The final two volumes, numbers IV and V, of the Oxford University Press edition of the Collected Works of Kurt Gödel appeared in 2003, thus completing a project that started over twenty years earlier. What I mainly want to do here is trace, from the vantage point of my personal involvement, the at some times halting and at other times intense development of the Gödel editorial project from the first initiatives following Gödel's death in 1978 to its completion last year. It may be useful to scholars mounting similar editorial projects for other significant figures in our field to learn how and why various decisions were made and how the work was carried out, though of course much is particular to who and what we were dealing with.

My hope here is also to give the reader who is not already familiar with the Gödel Works a sense of what has been gained in the process, and to encourage dipping in according to interest. Given the absolute importance of Gödel for mathematical logic, students should also be pointed to these important source materials to experience first hand the exercise of his genius and the varied ways of his thought and to see how scholarly and critical studies help to expand their significance.

Though indeed much has been gained in our work there is still much that can and should be done; besides some indications below, for that the reader is referred to.

The incompleteness theorems show that for every sufficiently strong consistent formal system of mathematics there are mathematical statements undecided relative to the system. A natural and intriguing question is whether there are mathematical statements that are in some sense absolutely undecidable, that is, undecidable relative to any set of axioms that are justified. Gödel was quick to point out that his original incompleteness theorems did not produce instances of absolute undecidability and hence did not undermine Hilbert's conviction that for every precisely formulated mathematical question there is a definite and discoverable answer. However, in his subsequent work in set theory, Gödel uncovered what he initially regarded as a plausible candidate for an absolutely undecidable statement. Furthermore, he expressed the hope that one might actually prove this. Eventually he came to reject this view and, moving to the other extreme, expressed the hope that there might be a generalized completeness theorem according to which there are no absolutely undecidable sentences.

In this paper I would like to bring the question of absolute undecidability into sharper relief by bringing results in contemporary set theory to bear on it. The question is intimately connected with the nature of reason and the justification of new axioms and this is why it seems elusive and difficult. It is much easier to show that a statement is not absolutely undecidable than to show either that a statement is absolutely undecidable or that there are no absolutely undecidable statements.

The concept of backtracking was introduced in [13]. Backtracking is a procedure that takes a non-λk(ξ)-free node of Tk, and returns it to λk(η)-free status for some η ⊃ ξ, with the same initial derivative. The concept, as manifested in [13], was intertwined with the validity of certain sentences, and so its analysis was substantially more complicated than we need, as we look at it without regard to the requirements assigned to the nodes. The lemmas we prove here can also be generally applied to systems of trees, as there is no dependence on the requirements to be satisfied.

This chapter is devoted to proving lemmas describing the behavior of the backtracking process. Throughout the chapter, we assume that we are dealing with a level ℓ construction.

Backtracking

Backtracking cannot always be carried out. In this section, we identify the sole obstruction to backtracking, and define a sequence generated by primary links that determines how backtracking is to be implemented, when possible.

Definition 9.1.1. Fix m ≤ ℓ, σ, ξ ∈ T0, and σm ∈ Tm. We say that 〈σm, σ〉 is weakly ξ-accessible if init(σm, σ) = σ and there is an η ∈ T0 such that η ⊇ ξ, init(σm, η) = σ and σm ⊆ λm(η); and if this is the case, we call η a witness to the weak ξ-accessibility of 〈σm, σ〉. We say that, 〈σm, σ〉 is ξ-accessible if there is an η ∈ T0 such that η is a witness to the weak ξ-accessibility of 〈σm, σ〉 and upm(η) = σm.

We will prove that 〈σm, σ〉, is weakly ξ-accessible, then iff there is no primary ξ-link restraining σ.

Kurt Gödel is almost as famous—one might say “notorious”—for his extreme platonist views as he is famous for his mathematical theorems. Moreover his platonism is not a myth; it is well-documented in his writings. Here are two platonist declarations about set theory, the first from his paper about Bertrand Russell and the second from the revised version of his paper on the Continuum Hypotheses.

The first statement is a platonist declaration of a fairly standard sort concerning set theory. What is unusual in it is the inclusion of concepts among the objects of mathematics.

The year 2006 marked the centennial of Kurt Gödel, who was born on 28 April 1906. The importance of Gödel's work for nearly all areas of logic and foundations of mathematics hardly needs to be explained to our readers.

