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Real Numbers: Models and The Proof of The Continuum Hypothesis

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by

Martin Flashman

Department of Mathematics

Humboldt State University

In memory of my mentor

Jean van Heijenoort

- To give some background on the Continuum Hypothesis. (CH)
- To outline the key concepts in constructing models for the real numbers.
- To indicate a specific model for the real numbers where the CH is false.
- To indicate why the CH is false in the specific model.
******************************************

- This presentation is only a rough indication of the organization and concepts needed to prove “the independence” of CH.
- Rigorous details will be omitted.

- JvH … TrotskyFrom Trotsky to Godel:
The Life of Jean van Heijenoortby Anita Fefferman

- Frida (movie cast)
Cast:Frida Kahlo Salma Hayek

Diego Rivera Alfred Molina

Leon Trotsky Geoffrey Rush

Jean Van HeijenoortFelipe Fulop

- A set S is countable if there is a function from N onto S.
- Any infinite subset of the natural numbers or the integers is countable.
- The set of rational numbers is a countable set.
- "Godel counting" argument.
25 38 : 5/8

- "Godel counting" argument.
- The algebraic numbers are countable.[ Another first type of diagonal argument.] 1874

- There is an uncountable set of real numbers.
- Any function from N to the interval [0,1] is not onto.
- A decimal based proof. (Similar to 1891 proof)Consider the set of real numbers with decimal expression: 0. a15a25 a35a4… and suppose this set is countableLet b= 0. b15b25 b35b4… where…
- There is no onto function from R, the set of real numbers, to P(R), the set of all subsets of the real number.
- There are sets which are "larger" than the set real numbers .

- The set of rational numbers between 0 and 1 has "measure" zero.
- Any countable set of real numbers has "measure" zero**************************************
Proof:For each element an of the countable set, choose the interval [an – z/4^n, an + z/4^n)] , n = 1,2,...

Then for any z>0, the union of the intervals has length < z, so the countable set has measure 0.

- The continuum hypothesis problem was the first of Hilbert's famous 23 problems delivered to the Second International Congress of Mathematicians in Paris in 1900.
- The Hilbert Problems of Mathematics challenged (and still challenge today ) mathematicians to solve these fundamental questions for the entire 20th Century.

- Two systems, i. e, two assemblages of ordinary real numbers or points, are said to be (according to Cantor) equivalent or of equal cardinal number, if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one definite number of the other. The investigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving. This is the theorem:
- Every system of infinitely many real numbers, i. e., every assemblage of numbers (or points), is either equivalent to the assemblage of natural integers, 1, 2, 3,... or to the assemblage of all real numbers and therefore to the continuum, that is, to the points of a line; as regards equivalence there are, therefore, only two assemblages of numbers, the countable assemblage and the continuum.
- From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage; the proof of this theorem would, therefore, form a new bridge between the countable assemblage and the continuum.

- Let me mention another very remarkable statement of Cantor's which stands in the closest connection with the theorem mentioned and which, perhaps, offers the key to its proof. Any system of real numbers is said to be ordered, if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that, if a is before b and b is before c, then a always comes before c. The natural arrangement of numbers of a system is defined to be that in which the smaller precedes the larger. But there are, as is easily seen infinitely many other ways in which the numbers of a system may be arranged.
- If we think of a definite arrangement of numbers and select from them a particular system of these numbers, a so-called partial system or assemblage, this partial system will also prove to be ordered. Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way, that not only in the assemblage itself but also in every partial assemblage there exists a first number. The system of integers 1, 2, 3, ... in their natural order is evidently a well ordered assemblage. On the other hand the system of all real numbers, i. e., the continuum in its natural order, is evidently not well ordered. For, if we think of the points of a segment of a straight line, with its initial point excluded, as our partial assemblage, it will have no first element.
- The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have a first element,i. e., whether the continuum cannot be considered as a well ordered assemblage--a question which Cantor thinks must be answered in the affirmative. It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor's, perhaps by actually giving an arrangement of numbers such that in every partial system a first number can be pointed out.

Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (1940)Kurt Gödel showed, in 1940, that

if the axioms of set theory are consistent,

then adding the Axiom of Choice and/ or the Continuum Hypothesis will not make the enlarged theory inconsistent.

