If P x1…, xn is a true proposition about the numbers x1,…, xn, then the formula P x1,…, xn is provable. If P x1…, xn is false, then P x1,…, xn is refutable. As one would expect, the formula P x1,…, xn constructed to numeralwise express P x1,…, xn also expresses P x1,…, xn under the usual interpretation of the symbolism.
To fix our ideas, let us consider formulas which contain free just the one particular variable v. For, if it is true, then by what it says it is false; and if it is false, then by what it says it is true. We escape paradox because whatever Hilbert may have hoped there is no a priori reason why every true sentence must be provable.
This conclusion is inescapable, if in the formal system only true sentences are provable.
Godel's Theorem in Focus
For, if Ap p were provable, then by what it says it would be false, contradicting our supposition that only true sentences are provable. So Ap p is unprovable, and then by what it says it is true. Under the same hypothesis, Ap p is also unprovable, since it is false. Rather than assume simply that only true sentences are provable, a formalist or metamathematician would prefer to substitute consistency properties. To establish that Ap p is unprovable, it suffices to assume that the system is simply consistent.
For this x, A p, x would be true; so, because A v, x numeralwise expresses A v, x , the and formula A p, x would be provable, and thence so would be Page 55 which is So both Ap p and would be provable, which would constitute a simple inconsistency. Incidentally, the formally undecidable proposition is arithmetical. For, if the enlarged system were would be simply inconsistent, then in the unenlarged system provable, whence Ap p would be provable; but Ap p is not provable if the system is simply consistent.
This shows that just the simple consistency of a system which is what Hilbert had in mind to prove would not guarantee its correctness under the interpretation when there are variables interpreted as ranging over the natural numbers. No sooner had we concluded that Ap p is unprovable than we knew it is true. What is lacking in the system that a formula we can recognize to be true is not provable in it?
Reactions to the Discovery of the Incompleteness Phenomenon
Consider our result more precisely. Page 56 simply consistent. All the reasoning to show that Ap p is unprovable and hence true, when carried out in full detail, is of the character of elementary number theory, except that we put that hypothesis into it. In brief, we have shown by elementary number-theoretic reasoning that: 1 If the formal system is simply consistent, then Ap p is true. From any contradiction in would follow.
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Hilbert and Bernays in Volume 2 of Grundlagen der Mathematik carried out this exercise in a similar system. Let us accept 2. For, if Consis were provable, then from it and 2 by modus ponens, Ap p would be provable; and we have seen that Ap p is not provable if the system is simply consistent.
This does not rule out absolutely the possibility of a finitary Page 57 consistency proof for a formalism embodying at least elementary number theory. Indeed, just this has happened. In September of , when I was about to spend a year at the Institute for Advanced Study, Hermann Weyl put in my hands a manuscript by Gerhard Gentzen which he wished me to read.
This manuscript contained a consistency proof for elementary number theory, of which a revised version was published in cf. Paul Bernays, Briefly, a numbertheoretic formal system consists of logic plus number-theoretic axioms. If the logic is complete but the whole is incomplete, the numbertheoretic axioms must be incomplete.
Thus he observed that they would hold good for the formal systems having the Zermelo-Fraenkel axiom system, or that of von Neumann, for set theory, or the Peano axioms for the natural numbers and the schema of primitive recursion, besides the rules of firstorder logic. In a note On completeness and consistency —2 , he starts with the last-mentioned system, and considers successive enlargements by adding higher-type variables for classes of Page 58 numbers, classes of classes of numbers, etc.
He thus obtains a sequence of formal systems continuable into the transfinite in which some undecidable propositions of earlier systems become decidable, while new undecidable propositions are constructible by the same procedure. In a paper On the length of proofs, he states that in systems with higher types of variables, and under the appropriate consistency assumptions, not only do some previously unprovable sentences become provable, but infinitely many of the previously available proofs can be very greatly shortened.
Church, , footnotes 3 and In these versions the formally undecidable propositions, which every formal system in Page 59 question must possess, are values of an arithmetical predicate P x picked in advance. Thus no collection of correct logical principles and mathematical axioms that one could ever be able to describe effectively will suffice for deciding the truth or falsity of P x for every x. Suppose we had a formal system which is complete and correct for the theory of a predicate P x.
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In brief, the existence of a complete and correct formalization of the theory of a predicate P x would impose on P x that it be expressible in the form Ey R x, y with R x, y general recursive. Post , , A. Markov , Kleene and J. Sheperdson and H. Sturgis That is, it can be proved rigorously that in every consistent formal system that contains a certain amount of finitary number theory there exist undecidable arithmetic propositions and that, moreover, the consistency of any such system cannot be proved in the system.
Ordinal numbers are associated with well-ordered sets, i. The ordinals themselves form a well-ordered collection. Hence, with the aim of making set theory consistent, systems of axioms have been formulated to say just what sets shall exist and just what shall be assumed about them. The first axiomatization of set theory was by Zermelo in a. This axiom asserts that the elements x of a given set y which have a definite property shall constitute a set.
Abraham A. Fraenkel proposed a remedy in , Skolem in , and Weyl even earlier , , dealt with the problem somewhat differently.
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For simplicity, I shall talk about a system of axioms usually called Zermelo-Fraenkel set theory, but which is closer to Skolem and Weyl on the point in question. Page 64 class and membership. Only sets can be members of sets; classes are not necessarily sets. For example, the class of the ordinals is definable by a formula of ZF, i.
The class of the ordinals is not a set; if it were, we would have the Burali-Forti paradox. A first crucial step concerns the axiom schema of comprehension. Indeed, iteration of eight operations, the full effect of the axiom schema of comprehension. This F is only a class; i. Cantor in , and in when he mistakenly thought that he had proved his continuum hypothesis cf.
Church, , was not thinking of its status relative to a list of axioms. After choosing an axiomatization, say ZFC, there are three possibilities respecting, say, the simple continuum hypothesis First, it is provable from the axioms. Second, it is refutable from the axioms. Third, it is undecidable from the axioms. Paul J. So is undecidable from the axioms of ZFC. Page 67 problems of the consistency or independence of various conjectures in set theory relative to this or that set of axioms are being investigated by constructing models.
I cannot imagine that Cantor ever dreamed that the subject he brought into the mathematical world just over a century ago would take on this aspect.
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What will it be like in another hundred years? I am using italic letters to name predicates and variable natural numbers of your and my language the metalanguage , Roman letters to name formulas and variables of the formal system, and bold face italic letters to name numerals. Note 2. Mathematische Annalen 99, — Journal of Symbolic Logic, 5, — Burali-Forti, Cesare Una questione sui numeri transfiniti. Rendiconti del Circolo Matematico di Palermo, 11, — Jahresbericht der Deutschen Mathematiker-Vereinigung, 1, 75—8.