The year 2006 saw several centennial observances. In particular, the program committee for the 2006 Association for Symbolic Logic annual meeting, which took place on 17–21 May at the Université du Québec à Montréal, commissioned a subcommittee to arrange a portion of the program that would commemorate the Gödel centennial. The subcommittee arranged three one-hour lectures, by Jeremy Avigad, Sy-David Friedman, and Akihiro Kanamori. It also arranged a two-hour special session on Gödel's philosophy of mathematics, with lectures by Steve Awodey, John Burgess, and William Tait. All of the lectures have led to papers in this volume. The volume contains one other new paper, “The Gödel hierarchy and reverse mathematics,” by Stephen G. Simpson. Other papers included in the volume are reprinted, in all but one case from The Bulletin of Symbolic Logic. We have included the papers presented at the 2004 ASL annual meeting at Carnegie-Mellon University, in a special session organized by the editors of Gödel's Collected Works, by Martin Davis, John W. Dawson, Jr., Cheryl A. Dawson, Solomon Feferman, Warren Goldfarb, Donald A. Martin, Wilfried Sieg, and William Tait. These appeared in the June 2005 Bulletin. Also reprinted are papers by Mark van Atten and Juliette Kennedy and by Charles Parsons that appeared in earlier issues of the Bulletin, as well as a paper by Peter Koellner that appeared in Philosophia Mathematica.

We now turn our attention to theorems whose proofs use Π2 constructions. We begin, in Section 5.1, by stating lemmas that isolate combinatorial properties of Π2 constructions. We then implement Π2 constructions to prove the existence of a high computably enumerable degree in Section 5.2, the Sacks Jump Inversion Theorem in Section 5.3, the Minimal Pair Theorem in Section 5.4 and an embedding of the pentagon into the computably enumerable degrees in Section 5.5.

Π2 Constructions

Requirements such as the thickness requirements discussed in Chapter 2 act to construct functionals that are total on given oracles, so require the declaration of infinitely many axioms. Such requirements cannot be handled by level 1 constructions, as a node of T1 will generally have the responsibility to declare only finitely many axioms for such functionals. This will introduce a coordination problem, as nodes having the responsibility to declare axioms on the same argument will appear on incomparable paths through T1.

Some requirements of the form Φ(A) = W can be handled by level 2 constructions. These constructions capture what are traditionally called infinite injury constructions. For those familiar with infinite injury constructions, we describe how these constructions are simulated by level 2 constructions.

A typical infinite injury construction will assign requirements to nodes of a tree T2, that can always be taken to be a binary tree. The nodes of the tree are prioritized, using the lexicographical ordering of these nodes. A computation of the current path through T2 is made at each stage of the construction, and action is carried out for nodes that lie along the current path.

“I once asked Carnap with which philosopher of the past he felt a close kinship,” writes Abraham Kaplan in a memoir. “His reply was Leibniz” (Kaplan [1991], p. 40). This feeling of kinship with Leibniz was something Gödel and Carnap had in common, and it distinguished them from the other members of the Vienna Circle and its following, who had less rationalist inclinations. However, Carnap and Gödel were in good company when we look more widely at the founders of modern logic and scientifically-oriented philosophy. Frege, Cantor, and Russell, among others, would likely also have named Leibniz if asked the same question. One obvious source of fascination for all these thinkers was Leibniz's idea of a universal deductive system, a calculus philosophicus or characteristica universalis, to connect all knowledge in a single, uniform deductive structure. The various efforts during the Enlightenment to classify and codify the new knowledge, in competition with theological and traditional lore, had overlooked or forgotten Leibniz's proposal, as had the subsequent positivist tradition. It was only with the more powerful logical tools developed in the later nineteenth century that Leibniz's dreams once again became relevant.

Of course these latter-day admirers of Leibniz diverged widely in their attitude toward this universal logical program. They diverged, for instance, along the left-right scale or “schema of possible philosophical world-views” that Gödel set up in a lecture written around 1961:

VB Quip

The world runs on databases. The modern world relies on them and would be lost without them. Business depends on them, from storing employee and payroll data, to customer data, supplies, manufacturing, transportation, and inventory – you name it. It seems like everything these days is on the computer. In fact, if it's not on the computer, it's probably old and outdated. The key then, is being able to access these data, manipulate them, update them, search them, sort them, and make them work for you. That's where LINQ to SQL comes in. It's the connection between your VB program and a database. In this case, it's an Access file, but the same principles hold true for access to almost any type of database. You'll get the basics to connect to a database, view it, update it, and use it. It takes several programming classes and a couple of database classes to get good at it, but here you'll get a taste of it and some general principles to get you started.

