$\begingroup$

Incompleteness, Undecidability, Independence and New Axioms

One major contemporary question in the philosophy of mathematics is "Does (and if so, to what extent) mathematics need new axioms?" This question falls directly out of the work done on undecidability and incompleteness. While Gödel's incompleteness theorems, Turing's theorem of the undecidability of the halting problem, the MRDP theorem, etc. give general examples of incompleteness that can never be "effectively" removed, due to the fact that the theorems are essentially recipes to create undecidable sentences in any given theory that satisfies certain criteria, there are still other statements that are undecidable in certain theories but for which there is hope those statements could be solved by more powerful axioms.

Gödel recognized (see "Remarks before the Princeton Bicentennial Conference on Problems in Mathematics" and "What is Cantor's continuum problem?") that some problems independent of the ZFC axioms of set theory might be able to be resolved on the basis of stronger axioms of infinity and the resulting research effort has been called Gödel's large cardinal program (see Steel "Gödel's Program"). A major portion of contemporary work in set theory is a result of the study of large cardinals. You can find two philosophical discussions of the results (and their limitations in regards to questions such as the continuum hypothesis) by Peter Koellner at Independence and Large Cardinals and Large Cardinals and Determinacy.

This topic has been debated and written about by many mathematical logicians, philosophers of mathematics, as well as contemporary set theorists who do not consider their work to be strictly within the field of logic any longer. An important piece of the debate is contained within the article Does Mathematics Need New Axioms?, which is a collection of four essays and rebuttals on this topic written by Solomon Feferman, Penelope Maddy, John Steel, and Harvey Friedman and presented at the Annual ASL meeting in 2000. From the abstract:

Does mathematics need new axioms? was the second of three plenary panel discussions held at the ASL annual meeting, ASL 2000, in Urbana- Champaign, in June, 2000. Each panelist in turn presented brief opening remarks, followed by a second round for responding to what the others had said; the session concluded with a lively discussion from the floor. The four articles collected here represent reworked and expanded versions of the first two parts of those proceedings, presented in the same order as the speakers appeared at the original panel discussion: Solomon Feferman (pp. 401-413), Penelope Maddy (pp. 413-422), John Steel (pp. 422-433), and Harvey Friedman (pp. 434-446). The work of each author is printed separately, with separate references, but the portions consisting of comments on and replies to others are clearly marked.

Feferman is ultimately skeptical of the program, he believes that there is no hard, pure mathematical reason for new axioms and that therefore the question is only philosophical in nature and thus has no definitive answer. More on Feferman's position (especially on his attitude that the continuum hypothesis is not a definite mathematical problem and is inherently vague) will be referenced further down.

Maddy argues from her naturalist philosophy of mathematics (cf. her book Naturalism in Mathematics for an overview and then especially Maddy's two powerful papers "Believing the Axioms I & II" on the topic of the nature of axioms), that extrinsic justifications for new axioms are not only viable but essential to the development of foundational mathematics.

Steel is of the position that there are still major open questions in set theory that have not been resolved and therefore Gödel's program cannot be shown to be defeated. He argues that there are still outstanding problems in descriptive set theory, as well as problems like the continuum hypothesis, that we have proven independent of ZFC and therefore, of course, we need new axioms to settle them.

And finally, Friedman has taken it upon himself to launch a program in mathematical logic which searches for examples of set theoretic undecidability in finite combinatorics (much more on this later). The justification of needing large cardinal axioms to prove finite combinatorial statements, he argues, is a fairly strong philosophical argument, given that our natural inclination is that these finite statements are indeed true (as in, nobody can doubt them based off of skepticism about infinity). His view can be seen as saying that there are major applications of large cardinals to concrete mathematics, and therefore we cannot throw them aside as purely philosophical and non consequential.

Additionally, in 2011-2012 Peter Koellner organized a project at Harvard titled Exploring the Frontiers of Incompleteness. The goal was:

to bring together some of the most prominent thinkers who have struggled with the following questions: (1) Do the questions that are independent of the standard axioms admit of determinate answers? (2) If so then what are those answers and how might we go about determining them?

