantitative counting) and the rationality (meaning the usefulness) has been extremely demonstrated as useful.
Mathematics, whether realized by its proponents or not, is actually based upon Definitional Logic and thus is incontrovertibly true as a system and due to extremely numerous examples, has been empirically demonstrated to be rational.
Anyone contending with mathematics is contending with rationality... and visa-versa.
Ape Cock Replies:
No, what Godel thinks he discovers, what I think he means and what I think all this stuff leads to is only one: any system of relationships, any set of Information Relationships, of Reciprocal Relationships and Interactions, any Action - Reaction Set, any system that is essentially talking to itself, that is interacting with itself (exactly like humanity, or a Man Brain or an Observer with the imagined "external" universe) will never be able to ground this system absolutely, definitely, will never be able to find and observe and declare its ultimate fundament: and this is simply because there is no fundament, the system is arbitrary from the outset, the system is a quirk and random and arbitrarily designed or configured system from the outset, not based on anything at all, no matter how hard you try to find some absolute necessity for it to be in the only way it is, no matter how hard you try to find the "God" behind it, no matter what, it is a completely random arbitrary fluke with no deeper necessity to being the way it is than anything else, than any random combination of rocks and sticks you find on any random street on the earth.
And in essense the very laws of physics, the set of the laws of physics themselves (and the corollary laws of logic, identity, non contradiction), taken as a set of reciprocal necessities and forced interactions, is, when seen from outside of themselves, completely arbitrary, a complete fluke, could have been any other laws at all, have non intrinsic necessity for they being the way they are except for they being as they are by pure random, blind chance.
In essence, you would have to be on the outside of the universe looking in, you would have to be such a generalized "Observer" to contain all possible Observer designs and all possible universes to be able to find what they all have in "common", you would have to be a super Observer outside of this or any universe able to see it all and find the common denominator and such: obviously impossible as even with the way our Man Brain is configured (only one possible design amongst 10^10000) and so on ) can only see very little and such: and such a plan falls apart as soon as you consider Observers and Universes (or both together) that don't follow any logic at all or any random logic where contradictions are ok, identity is iffy and so forth....
And so it is with the Man Brain, with our own structure as an Observer, as Natural Evolution evolved us, we are totally arbitrary, a totally arbitrary Action - Reaction Set, Reciprocal Information Relationship Set, a totally arbitrary, random fluke Reciprocal Action Reaction Set as to how an arbitrary Observer interacts with an arbitrarily delimited outside world.
Of course, as usual, I declare "All Contradictions are Operating" when I talk about this stuff, as it all leads to contradictions, as no logic can ever contain it all without falling apart somewhere...
Just in order to claim that, you have to assume that YOUR "system" or mind frame is capable of being "outside the system" that is your own mind.
If you want to get into what is or isn't "real" (which isn't what he was talking about), you know the real simply by what has affect. Nothing else matters.
Ape Head Replies:
That is why I said "All Contradictions Are Operating". Whatever. Once you establish definitions and rules of engagements, etc. you do create a kind of local absolute reference system, you create a kind of closed system. But you cannot prove the system from within the system, the system can only "talk to itself" so to say, but can't "get out of itself to prove itself". Godel discovered this but couldn't figure out what he really discovered: in fact no one still really understands that this is the real message of all the Godel stuff and such, I am the only one who has really understood the consequences of his work.
If I define "AWHEHDD" as a truth statement and invent another "$HTHTRRR" as a truth statement and say the second demonstrates the first (or any other kind of relationship I want), the definitions themselves create the proof, but the proof is only valid in this local universe with these two statements and me hardwiring them to be the way they are. It also implies that the entire universe and all of the universe consists of only these two statements, there is nothing outside of them. But Godel wanted to show that the startements were sufficient and hardwired the truth even considering what is outside: obviously this is not possible because you have trillions of things "Outside" of those two statements.
"being based on simple quantitative counting"
Counting is just one possible algorithm, and in fact truths don't even have to be algorithms: take a memory chip with 6 input bits and 4 (or 5 or whatever) output bits and you can program the contents of the memory to express the function of addition between the first 3 input bits and the second 3 input bits (you divide the bits arbitrarily as being 2 sets of 3 bits instead of one set of 6 bits, you delimit them arbitrarily, this is an example of "delimitations") giving you the addition result as the 4 bit output (but the addition is just the memory content the 6 bits address in reality). But you can decide to program the memory to express the function of multiplication between these of two sets: a 3 bit by 3 bit multiplier giving you the 4 bit output result (notice the old TTL 9344 4 bit by 2 bit multiplier is just a form of compression of the function into fewer gates, if you can't compress a function into fewer gates just use brute froce and write down the function you want by programming a memory etc.).
But an interesting thing happens with this memory: you can invent any new operator, like XZX and program the memory in such a way so as to express a new arbitrary and random function by associating the 3 bit XZX 3 bit giving you the 4 bit output according to the values written in the memory. But this new function is not addition or multiplication but some other new abstract function that may have some meaning for a new kind of Observer used to dealing with a world where this function makes snese and is natural just as our addition or multiplication is in ours. Now you can program the memory in 16^64 different ways (all of the possible combinations of the 64 4 bit value outputs: or like just one long number made up of 64 digits each having 16 values), each one expressing a new mathematical function valid for some Observer. Those are more than 10^76 different new functions. But for each of those Observers, he only deals with and uses a few of those trillions upon trillions of possibilities and can create a closed coherent and logically proved world within his universe: but he will never be able to prove it completely as he can't observe and use all of the other possibilities, hence functions and such.
Anyways, this is what the problem is ...
It seems that you are saying that a system cannot claim to know actual truth and yet you ARE a system claiming to know the actual truth of that claim.