The Quickest Gödel's Theory Explanation

Not exactly. There's a property of independency of axioms such as if {a_1, a_2, ..., a_n} are axioms, then for any two axiom is false that a_i → a_j.

By adding new axiom you either change your theory, or not. To prove it, that's true, you can use your computer to check it, while it may appear to be false or non-sequitur (i.e not inferenced), and this time the computer won't accept it as a true or as another axiom till you put him a command to accept it.

You loaded to a computer 10 axioms. And let's assume computer is not a mixer or a blender of axioms, but it checks any different statements to answer which of them are true, and along with it, which of them are axioms, which are not, etc. If you put a sequence to a computer it can reject it signalizing about it (somehow, or a master can discover the problem; if it's some upper level langs, then this task would be easier).

So, the first important things that the computer check different statements. It has to to be able to prove theorems. 

Also, it's not like that 10th axioms are true for a computer. Computer doesn't know what is true, or what is not. It can be like 1st axiom has a meaning of T, 2nd has a meaning of T, and so on. The important thing is to make a computer able to decide which statements can have T value (or meaning). 

A certain statement would have the value T if it could be derived from a set of statements which ones had the value of T. 

And we have to input also another statement (i.e. an axiom) to explain what statements will have T value, or by which mechanism we add T value to statements. 

Karl, every your notification - is indeed important to me. I mean it's just a bad style of mine to speak in such a manner (I hope I don't hurt anyone by it).

That's explanation wasn't strictly from Godel that it must've been seen. But one idea had been floating in my mind for years, and I couldn't find it to be very similar to the Godel's project. Besides, it was about the same: the completeness of a system and the value. - And again, I might be wrong, so all in the power of that idea. I'll try to explain it.

When we say that "water boils at 100 degrees celcius" is almost the same as to say "the 'water' and 'the process of boiling' = '100' and 'Celcius'". (Or in other words 'a property p of A' = 'a property q of B'.) The same would be in many other cases. (Surely, we can use some other operators, than the equality. The equality is a useful operator, and it has an important logical ability - it saves the value over some changes.)

So, ok what does it mean that "water boils at 100 degrees of celcius"? It mean we have to look at some meters or some measuring devices. Let's say it would be a plain thermometer. But what does it mean '100 degrees of Celcius'? It means when the water reaches the point the mercury liquid in the thermometer reaches some limit. But any of such things wouldn't save us from illusion or some mistake. We won't never find anything in the world that would be safe. All the ethalons or measuring ideals can be broken, dried out, or disformed. We have to determine everything: starting from 'water', and ending with 'numbers'. As we know it's impossible to finish: no a posteriori truth is a necessary truth. So, that's why a mistake might occur any time. 

But the same is about any theory since we have to express our ideas. I mean since we use language (i.e. names, sentences), we can confuse the names or sentences by the same reasons we confuse measuring ideals. (And we also can encounter some paradoxes with names like: "9 in necessary greater, than 7", "9 - is a number of planets", "The number of planet is necessary greater, than 7".) So, we don't have any insurances to prevent such weird cases. 

As soon as we have the predicate logic we would achieve a powerful tool that allows us to imply the formulas that aren't impossible to get in the propositional calculus: in the predicate logic we can deduce c from (a&b): (a&b)->c. But to do this we have to use quantifications. In other words, we achieve almost the same procedures we use in the situation with physics when we equal the boiling of water to the lenght of the thermometer; we can equal or disequal some values. If we're enough tricky we can use that form to imply some unusual and weird results. 

It is important - that in any step we can deduce the rules of the systems. It should be clear: if the rules of the systems are deducible, than they are true. So, even if we won't add any new extra axiom, it still means that we are able to imply the rules of the system. And as soon as it's possible we can deduce something about the rules themselves. And achieving reflexive forms of statements we can interpret them as self-contradiction statements: they would say that those statements had no T value.

In other words, by adding new extra axiom we just say something about our own system. I think that the most confusing part of the topic of mine was the word "axiom". I take a notion 'axiom' in a quite broad sense: it's the same as a statement that has a number < than a number n. (Considering that n - is a number of axioms. So, in this case if a computer would check a statement it would check a statement's number, and if it would satisfy x < n, then it would take the statement as something that has the value of T.)

It's possible. I'd also try to make my post simply to this:

It's impossible to define things using finite undefined definitions. 

This is exactly true - it cannot resolve the other algorithms. Let's say all what he can is to describe the work of the other computers (along with the work of something else that it's able to check) in the language it's been programmed to check. 

In other words, if a computer could re-axiomatized itself, he could resolve the other programms. So, that was exactly I was going to say. Thank you for introducing the computer programms example.