These results, though, were somewhat disappointing as they were obtained by highly non-constructive methods that provided no concrete method for eliminating cuts in a derivation. However, Girard showed in that simple type theory not only allows cut-elimination but that there is also a terminating normalization procedure. To quote Takeuti:. My fundamental conjecture itself has been resolved in a sense by Motoo Takahashi and Dag Prawitz independently.
Takeuti A major breakthrough that galvanized research in proof theory, especially ordinal-theoretic investigations, was made by him in This was my last successful result in this area. In PA one can define an elementary injective pairing function on numbers, e. With the help of this function an infinite sequence of sets of natural numbers can be coded as a single set of natural numbers. This is the basic idea underlying the ramified analytic hierarchy.
The problem of which ordinals could be used for the transfinite iteration led to the concept of autonomous progressions of theories. The general idea of progressions of theories is very natural and we shall consider it first before discussing the autonomous versions. He indicated also that it would allow the elimination of the induction principle—in exchange for dealing with infinite proofs. These considerations among others raised the issue of what constitutes a properly formal theory. At the very end he introduced the general recursive functions.
Church and Turing used their respective notion to establish the undecidability of first-order logic. In consequence of later advances, in particular of the fact that, due to A. Furthermore, the extensions of T are all supposed to be formal theories, i. To deal with both issues at once, one has to deal with ordinals in an effective way. That is what Turing did in his Princeton dissertation concerning, what he called, ordinal logics. There he considers two ways of achieving the effective representation of ordinals. Definition 5.
When it comes to theories T , quite unlike to other areas of logic e. Theorem 5. More details and other results on recursive progressions are discussed in Appendix B. R2 is called an extension by the local reflection principle , whereas R3 uses the uniform reflection principle.
For further discussion see Appendix B.
In the foregoing progressions the ordinals remained external to the theory. Autonomous progressions of theories are the proper internalization of the general concept of progressions. This idea of generating a hierarchy of theories via a boot-strapping process appeared for the first time in Kreisel , where it was proposed as a way of characterizing finitism and predicativism in mathematically precise way. It turned out that this ordinal can be expressed in a notation system developed by Veblen that will be presented next.
Ordinal representation systems utilized by proof theorists in the s first emerged in a purely set-theoretic context more than 50 years earlier. His method was to represent countable ordinals via increasing sequences of natural numbers and then to correlate a decimal expansion with each such sequence. Hardy used two processes on sequences: i Removing the first element to represent the successor; ii Diagonalizing at limits.
Veblen in extended the initial segment of the countable for which fundamental sequences can be given effectively.see
Logic Colloquium: symposium on logic held at Boston, 1972-73
The new tools he devised were the operations of derivation and transfinite iteration applied to continuous increasing functions on ordinals. After the work of Bachmann, the story of ordinal representations becomes very complicated. Feferman proposed an entirely different method for generating a Bachmann-type hierarchy of normal functions which does not involve fundamental sequences.
Buchholz further simplified the systems and proved their computability. For details we recommend the preface to Buchholz et al. The rules for the set quantifiers are as follows:. As per usual, the price one has to pay for rules with infinitely many premises is that derivations become infinite well-founded trees. The length of a derivation can then be measured by the ordinal rank associated with the tree. One also wants to keep track of the complexity of cuts in the derivation. The optimal result is the so-called second cut elimination theorem.
Here is a selection of such results: [ 24 ]. For the definitions of the theories in this theorem, see end of section 5. The investigation of such subsystems of analysis and the determined effort to establish their mathematical power led to a research program that was initiated by Friedman and Simpson some thirty years ago and dubbed Reverse Mathematics.
The objective of the program is to investigate the role of set existence principles in ordinary mathematics. Roughly speaking, by ordinary mathematics we mean main-stream, non-set-theoretic mathematics, i.
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Very often, if a theorem of ordinary mathematics is proved from the weakest possible set existence axioms, the statement of that theorem will turn out to be provably equivalent to those axioms over a still weaker base theory. For exact definitions of all these systems and their role in reverse mathematics see Simpson To get a sense of scale, the strengths of the first four theories are best expressed via their proof-theoretic ordinals:. This will require the much stronger representation to be introduced in Definition 5. There are important precursors of reverse mathematics.
The first theorem of genuinely reverse mathematics was established by Dedekind in his essay Stetigkeit und irrationale Zahlen It states that his continuity or completeness principle is equivalent to a well-known theorem of analysis, namely, every bounded, monotonically increasing sequence has a limit. He emphasizes,. This theorem is equivalent to the principle of continuity, i.
