Wilfried Sieg

Iterated inductive definitions and subsystems of analysis : recent proof-theoretical studies

Inductive definitions and subsystems of analysis.- Proof theoretic equivalences between classical and constructive theories for analysis.- Inductive definitions, constructive ordinals, and normal derivations.- The ??+1-Rule.- Ordinal analysis of ID?.- Proof-theoretical analysis of ID? by the method of local predicativity.

Fragments of arithmetic

Abstract We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman.

Step by Recursive Step: Church's Analysis of Effective Calculability

Alonzo Churchs mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was “Churchs Thesis” put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Godels general recursiveness, not his own λ-definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Churchs distinctive contribution to the analysis of the informal notion of effective calculability . In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Godels recursiveness for his Thesis . In Section 5, I contrast Churchs analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekinds investigation of continuity.

Computer environments for proof construction

Abstract Does the presentation and use of the search space matter for complex problem solving tasks? We address these questions for the construction of proofs in sentential logic. Using a fully computerized logic course, we isolated crucial features of computer environments and assessed their relative pedagogical effectiveness. After being given a pretest for logical aptitude, students were divided into three matched groups, each of which used a distinct computerized environment to construct proofs. All students were presented with identical course material on sentential logic for approximately five weeks. Students completed more than one hundred exercises during those five weeks and took a midterm at the end of the period. The group using the most informative and most flexible interface performed substantially better on the midterm— the difference was particularly striking for hard problems. In two follow‐up experiments we added strategic problem solving help; student performance improved again (entirely...

Dedekind’s Analysis of Number: Systems and Axioms

In 1888 Hilbert made his Rundreise from Königsberg to other German university towns. He arrived in Berlin just as Dedekind’s Was sind und was sollen die Zahlen? had been published. Hilbert reports that in mathematical circles everyone, young and old, talked about Dedekind’s essay, but mostly in an opposing or even hostile sense. A year earlier, Helmholtz and Kronecker had published articles on the concept of number in a Festschrift for Eduard Zeller. When reading those essays in parallel to Dedekind’s and assuming that they reflect accurately more standard contemporaneous views, it is easy to understand how difficult it must have been to grasp and appreciate Dedekind’s remarkably novel and thoroughly abstract approach. This is true even for people sympathetic with Dedekind’s ways. Consider, for example, the remark Frobenius made in a letter of 23 December 1893 to Dedekind’s collaborator and friend Heinrich Weber who was planning to write a book on algebra:

Church Without Dogma: Axioms for Computability

Churchs and Turings theses dogmatically assert that an informal notion of computability is captured by a particular mathematical concept. I present an analysis of computability that leads to precise concepts, but dispenses with theses. To investigate computability is to analyze processes that can in principle be carried out by calculators. Drawing on this lesson we owe to Turing and recasting work of Gandy, I formulate finiteness and locality conditions for two types of calculators, human computing agents and mechanical computing devices; the distinctive feature of the latter is that they can operate in parallel. The analysis leads to axioms for discrete dynamical systems (representing human and machine computations) and allows the reduction of models of these axioms to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for machine computations.

The AProS Project: Strategic Thinking & Computational Logic

The paper discusses tools for teaching logic used in Logic & Proofs, a web-based introduction to modern logic that has been taken by more than 1,300 students since the fall of 2003. The tools include a wide array of interactive learning environments or cognitive mini-tutors; most important among them is the Carnegie Proof Lab. The Proof Lab is a sophisticated interface for constructing natural deduction proofs and is central, as strategically guided discovery of proofs is the distinctive focus of the course. My discussion makes explicit the broader intellectual context, but also the pursuit of pedagogical goals and their experimental examination. The intellectual context includes i) the theoretical background for the proof search algorithm AProS and its use for a dynamic Proof Tutor, and ii) the programmatic expansion of the course to Computational Logic. (I recommend that the reader enter the virtual classroom of Logic & Proofs: the interactive components just cannot be properly reflected in a narrative. It is also very easy to download AProS and observe its ways of searching for proofs.) 1

Only two letters: The correspondence between Herbrand and Gödel

Two young logicians, whose work had a dramatic impact on the direction of logic, exchanged two letters in early 1931. Jacques Herbrand initiated the correspondence on 7 April and Kurt Godel responded on 25 July, just two days before Herbrand died in a mountaineering accident at La Berarde (Isere).1 Herbrands letter played a significant role in the development of computability theory. Godel asserted in his 1934 Princeton Lectures and on later occasions that it suggested to him a crucial part of the definition of a general recursive function. Understanding this role in detail is of great interest as the notion is absolutely central. The full text of the letter had not been available until recently, and its content (as reported by Godel) was not in accord with Herbrands contemporaneous published work. Together, the letters reflect broader intellectual currents of the time: they are intimately linked to the discussion of the incompleteness theorems and their potential impact on Hilberts

Foundations for analysis and proof theory

The title of my paper indicates that I plan to write about foundations for analysis and about proof theory; however, I do not intend to write about the foundations for analysis and thus not about analysis viewed from the vantage point of any “school” in the philosophy of mathematics. Rather, I shall report on some mathematical and proof-theoretic investigations which provide material for (philosophical) reflection. These investigations concern the informal mathematical theory of the continuum, on the one hand, and formal systems in which parts of the informal theory can be developed, on the other. The proof-theoretic results of greatest interest for my purposes are of the following form: for each F in a class of sentences, F is provable in T if and only if F is provable in T* where T is a classical set-theoretic system for analysis and T* a constructive theory. In that case, T is called REDUCIBLE TO T*, as the principles of T* are more elementary and more restricted.

Beyond Hilbert’s Reach?

Work in the foundations of mathematics should provide systematic frameworks for important parts of the practice of mathematics, and the frameworks should be grounded in conceptual analyses that reflect central aspects of mathematical experience. The Hilbert School of the 1920s used suitable frameworks to formalize (parts of) mathematics and provided conceptual analyses. However, its analyses were mostly restricted to finitist mathematics, the programmatic basis for proving the consistency of frameworks and, thus, their instrumental usefulness. Is the broader foundational quest beyond Hilbert’s reach? The answer to this question seems simple: “Yes & No”. It is “Yes”, if we focus exclusively on Hilbert’s finitism; it is “No”, if we take into account the more sweeping scope of Hilbert and Bernays’s foundational thinking. The evident limitations of Hilbert’s “formalism” have been pointed out all too frequently; in contrast, I will trace connections of Hilbert’s work, beginning in the late 19th century, to contemporary work in mathematical logic. Bernays’s reflective philosophical investigations play a significant role in reinforcing these connections. My paper pursues two complementary goals, namely, to describe a global, integrating perspective for foundational work and to formulate some more local, focused problems for mathematical work.