Background

Databases come in several forms, but by far the most popular database management system (DBMS) is a relational database.

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By now you've come to realize that the real bottlenecks of your programs are input and output. Some of that is because users can only type and click so much and some of it is because input is repetitive. You've created useful output but have been unable to hang on to it. So far, every program run has meant starting from scratch with your data, entering it as needed, calculating the results, and then losing all of that work with a click of the Exit Button. No more. File input and output gives you the ability to store data and retrieve it as needed. File input takes data stored in a file and puts it into your program. File output stores the results in a file so you can use the data at a later date. This chapter covers the details of opening a data file, reading the data, processing the data, displaying the output, and writing the results to a file.

The Basics of File Input and Output

A data file, for our purposes, is a text file. Each line in the file is a record. A record consists for one or more fields.

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Think of objects as tools. Better yet, think of them as building blocks. Imagine how hard a job would be if you had to create everything from scratch every time. Say you wanted to build a table. First, you'd need to make an axe to cut down the tree. Then you'd have to make a saw to cut the wood. You'd have to make your own nails to piece together the table, but that's only after you made your own hammer so you could pound the nails. Programming is like that sometimes – OK, usually. And it was certainly that way years ago. Everything was done from scratch and very little was reusable. However, there's been some progress since then. You saw reusable code when you learned procedures and functions. Objects and classes are that way, but on a grander scale. The controls in the Toolbox are objects, each specially created for a specific task. They're tools and, without them, development would be much tougher, and a whole lot less fun. Classes are code that defines an abstract data type – it's a thing, and the developer provides a description of it and determines how it can be used. This chapter shows you how to create objects with your code and how to define and use classes.

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We live in a world of specialization. Want a cup of coffee? There's a store that specializes in coffee, any way you like it. Need a doughnut to go with it? There's a company that makes and sells the best doughnuts. Need something to read while you're enjoying a bite? Pick from thousands of magazines or millions of books on any and all topics imaginable. Gone are the days of the local handyman – the resident “Mr. Fixit” – who could fix anything and get anything to work. Today's world has specialists. A mechanic specializes in car repair. The cable guy handles your TV problems. When the office copier is on the fritz, call the copier guy. Everyone's an expert.

Programming is the same way. Blocks of code are written for a particular purpose. Some do simple tasks, but do them often. It's easier to write a procedure once for a task and then call it multiple times as needed. Other blocks of code might do a complicated calculation or solve a particular problem. These provide a specific function. It's easier to write and store these functions and then use them when needed than it is to create them from scratch. That's the basis for procedures and functions.

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On a computer, memory is everything, or nearly so. Data are stored there and programs are loaded and run from memory. Memory is essential. Compared to what was available a few years ago, today's computers have a huge amount of memory. A few years ago that same comparison would have been true as well. Forty years ago we sent men to the moon with spaceships that had less memory than a good pocket calculator has today. All computers, old and new, have memory in common. Memory is where what the computer “knows” is stored. Much of programming comes down to allocating and managing memory. For the PCs of a generation ago, memory was a scarce and expensive commodity to be guarded closely and used judiciously. Programmers used it carefully and squeezed as much as they could from it. At the time, conserving memory gave you a leg up on development and helped make your programs faster and more efficient. And, today, while the amount of memory isn't as much of an issue, the management of memory is. When you work on your computer, you work with its “desktop.” These days, it's the size of a table. In the early days of programming, that “desktop” was the size of a postage stamp. Variables and constants are two important components of a computer program. Both are created in memory and are assigned values.

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“A picture is worth a thousand words,” or so said Frederick R. Barnard nearly ninety years ago. He was selling advertising for streetcars and couldn't have had computer graphics in mind. In fact, the first computer was still a generation away. But, he was pretty accurate because it just might take a thousand words of computer code to generate a good picture. Remember that the computer takes baby steps and every step has to be clearly described. The same is true with graphics. You can place pictures on your form and you can create graphics – beautiful graphics if you take the time – but you have to write some code to do it. Visual Basic can draw lines and shapes, it can work with text, and it's great with colors. It just takes a little work to create them and a little code to do it right. This chapter introduces the basics for graphics, sound, and multimedia. When it's over, you'll be able to play sound and video, create simple shapes, and draw and paint with the computer.

Graphic Basics

There are several ways to add graphics to a program. You've already worked with the PictureBox to add graphics. But there's so much more.