This subject immediately touches on the pluralism vs. non-pluralism debate that Joel Hamkins has outlined in his answer (of course, he is also a major player in the debate and was part of the project). The multimedia page for the project contains links to preprints, slides, and videos of the lectures of the talks given by each of the participants. Some of the highlights that I think you should consider when going over this topic are:

Hugh Woodin's two papers The Realm of the Infinite and Strong Axioms of Infinity and the search for V. Woodin's work on this area of set theory has shown that in a strong sense, if there is an L like model of a supercompact cardinal, then in the strong sense there is an L like model of every known to be consistent large cardinal. This would give the result that it is possible to form an ultimate $V = L$ like axiom that would resolve many if not all outstanding independence problems, including the continuum hypothesis. There is much in the literature on this topic, I would suggest reading Koellner's response that was included in the project, Woodin on “The Realm of the Infinite”, as well as Colin Rittberg's How Woodin changed his mind: new thoughts on the Continuum Hypothesis.

Feferman was also present and he touched on the same topics as his paper outlined earlier. The highlights from this project are Is the Continuum Hypothesis a definite mathematical problem?, Infinity in mathematics, is Cantor necessary?, The philosophy of mathematics. 5 questions, and What's definite? What's not? as well as Koellner's response Feferman on the Indefiniteness of CH.

Again, Joel Hamkins has provided an answer which outlines his stance in the pluralism vs. non pluralism debate and I agree with his suggestion to follow the links he provided. Here is another link to his The set-theoretic multiverse paper, and Koellner's response from the project Hamkins on the Multiverse.

Charles Parsons outlines the impact that the study of the consistency and relative interpretability hierarchies of formal theories has had on these foundational questions in his paper Evidence and the hierarchy of mathematical theories.

Philip Welch's paper Global Reflection Principles and Koellner's response On Reflection Principles outline the application of reflection principles to the foundation of mathematics and the search for new axioms. To quote the abstract of Koellner's response: "Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified."

Finally the last of my highlights is once again John Steel's work. His four slides The Triple Helix, Gödel's program, Gödel's Legacy, and Generic Absoluteness and the Continuum Problem (I won't link them because they are automatic downloads instead of being hosted on Harvard's server, but they are available on the webpage under his name) give great historical overview of Gödel's large cardinal program, its prominence in contemporary set theory and its lasting philosophical impact.

Finally, I would like to mention a few more points about Harvey Friedman's work. His section of the Does Mathematics Need New Axioms? paper was written during a time when a lot of his work on this subject was first being started. In the almost two decades since, his work has exploded with a plethora of examples of concrete independence of combinatorial statements from ZFC. He wrote an online book titled Boolean Relation Theory, which explains all of the fundamental results of this work. Now, he is currently expanding it into a larger book called Concrete Mathematical Incompleteness which will also discuss the place of incompleteness within logic, mathematics, and philosophy. He wrote an paper for the collection Kurt Gödel and the Foundations of Mathematics Horizons of Truth titled My Forty Years on his Shoulders where he discusses the nature of incompleteness and the impact that Gödel's work has had on mathematics as a whole. In the ending section titled "Incompleteness in the Future" he states:

Mathematics as a professional activity with serious numbers of workers is quite new, let’s say, one hundred years old, although even that is a stretch. Assuming that the human race thrives, what is this compared to, say, a thousand more years? It is probably merely a bunch of simple observations in comparison. Of course, a thousand years is absolutely nothing in evolutionary or geological time. A more reasonable number is a million years. What does our present mathematics look like compared with mathematics in a million years’ time? These considerations should apply to our present understanding of the Gödel phenomena. We can, of course, take this even further. A million years’ time is absolutely nothing in astronomical time. Our sun has several billion good years left (although the sun will cause a lot of global warming). Mathematics in a billion years’ time? Who can know what that will be like. However, I am convinced that the Gödel legacy will remain very much alive – at least as long as there is vibrant mathematical activity.

I believe this is a beautiful sentiment which places the concepts of incompleteness, undecidability, independence and new axioms of set theory within their copacetic position in the philosophy of mathematics.