As Morris Kline said in his book "Mathematics and the Loss of Certainty":By "safe logical principles", Kline is referring to first-order logic and finitary proof theory, which was regarded as "safe" by the classical formalists."Godel's result on consistency says that we cannot prove consistency in any approach to mathematics by safe logical principles...
Here is a quote from John Stillwell's book "Mathematics and its History":By "system", Stillwell is again assuming a particular class of formalization. To provide my own quote, Kleene, "Introduction to Metamathematics", (1971 sixth reprint), Ch. XV, sec 79, pp478,479 says:"If S is any system that includes PA, then Con(S) [the consistency of S] cannot be proved in S, if S is consistent." [Godel's 2nd theorem]
"Gentzen's discovery is that the Goedel obstacle to proving the consistency of number theory can be overcome by using transfinite induction up to a sufficiently great ordinal... The original proposals of the formalists to make classical mathematics secure by a consistency proof did not contemplate that such a method as transfinite induction up to eps0 would have to be used. To what extent the Gentzen proof can be accepted as securing classical number theory in the sense of that problem formulation is in the present state of affairs a matter for individual judgment..."The basic reason why we haven't simply declared Hilbert's program and first-order predicate logic and finitary proofs an interesting failure and moved on, is from a wonderful bit of serendipity: that through Goedel's work on recursive functions, and the later work by Church, Turing, Kleene, and a host of others, we found out that first-order predicate logic and finitary proofs is exactly the right mathematical model for computability and our modern computer systems. Goedel's results thus becomes a theorem on computability (E.G. Turing's Halting Problem formulation).
Unfortunately, we have now confused several aspects of the theory. It shows there are limits to A) computability of digital computers, and B) the ability to prove consistency of mathematics when limited to a particular toolbox of proof techniques. It says nothing about C) the ability to prove the consistency of mathematics, for which we do have proof techniques that work for number theory at least.
As Morris Kline said in his book "Mathematics and the Loss of Certainty":So what? Kline doesn't know what he is talking about. That is a very very very very very very pathetic and ignorant book you are referring to, filled with mistakes, howlers, gibberish and all round stupidity.... Your quotations are irrelevant, and merely show that you have bought the pop-poop version, the kind that sells."Godel's result on consistency says that we cannot prove consistency in any approach to mathematics by safe logical principles...
There is a consistency proof of PA, and it is quite elementary, using standard mathematical techniques (ie, ZF). It consists of exhibiting a model.
IF Euclidean geometry is consistent
then non-Euclidean geometry is consistent.
IF ZF is consistentRelative consistency proofs assert nothing about the absolute consistency of any system, they just relate the consistency of one system to that of another. Here's what the Encyclopedic Dictionary of Mathematics (2nd Ed) says on the subject:
then ZFC is consistent.
"Hilbert proved the consistency of Euclidean geometry by assuming the consistency of the theory of real numbers. This is an example of a relative consistency proof, which reduces the consistency proof of one system to that of another. Such a proof can be meaningful only when the latter system can somehow be regarded as based on sounder ground than the former. To carry out the consistency of logic proper and set theory, one must reduce it to that of another system with sounder ground. For this purpose, Hilbert initiated metamathematics and the finitary standpoint...Let S be any consistent formal system containing the theory of natural numbers. Then it is impossible to prove the consistency of S by utilizing only arguments that can be formalized in S...."This implies to me that all known consistency proofs of arithmetic rely on something like transfinite induction (or possibly primitive recursive functionals of finite type), the consistency of which is no more self- evident than that of arithmetic itself."In [these] consistency proofs of pure number theory..., transfinite induction up to the first e-number e_0 is used, but all the other reasoning used in these proofs can be presented in pure number theory. This shows that the legitimacy of transfinite induction up to e_0 cannot be proved in this latter theory."
Kevin Brown says:I used to argue this line until I was set straight by some logicians. (Is Torkel available?:-) There is no proof in *first-order* logic that arithmetic is consistent. But that has more to do with the limitations of first-order logic than anything else and there are other more general types of logic in which proofs of the consistency of arithmetic are available.
" ...we have no rigorous proof that the basic axioms of arithmetic do not lead to a contradiction at some point."