However, it was not clear whether there was a constructive foundation of these functionals along the lines of hereditarily continuous functionals that can be represented by computable functions akin to Kleene ; Kreisel which would make them acceptable on intuitionistic grounds. One reason for this interest was surely that the intuitionistic versions corresponding to the accessible i. If one adds a new 1-place predicate symbol P to the language of arithmetic, one can describe the so-called positive arithmetical operators.
They are of the form. The topic of theories of iterated inductive definitions was flourishing at the conference on Intuitionism and Proof Theory in Buffalo see Kino et al. This was all the more desirable because of known reductions of important fragments of second order arithmetic to theories of the former kind. Therefore we intersperse a brief account of how to proceed onwards, adumbrating the main ideas. To obtain ordinal analyses of ever stronger theories one has to find new ways of defining ordinal representation systems that can encapsulate their strength.
The latter goes hand in hand with the development of new cut elimination techniques that are capable of removing cuts in infinitary proof systems with strong reflection rules. Ordinal representations, however, appear to pose a considerable barrier to understanding books and articles in this research area. However, such strong set-theoretic assumptions can be avoided.
The pattern of definition exhibited in Definition 5. Analogies between large set-theoretic ordinals cardinals and recursively large ordinals on the one hand and ordinal representation systems on the other hand can be a fruitful source of inspiration for devising new representation systems. More often than not, hierarchies and structural properties that have been investigated in set theory and recursion theory on ordinals turn out to have proof-theoretic counterparts.
Using an extended version of the representation system from Definition 5. A generalized treatment of theories of iterated inductive definitions for arbitrary well-orderings and of autonomous iteration was carried out in Rathjen , Indeed, the easiest way to build an extended ordinal representation system sufficient unto this task modeled on Definition 5.
The goal of giving an ordinal analysis of full second order arithmetic has not been attained yet. Feferman To get a sense for the enormous difference, it seems advisable to work in admissible set theory and consider a hierarchy of recursively large ordinal notions wherein these comprehension schemes correspond to the bottom and the top end of the scale, respectively. That is discussed in Appendix D. In particular, these theories prove the same theorems in the negative arithmetic fragment.
Proof theory has become a large subject with many specialized branches that can be mathematically quite complex. So we have tried to present developments close to the main artery of its body, starting with its inception at the beginning of the twentieth century and ending with results from the close of the century. Some additional contemporary proof theoretic developments are described in appendices D , E and F. First steps into ordinal analysis are taken in Pohlers Let us also report on progress on the methodological issues the finitist consistency program was to address.
First of all, due to quite a bit of important historical work, we have a much better grasp of the evolution of the program in the s and its roots in the development toward modern structuralist mathematics in the nineteenth century. Secondly, as to the properly methodological issues, we presented some broad considerations in section 4. We did not make any remarks about what is characteristic of a constructive perspective and why such a perspective has not only a mathematical, but also a philosophical point. There is, of course, a very rich literature.
Back to proof theory: We have to admit that we neglected some classical topics. They also do not cover proof systems for temporal and modal logic, neither are substructural logics presented.
Logic Colloquium: symposium on logic held at Boston, - Rohit Parikh - Google книги
A second omission concerns Bounded Arithmetic, where feasibility issues are a central focus: one studies formal theories with provably recursive functions that form very small subclasses of the primitive recursive ones. A third omission concerns proof mining; that line of deep mathematical investigations using proof theoretic tools was initiated by Kreisel , and Luckhardt , but really perfected only by Kohlenbach We hinted at the work of his school at the very end of section 4.
Proof theory, as we described it, deals primarily with formal proofs or derivations. Hilbert aimed, however, as we pointed out in section 1 , for a more general analysis of ordinary, informal mathematical proofs. The aim of Hilbert and his collaborators was undoubtedly to achieve a deeper mathematical and conceptual understanding, but also to find general methods of proof construction in formal calculi.
This is now being pursued in the very active area of using powerful computers for the interactive verification of proofs and programs as well as the fully automated search for proofs of mathematical theorems. It is clearly in the spirit of Hilbert who articulated matters in his second Hamburg talk of as follows:. The formula game … has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed.
Symposium on Logic held at Boston, 1972-73
These rules form a closed system that can be discovered and definitively stated. Then he continues with a provocative statement about the cognitive goal of proof theoretic investigations. The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds.
Hilbert [ ]. It is clear to us, and it was clear to Hilbert, that mathematical thinking does not proceed in the strictly regimented ways imposed by an austere formal theory. Though formal rigor is crucial, it is not sufficient to shape proofs intelligibly or to discover them efficiently, even in pure logic.