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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

1Actually, 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 ℵ0 members, the set of real numbers has 2ℵ0 members, the powerset of real numbers has 22ℵ0 members, etc.

◮ The continuum hypothesis (CH) is the claim that there are no

infinities between ℵ0 and 2ℵ0.

◮ The generalized continuum hypothesis (GCH) is the claim

that there are no infinities between an infinite set S and its powerset 2S. 2

2If 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.

3This 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 Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

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, ℵ0, and the infinity represented by the real numbers, 2ℵ0.

◮ If ¬CH, then there is some other infinity between ℵ0 and 2ℵ0. ◮ 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 Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

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

• ther 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 Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

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 Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Proof that the CH is true

◮ The claim that CH is true within ZFC is credited to Kurt

G¨

• del 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¨

• del’s result since it would have

required a thorough understanding of constructible sets.

◮ G¨

• del 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 Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

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 Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

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 2S, 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 .

4As will be shown later, this axiom is actually more restrictive than this

statement would suggest.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

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

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Quick thoughts on the Axiom of Choice

◮ It is worth noting that the Axiom of Choice (AoC) is

somewhat controversial. Much of the controversy surrounds the properties that some sets have given the AoC [Pot04].

◮ That being said, this claim should appear familiar to a

computer scientist.

◮ Certainly, all of the sets that computer scientists are generally

interested are ones that have choice functions - decidable and computable sets.

◮ It is worth asking how non-computable sets, like ¬K, might fit

within an axiomatic system that includes the AoC.

◮ It seems like it goes without saying that ¬K, for example,

doesn’t have a choice function; if it did, of course, then this function could be computed by a Turing machine.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Models (via groups)

◮ Perhaps the best way to begin describing models is through

an analogy. The best analogy is the concept of a group from Group theory [Cho07].

◮ A group is some operation, like multiplication, addition, etc,

combined with a set.

◮ For simplicity, assume that the operation is ⋆, the set is S,

and thus the group is (⋆, S).

◮ (⋆, S) is truly a group if it exhibits four properties. ◮ Note that these properties are analogous to the axioms of

ZFC.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Models (via groups)

The properties of groups are: Identity There must exist some i ∈ S such that whenever the

• peration ⋆ is applied to i and any x ∈ S, then the

result is the other member x, i.e. x = x ⋆ i = i ⋆ x. Inverse For any x ∈ S, there must exist some j ∈ S such that whenever the operation ⋆ is applied to x and j, then the result is the identity element i, i.e. i = x ⋆ j = j ⋆ x. Closure For any x, y ∈ S, x ⋆ y ∈ S. In other words, the result of the ⋆ operation on any two members of S must also be a member of S. Associativity For any x, y, z ∈ S, x ⋆ (y ⋆ z) = (x ⋆ y) ⋆ z. Associativity basically describes how multiple (more than two) members of a group can be related.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Models (via groups)

◮ Consider the following ‘putative’ group: (+, Z). (+, Z) will

• fficially be a group if it has the four properties of a group.

◮ The identity element of (+, Z) is 0 since the result of adding

any integer x and 0 is just x.

◮ The inverse of any x ∈ Z is just −x since the result of adding

any integer x to −x is 0, the identity element.

◮ Given any x, y ∈ Z, x + y ∈ Z so (+, Z) is closed. ◮ (+, Z) is associative since, for x, y, z ∈ Z,

x + (y + z) = (x + y) + z.

◮ Therefore, (+, Z) is officially a group. ◮ It is worth acknowledging that if this group were defined over

N instead of Z, then it would not be a group since, within (+, N), there is no identity element.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Models (foreshadowing forcing via groups)

◮ Let S = Z. ◮ Given the group (+, S), how would S have to change in order

to add 1/2 to it, i.e. S = S ∪ {1/2}?

◮ The element 1/2 still works with the group identity since

1/2 + 0 = 1/2 so that is a good start.

◮ However, given some odd element of Z like 3, 1/2 + 3 = 3/2.

Of course, 3/2 is not (yet) in S (remember S is currently Z ∪ {1/2}).

