Question about Godel's Incompleteness Theorems

[galoisjr]

First I would like to say that I read over the statement for the first time, and I don't pretend to fully understand it, but I think that I do understand the basic idea. If you have a certain system of axioms that establish at least the intuition of arithmetic then there are statements about the set or sets that this system acts on that are unprovable. This got me thinking, and I am pretty sure that this statement is incomplete and that there is actually a way around it. Firstly, Godel assumes that the system of axioms have been assigned the truth value of being true. But from basic ideas of logic if this is not the case and the system is given the truth value of false then relatively anything can be proven from them about any set concerning the system. Bertrand Russel had a very elegant way of showing this in a number of cases. So what I am saying is that it is not what is assumed to be false that limits provability. What limits provability is the fact that we require these statements to be true. And this makes sense if you think about it in terms of deduction from process of elimination, or trial and error. For example, say I was standing a football field away from you and I told you to walk to me. If you decided to do it then you would walk straight towards me. Now did you do this because you automatically knew that a straight line is the shortest distance between two points? Or sometime in your past did you, maybe subconciously, by deductive reasoning, eliminate the infinitude of other possible ways to get travel the distance between two points by choosing the shortest? See if you automatically can choose the right answer to this then the question has to be asked of what do you not automatically know the right answer to. And we as humans obviously do not know the correct answers in every situation. So it has to be the second that it was a logical progression by deduction. For further clarification, say that the distance between two points is defined to be the straight line between them, and I asked you to walk to me by taking the longest route. What would you do? I don't know either. Probably just walk away. But what I am saying is that there are things about the naturals that are unprovable by its axioms because we assume them to be true. If conversely, we were logical and arrived at conclusions by deduction instead of from the axioms then there could be no limit on provability because nothing was previously assumed. While this is an exhaustive and tedious method, I'm sure there is a greater mind than mine that can condense the idea. We could, for example, define the axioms to be only what is not true.

 

[galoisjr]

I guess that a very brief sketch of this would be that since T limits provability then this implies that not T does not limit provability. If we assume that the double negative law from English holds in this statement then the phrase "limits provability" should be logically equivalent to "does not limit unprovability" and thus by the the same reasoning "does not limit provability" is logically equivalent to the statement "limits unprovability." Here I am assuming that the set of all conclusions are either provable or unprovable so this should hold. Then the second statement can be rewritten as F limits unprovability.

 

[HallsofIvy]

I have no idea what you mean by "Godel assumes that the system of axioms have been assigned the truth value of being true". That's pretty much the definition of "axiom" isn't it?

 

[galoisjr]

Yes and no. Our axioms contain only true statements that is true. But an axiom can be anything that is considered necessary to the subject. False statements are just as essential to logic as true statements are.

But I thought about it a little bit more and I think that in setting a system of statements that are false we are actually accepting that it is true that they are false. So my theory needs a little work.