[This will not be discussed here.]

- In 1963 Paul Cohen proved that the Axiom of Choice is independent of the other axioms of set theory. Cohen used a technique called "forcing" to prove the independence of the axiom of choice and/or of the generalized continuum hypothesis from the conventional axioms for set theory.
- In 1967 Dana Scott [and Robert Solovay] published Models for the real numbers based on Probability-Measure Theory.
- A proof of the independence of the continuum hypothesis, Mathematical Systems Theory, volume 1 (1967), pp. 89-111.
- This paper demonstrated the independence of the CH using a probability based model for the real numbers.

- A Formal System uses symbolic logic with predicates and quantifiers to try to capture and express completely and uniquely the totality of statements of a mathematical theory.
- Key issues for such a formal system are
- Is the system of logically related propositions sound?
- Is the system consistent?
- Does the system contain all the propositions of the mathematical theory as theorems…. Is it complete?
- A (set theoretic) model for a formal system is an interpretative correspondence between a part of set theory and the constants, variables, predicates, and other aspects of the formal system. In the model’s interpretation every theorem (proven statement) of the system is true.

Suppose P and S are sets.

We say a set S is P countable if there is a function from P onto S.

The PR- Hypothesis (PRH):

Suppose P is a subset of R. If X is a subset of R and X is not P countable, then R is X countable.

- Note: With P = the natural numbers and
R= the real numbers, PRH = CH.

- Model 1: Let P = {1,2,3,4,5} and R = { 1,2,3,4,5,6}.
Then Ris the only subset of R that is not P countable and PRH is true for this model.

Thus- PRH is consistent with formal set theory (including the axiom of choice).

- Model 2: Let P = {1,2,3,4,5} and R = { 1,2,3,4,5,6,7}
Then let X = { 1,2,3,4,5,6}. Xis not P countable but R is not X countable. So the PRH is false for this model.

Thus- the negation of PRH is consistent with formal set theory (including the axiom of choice).

SO in general: If formal Set Theory (including the axiom of choice) is sound (consistent), PRH cannot be proven as a result of Formal Set Theory, i.e.,

PRH is independent of the axioms of formal set theory.

Is built using a well established formal system for set theory. A formal system for the real numbers must have enough features to makes sense of at least such concepts as

- the natural numbers
- the rational numbers
- the operations of addition and multiplication
- the relations of equality and inequality
- functions and functionals.

- Usual treatment given in many high school courses and justified more carefully in a university level real analysis course.
- Natural numbers connected to cardinal numbers of sets.
- Integers and rational numbers as classes of natural numbers.
- Real numbers can be understood as represented by infinite decimals or convergent sequences of rational numbers.
- Number equality explains why 1 =.9999999…
- Operations are based on sums and disjoint unions of sets.
- Functions and functionals are based on ordered pairs.

- Definition: A random real is a measurable function from a probability sample space, Ω, to the real numbers, R: i.e.,r:Ω -> R so that for any a < b, the probability of the set {s: a<r(s)<=b} is measurable or {s: a<r(s)<=b} is a measurable sub set of Ω .
- Note: The total measure of Ω is 1, and Ω can have sets of measure 0.
- In particular Ω can be the cartesian product of a large number of copies of the interval [0,1]. {We'll decide how large later.}
- Think about Ω = [0,1]x[0,1] as an example. There are several random reals on Ω :
- Constant random reals with the natural numbers. 0(s)=0 1(s)=1,2(s)=2, etc.
- Projection random reals:p1(s) = x and p2(s) = y where s = (x,y).

- Consider how random real numbers might satisfy key formal properties of the usual real numbers.
- For example, one key property that we can use as a TEST STATEMENT about the real numbers is
- If a*b= 0 then either a=0 or b=0.
- Unfortunately, if a and b are random real numbers then the fact that a*b=0 doesn't imply that a=0 or b=0.Here is a specific counterexample:
- Let a(x,y)=0 when y<=.5 and a(x,y) = 1 when y >.5 and b(x,y) = 1- a(x,y). Then, for any s=(x,y), either a(s)=0 or b(s)=0, so a*b(s) =a(s)*b(s)=0, but neither a= 0 nor b = 0.