◮ Therefore, 3/2 would need to be added to S as well as 5/2,

7/2, etc.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Models (foreshadowing forcing via groups)

◮ Also, for any of these additional elements, their respective

negatives (−1/2, −3/2, −5/2, −7/2, etc) must be added as well in order for each new element to have an inverse.

◮ It is obvious that amending a group to include a new member

is not a trivial task.

◮ The thought process required to amend a group is similar to

the thought process required to amend models of ZFC.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Models

◮ A model in set theory is a set combined with the axioms that

determine its membership.

◮ Unlike groups, models are highly theoretical entities such that

it is a matter of debate whether any models exist at all [Cho07].5

◮ Yet, even though no models may exist it is still possible to

visualize them using an entity that is ”close enough”.

◮ The best ”close enough” example is the class of all sets; call it

V . As it turns out all of the axioms of ZFC hold for V [Eas07].

◮ Let model M = V .

5As you might expect, models of ZFC exist if and only if ZFC is consistent.

This is a consequence of G¨

• del’s completeness theorem

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Models

◮ Another property of M is that it is standard and transitive. ◮ Transitive means that, for any set S ∈ M, ∀x ∈ S → x ∈ M. ◮ M is standard because it consists only of well-founded sets. ◮ Well-founded sets are built, from the bottom up, using sets.

So, the number ‘0’ is represented as the empty set, ∅; building

• n ‘0’, ‘1’ is {∅}, ‘2’ is {∅, {∅}}, etc.6 7

◮ A model can be thought of as a ‘universe’ of sets where some

claims will be true, based upon its membership, and some claims will be false.

6Since M is assumed to be transitive, then the ∅ will eventually be reached

and will, of course, be contained within M.

7Well-founded sets are meant to be constructed in such a way as to prevent

any self-reference.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Models (countability and powersets)

◮ Besides being transitive and standard, M must also be

countable in the sense that ℵ0 is countable.

◮ However, this creates a problem given the axioms of ZFC. ◮ Note that a great deal of the set theoretic objects that exist in

V , the set of all sets, have analogues in M. For the most part, they are equal [Cho07].

◮ An important exception to this statement is the powerset. ◮ ℵ0 is in M and, so, by the powerset axiom, should its

powerset, 2ℵ0. Right?

◮ Yet, if 2ℵ0 is within M, then M cannot be countable since 2ℵ0

isn’t countable.

◮ What is going on?

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References CH Proofs of independence ZFC Models

Models (countability and powersets)

Axiom of Powerset

Given any x ∈ M, y is the powerset of x if ∀z ∈ M (z ∈ y iff ∀w ∈ M that satisfies w ∈ z if w ∈ y) [Eas07]

◮ The notable feature of this axiom is that the powerset of x

consists of only those members of x already in M.

◮ Thus, in order to conform the powerset axiom, no new

members need be added to M.

◮ Most importantly, the powerset of ℵ0 is uncountable within

M but countable from the perspective of V (as is M).

◮ Thus, cardinality relationships are preserved within M.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

M[G]

◮ Remember, in order to show that ¬CH, a model M will need

to be built in which ¬CH holds.

◮ What sort of entities need to be added to M? ◮ In order for ¬CH to be true, the following inequality must

hold within M: ℵ0 < ℵ1 < ℵ2, where ℵ2 ≤ 2ℵ0. This puts ℵ1 in the role of contradicting the CH.

◮ Let G be the set containing whatever is to be added to M in

• rder for ¬CH to be true.

◮ Thus, M[G] is the model that needs to be constructed.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

A few quick and important notes

◮ Whether CH or ¬CH is true, there must exist cardinals ℵ0,

ℵ1, ℵ2, etc within M[G].

◮ If CH is true, then the following will be true within M[G].

◮ ℵ0 = ℵ0 ◮ ℵ1 = 2ℵ0 ◮ ℵ2 = 22ℵ0

◮ If ¬CH is true, then the following will be true within M[G].

