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Axioms as Definitions: Revisiting Poincaré and Hilbert
Abstract
A fondamental problem in the discussion on the foundations of mathematics is to clarify what an axiom is. This is especially important in the light of the most recent advances in set theory where new axioms have been proposed whose legitimacy is highly controversial (for example, large cardinal axioms); this paper is a contribution to this discussion. By analysing the view of Poincaré and Hilbert on axioms, we observe that, despite the deep differences in their philosophical thinking, the two logicians came to the same conception of the axioms of geometry as definitions in disguise. We revisit and generalise this view by arguing that any axiomatic system (set theory in particular) is the definition of some concepts.
I shall argue that the commonly held V not equal L via maximize position,
which rejects the axiom of constructibility V = L on the basis that it is
restrictive, implicitly takes a stand in the pluralist debate in the philosophy
of set theory by presuming an absolute background concept of ordinal. The
argument appears to lose its force, in contrast, on an upwardly extensible
concept of set, in light of the various facts showing that models of set theory
generally have extensions to models of V = L inside larger set-theoretic
universes.
The multiverse view in set theory, introduced and argued for in this article,
is the view that there are many distinct concepts of set, each instantiated in
a corresponding set-theoretic universe. The universe view, in contrast, asserts
that there is an absolute background set concept, with a corresponding absolute
set-theoretic universe in which every set-theoretic question has a definite
answer. The multiverse position, I argue, explains our experience with the
enormous diversity of set-theoretic possibilities, a phenomenon that challenges
the universe view. In particular, I argue that the continuum hypothesis is
settled on the multiverse view by our extensive knowledge about how it behaves
in the multiverse, and as a result it can no longer be settled in the manner
formerly hoped for.
§0. Introduction. Ask a beginning philosophy of mathematics student why we believe the theorems of mathematics and you are likely to hear, “because we have proofs!” The more sophisticated might add that those proofs are based on true axioms, and that our rules of inference preserve truth. The next question, naturally, is why we believe the axioms, and here the response will usually be that they are “obvious”, or “self-evident”, that to deny them is “to contradict oneself” or “to commit a crime against the intellect”. Again, the more sophisticated might prefer to say that the axioms are “laws of logic” or “implicit definitions” or “conceptual truths” or some such thing.
Unfortunately, heartwarming answers along these lines are no longer tenable (if they ever were). On the one hand, assumptions once thought to be self-evident have turned out to be debatable, like the law of the excluded middle, or outright false, like the idea that every property determines a set. Conversely, the axiomatization of set theory has led to the consideration of axiom candidates that no one finds obvious, not even their staunchest supporters. In such cases, we find the methodology has more in common with the natural scientist's hypotheses formation and testing than the caricature of the mathematician writing down a few obvious truths and preceeding to draw logical consequences.
The central problem in the philosophy of natural science is when and why the sorts of facts scientists cite as evidence really are evidence. The same is true in the case of mathematics. Historically, philosophers have given considerable attention to the question of when and why various forms of logical inference are truth-preserving. The companion question of when and why the assumption of various axioms is justified has received less attention, perhaps because versions of the “self-evidence” view live on, and perhaps because of a complacent if-thenism.
Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own development of the meta-mathematics of weak systems of arithmetic to show that the true philosophical significance of Hilbert's program is that it makes the autonomy of mathematics evident. The result is a vision of the early history of modern logic that highlights the rich interaction between its conceptual problems and technical development.
[t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is greater than א0, conjectured that it is א1. An equivalent proposition is this: any infinite subset of the continuum has the power either of the set of integers or of the whole continuum. This is Cantor’s continuum hypothesis. … But, although Cantor’s set theory has now had a development of more than sixty years and the [continuum] problem is evidently of great importance for it, nothing has been proved so far relative to the question of what the power of the continuum is or whether its subsets satisfy the condition just stated, except that … it is true for a certain infinitesimal fraction of these subsets, [namely] the analytic sets. Not even an upper bound, however high, can be assigned for the power of the continuum. It is undecided whether this number is regular or singular, accessible or inaccessible, and (except for König’s negative result) what its character of cofinality is. Gödel 1947, 516-517 [in Gödel 1990, 178]
This is a continuation of Believing the axioms . I, in which nondemonstrative arguments for and against the axioms of ZFC, the continuum hypothesis, small large cardinals and measurable cardinals were discussed. I turn now to determinacy hypotheses and large large cardinals, and conclude with some philosophical remarks.
Determinacy is a property of sets of reals. If A is such a set, we imagine an infinite game G ( A ) between two players I and II. The players take turns choosing natural numbers. In the end, they have generated a real number r (actually a member of the Baire space ω ω ). If r is in A , I wins; otherwise, II wins. The set A is said to be determined if one player or the other has a winning strategy (that is, a function from finite sequences of natural numbers to natural numbers that guarantees the player a win if he uses it to decide his moves).
Determinacy is a “regularity” property (see Martin [1977, p. 807]), a property of well-behaved sets, that implies the more familiar regularity properties like Lebesgue measurability, the Baire property (see Mycielski [1964] and [1966], and Mycielski and Swierczkowski [1964]), and the perfect subset property (Davis [1964]). Infinitary games were first considered by the Polish descriptive set theorists Mazur and Banach in the mid-30s; Gale and Stewart [1953] introduced them into the literature, proving that open sets are determined and that the axiom of choice can be used to construct an undetermined set.
this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.
Leipzig: Teubner, English translation in Foundations of Geometry
- David Hilbert
Hilbert, David [1899], Grundlagen der Geometrie, Leipzig: Teubner, English
translation in Foundations of Geometry, L. Unger (ed.), La Salle: Open
Court Press, 1971.
An essentially undecidable axiom system
- Raphael M Robinson
Robinson, Raphael M. [1950], An essentially undecidable axiom system, in:
Proceedings of the International Congress of Mathematics, 729-730.
Implicit definitions and abstraction, for Abstraction Day 99
- Stewart Shapiro
Shapiro, Stewart [1999], Implicit definitions and abstraction, for Abstraction
Day 99, University of St Andrews.