Introduction Preliminaries Proof that ¬ CH What are some of the takeaways References Forcing and Relativization, Side-by-Side Derek Kern Department of Computer Science University of Colorado at Denver derek.kern@ucdenver.edu December 16, 2012 Kern, Derek Forcing and Relativization, Side-by-Side
Introduction Preliminaries Proof that ¬ CH What are some of the takeaways References My poor choice of title I know it is a misnomer. However, it has a euphonious quality that I found too tempting. For the record, both of the proofs in this work use both relativization and forcing. 1 1 Actually, Baker-Gill-Solovay’s proof doesn’t use forcing as it is defined in set theory. Kern, Derek Forcing and Relativization, Side-by-Side
Introduction Preliminaries Proof that ¬ CH What are some of the takeaways References Quick description of the continuum hypothesis (CH) ◮ Cantor established that there an infinity of infinities. ◮ Within this ‘hierarchy’ of infinities, the set of natural numbers has ℵ 0 members, the set of real numbers has 2 ℵ 0 members, the powerset of real numbers has 2 2 ℵ 0 members, etc. ◮ The continuum hypothesis (CH) is the claim that there are no infinities between ℵ 0 and 2 ℵ 0 . ◮ The generalized continuum hypothesis (GCH) is the claim that there are no infinities between an infinite set S and its powerset 2 S . 2 2 If GCH is true, then is the hierarchy of infinities countable or not? Kern, Derek Forcing and Relativization, Side-by-Side
Introduction Preliminaries Proof that ¬ CH What are some of the takeaways References Quick description of the relativized P ?= NP question ◮ The P ?= NP question has dogged the theory of computation for over 40 years. ◮ Since many of the most important theorems of the theory of computation have been proven using the technique of diagonalization, it is reasonable to assume that the P ?= NP question might be settled by it as well. ◮ It is generally accepted that if some proposition Q is true categorically, then there will be no (oracle-based) relativization where Q is false. 3 ◮ Baker-Gill-Solovay show that the P ?= NP question can be relativized. 3 This relativization claim is not accepted universally and there are good reasons to doubt it. Kern, Derek Forcing and Relativization, Side-by-Side
Introduction Preliminaries Proof that ¬ CH What are some of the takeaways References Quick description of the relativized P ?= NP question ◮ The upshot of this demonstration is generally understood to call into question the efficacy of diagonalization for settling the P ?= NP question. ◮ This claim can also be understood as the claim that the P ?= NP question is independent of the technique of diagonalization. Kern, Derek Forcing and Relativization, Side-by-Side
Introduction Preliminaries Proof that ¬ CH What are some of the takeaways References My goal Personally, I found both of these proofs challenging. When taking Tom’s theory class, I found putting diagonalization proofs into juxtaposition to be very useful. Both of these proofs use similar techniques, like forcing, to achieve their purpose. So, I hoped that by putting them side-by-side, they both could be understood better. That being said, this presentation was becoming way too big. Since we discussed it earlier in the semester, we will not cover P ?= NP (aside from a few mentions). Get ready for lots of set theory. If you’d like to see the full paper, I’d be happy to provide it. Kern, Derek Forcing and Relativization, Side-by-Side
Introduction Preliminaries Proof that ¬ CH What are some of the takeaways References Outline Introduction Preliminaries CH Proofs of independence ZFC Models Proof that ¬ CH M [ G ] Defining P Forcing P into M [ G ] What are some of the takeaways References Kern, Derek Forcing and Relativization, Side-by-Side
Introduction CH Preliminaries Proofs of independence Proof that ¬ CH ZFC What are some of the takeaways Models References The continuum hypothesis (CH) Continuum Hypothesis (CH) Suppose that X ⊆ R is an uncountable set. Then there exists a bijection π : X → R [Jec11] . ◮ The upshot of this claim is that there is no infinity between between the countable infinity, ℵ 0 , and the infinity represented by the real numbers, 2 ℵ 0 . ◮ If ¬ CH, then there is some other infinity between ℵ 0 and 2 ℵ 0 . ◮ It was famous (infamous, perhaps) enough by 1900 to make David Hilbert’s list of 23 unsolved problems in mathematics. Kern, Derek Forcing and Relativization, Side-by-Side