◮ ℵ0 = ℵ0 ◮ ℵ1 = ? ◮ ℵ2 ≤ 2ℵ0

◮ So, showing ¬CH amounts to showing that ℵ1 = 2ℵ0.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

Setting the stage

◮ Let 2ℵ0 relativized to M be known as ℵM[G] α

.

◮ What is needed an entity to play the role of ℵ2 in M[G]. ◮ Remember that since M is a countable model, it must remain

so, even when it contains ℵM[G]

α

.

◮ Fear not, the version of 2ℵ0 (i.e. ℵM[G] α

) within M need only contain the elements of 2ℵ0 (i.e. ℵM[G]

α

) that are already within M. (Remember the Axiom of Powerset)

◮ Therefore, within the context of V , M remains countable. ◮ Furthermore, within M, ℵM[G] α

remains uncountable since there will exist no bijection between it and ℵ0.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

Picking something uncountable

◮ We need some actual entity that is on the order of 2ℵ0 to be

ℵM[G]

α

.

◮ We know of lots of uncountable sets, like R, 2ℵ0, etc ◮ Since M[G] is well-founded and transitive, a good candidate is

the set of all boolean functions. (Remember the diagonalization proof that the set of all boolean functions is uncountable from Tom’s class?)

◮ Remember that these functions can be thought of as

f : ℵ0 → {0, 1}.

◮ Within M, both ℵ0 and {0, 1} are known to exist. ◮ Let this set of functions just be the contents of ℵM[G] α

.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

We need an injection

◮ Therefore, if a mapping can be defined between ℵM[G] 2

and ℵM[G]

α

, then ¬CH will hold.

◮ Remember, sets S and T have the same cardinality iff there is

a bijection between them. Remember Cantor used diagonalization in order to show that no bijection could exist between Z and R.

◮ However, we need only show that ℵM[G] 2

≤ ℵM[G]

α

so, in this case, no more than an injection must exist between them.

◮ If an injection f exists between sets S and T such that

f : S → T, then |S| ≤ |T|.8

8An injection f : S → T basically maps all of the elements of S to unique

elements of T until the elements of S are exhausted.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

Defining the injection: P

◮ The necessary injection can be represented mapping ℵM[G] 2

into ℵM[G]

α

.

◮ Let P = ℵM[G] 2

× ℵ0 into {0, 1} [Cho07].

◮ Note: ℵM[G] α

has just been rearranged in P. P still represents an injection from ℵM[G]

2

into ℵM[G]

α

.

◮ So, if P can be made to exist within M[G], then ¬CH and we

are done.

◮ Also note: Given its definition, some of P is already within

M[G].

◮ Ok. Let the forcing begin!

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

Forcing

Fundamental Theorem of Forcing

If M is a countable, transitive model where G ⊆ P ∈ M, then the set M[G] is a countable, transitive model [Cho07].

◮ The theorem above tells us that, if we go through the work of

specifying G, then the desired properties of resulting model M[G] will be preserved from M.

◮ Quick quiz: Is G ⊆ M? Hint: Think back to the discussion of

the Axiom of Powerset.9

9And the answer is: No. It is safe to assume that some members of G are in

• M. However, since M is countable and the 2P is not countable, then not all

subsets of P can be in M. It is some of these members that will be added to M[G].

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

Forcing

Definition of Forcing

Let p ∈ P and φ be a sentence that can be true or false. p | = φ, if, for any G ⊆ P, p ∈ G → φ in M[G] [Cho07].

◮ p |

= φ can be thought of in two ways: “p forces φ” or “φ satisfies p”.

◮ Example:

◮ Let p = ∅ ◮ Let φ be the sentence “G = ∅” ◮ So, p |

= φ means that p | = “G = ∅”

◮ In other words, the fact ∅ is a member of G means that

“G = ∅” must be true.

◮ S |

= “2S ⊆ G”?

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

Forcing rules

◮ The example on the previous slide is a forcing rule. There are

many forcing rules and they can be combined.

◮ Some of the forcing rules are:

◮ If ∀q ∈ p, q |

= b, then p | = ¬b.

◮ If p |

= b and p | = c, then p | = b ∧ c.

◮ If p |

= b or p | = c, then p | = b ∨ c.

◮ If p |

= b or p | = ¬a, then p | = a → b.

◮ If p |

= a → b or p | = b → a, then p | = a ↔ b.

◮ As you can see, by combining forcing rules, some fairly

complex conditions can be specified on the membership of G.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

Forcing wrap-up

◮ The upshot of the forcing process is that the membership of

G can be specified by adding necessary members and then ‘forcing’-in auxiliary members.

◮ In order to complete his proof, Cohen forces the membership

• f M[G] such that enough of P is within M[G] so that

ℵ2 ≤ |P| (and thus ℵ1 < |P|) holds.

◮ There are a lot more technical details of forcing, models, and

ZFC that I am omitting because of time (and knowledge).

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References M[G] Defining P Forcing P into M[G]

Final thoughts on forcing

◮ There are other ways to ‘force’ memberships that do not use

the forcing process itself. An example is the use of boolean-valued models (BVM).

◮ When using BVMs, the membership of a set may be ‘forced’,

but this is not forcing, per se.

◮ Baker-Gill-Solovay’s relativization of the P ?= NP question is

similar in this regard. It does essentially ‘force’ the membership of oracle B, but it does not use the forcing process.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References

Takeaways

◮ Of course, the basic upshot of Cohen’s proof is that the CH

cannot be settled within ZFC.

◮ As with any proof, its conclusion is only as good as its

premises.

◮ There are number of tools (i.e. premises) that were used in

Cohen’s proof that are still in dispute today.

◮ The generalized form of the continuum hypothesis (GCH) has

also been shown to be independent of ZFC.

◮ Interestingly, the GCH entails the Axiom of Choice (AoC). So,

by modus tollens, ¬AoC → ¬GCH.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References

Takeaways

◮ There are, at least, five approaches to resolving the

independence of CH from ZFC:

• 1. Show that ZFC is not consistent, i.e. one or more of the

axioms of ZFC result in a contradiction that ultimately allows both CH and ¬CH to be safely derived.

• 2. Show that models of ZFC truly cannot exist. As stated earlier,

whether any actually exist is a matter of dispute. If they don’t exist, then Cohen’s proof falls apart.

• 3. Assume that ZFC is lacking an axiom or axioms that would

preclude either CH or ¬CH and attempt to find them.

• 4. Assume that, from some reason, no axiomatic system can

settle the CH/¬CH question.

• 5. Forget about it and have a beer.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References

Takeaways

◮ Concerning 1, the Axiom of Choice is still controversial so this

approach is still in play.

◮ Concerning 3, this approach is still the one that has garnered

the most adherents. As stated above, G¨

• del followed this path
• himself. I am aware of work on this approach as late as 2000

(see [Wo00a]).

◮ Concerning 4, there are those that think that the CH/¬CH

question simply cannot be settled. In this camp are those that think that the CH is ‘inherently vague’ [Pot04].10 This position amounts to a retreat. It implies that there is no amount of refinement that can be made to this question that would result in more perspicuity.

10In philosophy, many of the most notable paradoxes, like the liar paradox or

the soroties paradox, are thought by many to be inherently vague.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References

References

[Cho07] Chow, T. A beginner’s guide to forcing, 2007. [Eas07] Easwaran, K. A Cheerful Introduction to Forcing and the Continuum Hypothesis, 2007. [Jec11] Jech, Thomas Set Theory The Stanford Encyclopedia of Philosophy, 2011. [Pot04] Potter, M.D. Set Theory and Its Philosophy: Critical Introduction Oxford University Press, 2004.

Kern, Derek Forcing and Relativization, Side-by-Side

Introduction Preliminaries Proof that ¬CH What are some of the takeaways References

References

[BGS75] Baker, T. and Gill, J. and Solovay, R. Relativizations of the P =?NP Question SIAM Journal on Computing, 4-4, 1975. [Wo00a] W. Hugh Woodin The Continuum Hypothesis, Part 1 Notices of the AMS, 48-6, 2001.

Kern, Derek Forcing and Relativization, Side-by-Side