- Definition: We will say that a simple arithmetical/algebraic (formal) statement P(x) about a real number x is true in this probability model ( M- true) for the random real rif the probability of the set { s : P(r(s)) is true} is 1 and is false in this probability model (M-false) if the probability of the set { s : P(r(s)) is true} is 0.
- For example, the function defined by
f(s)=0 when s is rational and f(s)=1 when s is not rationalis a random real for the sample space [0,1] and the statement that f = 1 is M-true in this model. [Any countably infinite subset of real numbers has measure 0.]

- Even using this standard for truth, our test statement for the random reals to model the real numbers is not true. The same counter example can be used. a*b = 0 is M- true but a=0 is not M-true and b=0 is also not M-true.
- What we need is an interpretation not only of the real numbers, arithmetic, and equality, but a different interpretation in this model for the logical connectives and quantifiers used in the formal statements describing the real numbers.

- We'll say that value of a formal statement L(x) about a real number x, v(L), is the probability of the subset of Ω {s:L(r(s)) is true in the common meaning for a random real r}.
- We'll say that a statement isP-true if its value is 1,P-false if its value is 0.
- Consider the example random real a.Then the statement a=0 is not P true but is also not P false!
- For more complicated statements we use the following procedures to evaluate a statement:
- v(A&B)= prob{s: A(s) and B(s) are true.}
- v(A or B) = prob{ s:A(s) or B(s) (or both) is true}
- v(not A) = prob {s: not A(s) is true}
- Notice: the value of the statement F(a): “Either a=0 or it is not the case that a=0” is determined by the probability of {s: a(s)=0 or it is not the case that a(s)= 0}. This set is Ω, so the probability is 1 and this statement is P-true.

- Now let's look at the TEST STATEMENT RESTATED using negation and “or”:
- Either a=0, b=0, or it is not the case that a*b=0.
- To determine the value of this statement we consider the probability of the set{ s: a(s)=0, b(s)=0, or not a(s)*b(s) =0 is true.}
- But for any s in Ω, if a(s)*b(s)=0, then either a(s)=0 or b(s)=0 is true. So the set under consideration is Ω, and the probability is 1. So the test statement is P-true.

- With more work extending the structures and logic, Scott showed that the random real numbers for any particular probability measure space would provide a consistent model for the reals. [Assuming Set Theory including the axiom of choice is already consistent.]
- Now the consistent model we want is one in which the continuum hypothesis fails to be true in some way, in particular the Continuum Hypothesis will not be P-true in real number model based on Random reals as just outlined.

- Lemma: There is no onto function from R, the set of real numbers, to P(R), the set of all subsets of the real number. Proof: Suppose f: R -> P(R). Let B= {x such that x is not an element of f(x)}. Suppose B=f(b) for some b. If b is in B then b is not in f(b)=B, which is a contradiction. So b is not in B, but then b is not in f(b), so b is in B! Thus B is not f(b) for any b, and f is not onto.
- Thus: There are sets which are larger than the reals.
- Use Ω = the product of one copy of the interval [0,1] for every subset of the real numbers.
- It is a result of measure theory using the Axiom of Choice, that this Ω is a sample space for a probability measure and any of the projection functions are random reals.

- Let the set T contain precisely those random reals that correspond to the projections for the single element subsets of the reals.
- The following can then be shown:
- The set of random reals that correspond to the natural numbers in this model cannot count (be mapped onto) the set T.
- The set T cannot map onto the set of all projection random reals, so it cannot count (be mapped onto) all the random reals.
- THUS, the continuum hypothesis fails to be true in this probability model for the formal system of real numbers.

- Philosophical Introduction to Set Theory by Stephen Pollard
- The Mathematical Experience by Philip J. Davis and Reuben Hersh
- P. J. Cohen, The independence of the Continuum Hypothesis. I. Proc. Nat. Acad. Sci., U.S.A. 50 (1963) 1143-1148, and II. ibid. 51 (1964) 105-110.
- Dana Scott, A proof of the independence of the continuum hypothesis, Mathematical Systems Theory, volume 1 (1967), pp. 89-111.
- What is mathematical logic? by J.N.Crossley et al.
- Set Theory and the Continuum Hypothesis by Raymond M. Smullyan and Melvin Fitting
- Intermediate Set Theory by F.R. Drake and D. Singh

The End