Introduction CH Preliminaries Proofs of independence Proof that ¬ CH ZFC What are some of the takeaways Models References Proofs of independence ◮ A proof of independence involves, for the most part, showing that some claim C can be both true and false within some axiomatic system S without violating any of the axioms of S . ◮ Therefore, there is usually a proof that, given S , C ; and some other proof that, given S , ¬ C . ◮ Thus, both C and ¬ C hold within S . ◮ Assume that the proof is sound, this can mean one of two things: 1. S is inconsistent, i.e. some of the axioms of S ultimately contradict each other. 2. S is not complete, i.e. it lacks some axiom that would be consistent within S and would settle the matter of C or ¬ C . This situation is known as ‘independence’. Kern, Derek Forcing and Relativization, Side-by-Side
Introduction CH Preliminaries Proofs of independence Proof that ¬ CH ZFC What are some of the takeaways Models References Proofs of independence ◮ Both of proofs (well, one now) in this work result in some kind of independence, they both have two sides. ◮ For the P ?= NP question, the two sides of the proof are that there exists a relativization where P = NP and there exists a relativization where P � = NP. ◮ For the CH question, the two sides of the proof are that CH is true within ZFC and the proof that it is not. ◮ We will be covering the proof that ¬ CH is true within ZFC. Kern, Derek Forcing and Relativization, Side-by-Side
Introduction CH Preliminaries Proofs of independence Proof that ¬ CH ZFC What are some of the takeaways Models References Proof that the CH is true ◮ The claim that CH is true within ZFC is credited to Kurt G¨ odel and his demonstration using constructible sets [Cho07] . ◮ Note that the claim that “CH is true in ZFC” really amounts to saying that “ZFC cannot be used to disprove CH”. ◮ I didn’t have time to review G¨ odel’s result since it would have required a thorough understanding of constructible sets. ◮ G¨ odel worked until his death on resolving the CH. He is known to have written a paper that added a number of new axioms to ZFC. At one point, he even submitted this paper for publication, but withdrew it prior to his death [Pot04] . Kern, Derek Forcing and Relativization, Side-by-Side
Introduction CH Preliminaries Proofs of independence Proof that ¬ CH ZFC What are some of the takeaways Models References ZFC ◮ A cursory understanding of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) is required before moving on. ◮ ZFC is one of many axiomatic systems developed in response to numerous problems in na¨ ıve set theory, like Russell’s Paradox, encountered in the early 20th century [Jec11] . Some of the axioms of ZFC are below: Axiom of Extensionality This axiom states that two sets are the same if they have the same members. Axiom of Null Set This axiom states the existence of a null set. Axiom of Pairs This axiom states that, for any two sets, S and R , there exists a pair set { S , R } . Kern, Derek Forcing and Relativization, Side-by-Side
Introduction CH Preliminaries Proofs of independence Proof that ¬ CH ZFC What are some of the takeaways Models References ZFC Axiom of Union This axioms states that for any set S , there exists a set R containing the members of the members of S . Axiom of Power Set This axiom asserts, for any set S , there exists the 2 S , i.e. the powerset of S . 4 Axiom of Infinity This axiom asserts the existence of an inductive set, i.e. a set in which the Principle of Well-Ordering applies. Axiom of Choice Given the definition of a choice function as a function f where f ( S ) ∈ S , ∀ S ∈ dom ( f ), this axiom asserts that all sets have a choice function f . 4 As will be shown later, this axiom is actually more restrictive than this statement would suggest. Kern, Derek Forcing and Relativization, Side-by-Side
Introduction CH Preliminaries Proofs of independence Proof that ¬ CH ZFC What are some of the takeaways Models References ZFC ◮ Beyond basic idea behind stating these axioms is that they determine the membership of any ‘model’ that is said to obey ZFC. ◮ In other words, models of ZFC must obey these axioms. ◮ For example, if a sets A and B are part of a model M of ZFC, then, by the Axiom of Pairs, { A , B } must also be within M . Kern, Derek Forcing and Relativization, Side-by-Side
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