Is the Continuum Hypothesis true or false? Michael Stob Calvin College

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1 Is the Continuum Hypothesis true or false? Michael Stob Calvin College

2 Answers: Panel at 2005 AMS Meeting Paul Cohen (Stanford) Hugh Woodin (Berkeley) Tony Martin (UCLA) no yes, false I have no idea

3 Continuum Hypothesis Question: How large is the set of real numbers?

4 Different sizes of sets Two sets are of the same size if there is a one-to-one correspondence between their elements

5 Cantor knew: 1. N is the smallest infinite set (every infinite subset, such as {2, 3, 5,... } can be put in one-to-one correspondence with N), 2. Q has the same size as N, but 3. R is larger than N (diagonal argument)

6 Cantor s Continuum Hypothesis Every infinite set of real numbers X is either of the same size as N (ℵ 0 ) or of the same size as R (2 ℵ 0 ) There is no infinite number intermediate between N and R.

7 The axioms for Set Theory Adopted over the first third of the twentieth-century Zermelo-Frankel Axiom of Choice ZF AC The theorems of ZFC are the theorems of set theory.

8 The Problem Gödel (1940): CH is consistent with ZFC. Cohen (1966): The negation of CH is consistent with ZFC. An instance of Gödel s Incompleteness Phenomenon

9 Can CH be decided? Possible answers: Set theory needs new axioms ZFC exhausts our knowledge/intuition about infinity so that CH is inherently vague We are free to adopt both CH and not-ch (though not at the same time) there is an obvious formalist reading of this strategy but there is a realist reading - plentitudinous Platonism

10 Sets of Reals Open intervals. Every open interval (a, b) of reals has cardinality 2 ℵ 0. Closed sets. A closed set is either countable or has a perfect subset. Every perfect set has cardinality 2 ℵ 0. Borel sets. Every Borel set is either countable or has a perfect subset.

11 Projective sets of reals A set X of reals is projective if it can be defined in a reasonable language for defining sets of reals. Definitions of sets of reals: Q: {x n m((mx = n) (m N) (n N) (m 0))} (a, b): {x (a < x < b)}

12 Projective sets include all sets that arise in everyday mathematics except those pathological sets constructed by the axiom of choice. ZFC cannot prove or refute that every projective set is either uncountable or has a perfect subset. (Alternate definition of projective sets: the smallest collection of sets containing the open sets and closed under complementation and continuous images.)

13 Evaluation of new axioms 1. Intrinsic justification. Appeal to our intuition about or observation of sets. Self-evidence Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory as is seen from the fact that the axioms force themselves upon us as being true. I don t see any reason why we should have less confidence in this kind of perception, i.e; in mathematical intuition

14 2. Extrinsic justification. (List due to P. Maddy) confirmation by instances (consequences known) prediction (consequences later proved) new proofs of old theorems extending patterns unifying new results with old (old results special cases) proofs of statements previously conjectured explanatory power fills in a gap in a false but natural proof intertheoretic connections

15 There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole discipline, and furnishing such powerful methods for solving problems (and even solving them, as far as that is possible, in a constructive way) that quite irrespective of their intrinsic necessity they would have to be assumed in the same way as any well-established physical theory. Gödel (1947)

16 Secondly, however, even disregarding the intrinsic necessity of some new axiom, and even in case it has no intrinsic necessity at all, a probable decision about its truth is possible also in another way, namely inductively by studying its success. Gödel (1947)

17 New Axioms for Set Theory Large cardinals. (big numbers not portly prelates) Determinacy.

18 Axioms of Determinacy Two person game: Player I and Player II alternate playing the digits of a decimal number. Player I Player II A is a predetermined set of real numbers. Player I wins the game if the number created, is in the set A.

19 The Axiom of Determinacy A set A of real numbers is determined if one or the other of the two players has a winning strategy. Axiom (Axiom of Determinacy). Every set of real numbers is determined. The Axiom of Determinacy contradicts the Axiom of Choice and so is (today) false.

20 True instances of determinacy Every finite set is determined (and a win for Player II) Every open interval is determined (for one or the other Player)

21 Projective Determinacy Axiom (PD). Every projective set of real numbers is determined. PD is not provable from ZFC. There is no intrinsic justification for accepting PD. It is not even provable that if ZFC is consistent then so is ZFC+PD Yet, many set theorists consider PD to be true.

22 Consequences of Projective Determinacy 1. Every projective set is Lebesque-measurable 2. Every projective set is either countable or has a perfect subset. 3. There is no projective counterexample to the Continuum Hypothesis. 4. All reasonable questions about projective sets are settled. 5. Projective determinacy is implied by certain large cardinal axioms. 6. By translation to P(N), second-order number theory is determined.

23 The argument that CH may yet be solved 1. ZFC does not settle many natural questions about projective sets of reals. 2. PD settles all natural questions about the projective sets and in the right way. 3. This is (extrinsic) evidence for accepting PD. 4. Therefore it is plausible that there are axioms that we might accept that further determine what is true about sets of reals.

24 Woodin s program Hugh Woodin has a program that is the next step after studying the projective sets that could yield axioms showing that CH is false.

This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },