Can Formal Systems Ever Truly Guarantee Consistency?

[ivan]

hi everybody

i recently read a book about Godel's theorem. book was discussing the problem of theorem in formal systems. it talked about two general ways to insure consystency: a method of modeling, when one takes a new system's postulates and converts into valid theorems of already known system.(say, you come up with a model within euclidian geometry and try to see that a noneuclidian geometry's postulates can be converted into valid theorems of euclidian geometry) then author points out the problem with such aproach, namely you still have to show the consistensy of the already known system, in the case above of euclidian geometry. second aproach, he talks about, is to completely formalize the system.

anyways, in both methods i could not see how they would guarantee consistensy. could you please help me out and tell me if there's really any logical way to prove that a given formal system will be consistent, for example in number theory there's no way one can prove nonexistanse of the highest prime number. and i do not mean it has been proved such number does not exist(i know how to prove it) but that there is no way from infinite number of theorems within the system one can show they will not combine in such a way that conclusion will be highest prime number exist.

please, give more laymen's explanation since i am not professional logician/mathematician.

thank you all

 

[Hurkyl]

You don't guarantee consistency -- you prove relative consistency.

In the author's first example, he proves the following metamathematical theorem:

(Euclidean geometry is consistent) ==> (Hyperbolic geometry is consistent)

And the point is that if we're willing to accept Euclidean geometry is consistent, then we must further accept hyperbolic geometry as being consistent as well.



When we formalize a theory, and prove the consistency of the formal theory, what we are really doing is proving the metamathematical theorem:

(Formal logic is consistent) ==> (The original theory is consistent)

 

[ivan]

point is well taken; so, we will never know if a formal system is consistent, since you can never proove within that system that the formal system is consistent. we can rely on the metamathematical theorem

(Formal logic is consistent) ==> (The original theory is consistent)

and the hopeful consitensy of formal logic. but the question still remains whether formal logic is consistent, right? and for the same reason, as stated above regarding proving consistency from within the system, you cannot prove consistency of formal logic without referring to something else. so, this would mean we should be hoping formal logic's consitensy, am i wrong?

second, if i understood you right that consistensy of formal logic cannot be proven, do we really need to refer to that theorem as long as you formalize a system? why do we need to refer to formal logic and those theorems if logics consitensy cannot be proved? to explain my point i'll give you an example. euclidian geometry has been around for more than 2 millenia and we think it has been "consistently" consistent. while those theorems in formal logic is recent developments. can't we just follow axiomatic method? if axiomatic method works well for euclidian geometry why shoud it not for other systems?

thanks agian Hurkyl

 

[robert Ihnot]

ivan: it talked about two general ways to insure consystency

ivan did not have any thing to say about the second method of proving consistency, which I take to be exhaustion of all possible cases. This can be done with syllogisms where it can be shown that only valid statements can be deducted from the premises by modus ponus.

Now some things like the rotations of a square can be either shown to be consistent by exhausting all the possible rotations or by examining all cases of the representation by group theory.

However, the matter becomes more difficult as the system grows more complex. All the cases can not be exhausted, such as the integers, where some theorems are undecidable.

I read that Hilbert proved the consistency of Euclidean Geometry. http://mathworld.wolfram.com/EuclideanGeometry.html

 

[heels_overhead]

well the degree of the efficiency is alway that to the sum of the 10th value thus the subratction of pie to the 2nd degree. Hope it helps.