Models, Numbers, and Measure Theory: The Independence of The Continuum Hypothesis

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Models, Numbers, and Measure TheoryThe
Independence of The Continuum Hypothesis

• by
• Martin Flashman
• Department of Mathematics
• Humboldt State University
• In memory of my mentor
• Jean van Heijenoort

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A Sequel to .

• "The Continuum Hypothesis A Look at the 20th
  Century History of the Real Numbers from Cantor
  to Cohen/Scott/Solovay" Sept. 14, 2000
• "The Continuum Hypothesis  A Look at the History
  of the Real Numbers in The Second Millennium. ",
  MAA Jan. 7, 2002 

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This presentation will attempt

• To give some background on the Continuum
  Hypothesis.
• 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.

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Personal AnecdotesSpring, 1965Bates College

• Topology Ed Baumgartner
• The set of real numbers is uncountable.
• Using the usual ordering of the Natural Numbers,
  every non-empty set of natural numbers has a
  "first" element, i.e., the usual ordering of the
  Natural Numbers is a well ordering.
• Problem Try to construct an ordering of the real
  numbers for which every non-empty set of real
  numbers has a "first" element by this ordering,
  i.e., a well ordering of the real numbers

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1965-1966Brandeis University

• Real Analysis Paul Monsky
• The well ordering problem for the real numbers
  was not solvable!
• Paul Cohen had shown this was an axiom of set
  theory.
• Logic. Jean van Heijenoort
• Introduction to syntax (formal and symbolic) and
• semantics (interpretations of symbols- sets,
  etc.)

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1966-1967Brandeis University

• Graduate Algebra and Topology
• Structural axiomatic mathematics
• Senior Tutorial with JvH.
• Metamathematics by S. Kleene
• Godels Theorem on the Incompleteness of
  Arithmetic
• Introduction to independence proofs using models.
  
• Undecidable statements in arithmetic.
• From Frege to Godel by JvH.

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1968 - 1969Brandeis University

• Graduate Analysis Michael Spivak
• Measure Theory
• Jean van Heijenoort gives me a copy of the Scott
  paper on the independence of the Continuum
  Hypothesis.

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BreakJean Van Heijenoort and the Revolution

• JvH TrotskyFrom Trotsky to Godel
• The Life of Jean van Heijenoort by Anita
  Fefferman
• Frida (movie cast)
• CastFrida Kahlo Salma Hayek
• Diego Rivera Alfred Molina
• Leon Trotsky Geoffrey Rush
• Jean Van Heijenoort Felipe Fulop

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Cantor 1845-1918Infinite Sets

• 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
• The algebraic numbers are countable. Another
  first type of diagonal argument. 1874

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CantorUncountable Infinite Sets

• 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 .

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Sets and Measure

• The set of rational numbers between 0 and 1 has 
  "measure" zero.
• Any countable set of real numbers has "measure"
  zero
• ProofFor each element an of the countable set,
  choose the interval an z/4n, an z/4n) , n
  1,2,...
• Then for any z0, the union of the intervals has
  length

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The Continuum HypothesisDavid Hilbert
(1862-1943)

• 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.

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From Hilbert's original paper "MATHEMATICAL
PROBLEMS.Problem 1A

• 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.    

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Hilbert Problem 1BWell Ordering The Real Numbers

• 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.
   

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Godel (1906-1978)Consistency of CH (and Axiom
of Choice)

• 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.

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Cohen (1934- )Independence of the CH(and the
Axiom of Choice)

• 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.

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Models for Formal Mathematical Logical Systems

• 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 models interpretation every
  theorem (proven statement) of the system is true.

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How to show the CH is not provable from formal
set theory even with the axiom of choice An
example of the argument.

• Suppose P,S, and Q are sets and P and S are
  subsets of Q.
• We say a set S is P countable if there is a
  function from P onto S.
• We say a set Q is S-countable if there is a
  function from S onto Q.
• The Q- Hypothesis (QH)
• Suppose X is a subset of Q and X is not P
  countable, then Q is X countable.
• Note With P the natural numbers and
• Q the real numbers, QH CH.

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Two Models For QH

• Model 1 Let P 1,2,3,4,5 and Q
  1,2,3,4,5,6.
• Then Q is not P countable and QH is true for this
  model.
• Thus- QH is consistent with formal set theory
  (including the axiom of choice).
• Model 2 Let P 1,2,3,4,5 and Q
  1,2,3,4,5,6,7
• Then Q is not P countable but the QH is false for
  this model.
• Thus- the negation of QH 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), QH
  cannot be proven as a result of Formal Set
  Theory, i.e.,
• QH is independent of the axioms of formal set
  theory.

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A Formal Systemfor the Real Numbers

• Is built using a well established formal system
  for set theory. A formal system for the real
  numbers must have enough 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.

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A standard model or the real numbers

• 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.

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A Random Real

• Definition A random real is a measurable
  function from a probability sample space, O, to
  the real numbers, R i.e., r O - R so that for
  any a ameasurable sub set of O .
• Note The total measure of O is 1, and O can
  have sets of measure 0.
• In particular O can be the cartesian product of a
  large number of copies of the interval 0,1.
  We'll decide how large later.
• Think  about O 0,1x0,1 as an example.
  There are several random reals on O
• Constant random reals with the natural numbers.
  0(s)0 1(s)1,2(s)2, etc.
• Projection random realsp1(s) x and p2(s)
  y   where s (x,y).

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The Formal Real NumbersMaking a Model Using the
Random Reals

• 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 ab 0  then either a0 or b0.
• Unfortunately, if a and b are random real
  numbers then the fact that ab0 doesn't imply
  that a0 or b0.Here is a specific
  counterexample
• Let a(x,y)0 when ywhen 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 ab(s)
  a(s)b(s)0, but neither a 0 nor b 0.

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Simple Statements that are true for The Random
Reals Model

• 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 r if 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. ab 0 is M- true but a0 is not
  M-true and b0 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.

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Logic for the Model

• 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 O sL(r(s)) is true in the
  common meaning for a random real r.  
• We'll say that a statement is P-true if its
  value is 1, P-false if its value is 0.
• Consider the example random real a. Then the
  statement a0 is not P true but is also not P
  false!
• For more complicated statements we use the
  following procedures to evaluate a statement  
• v(AB) probs A(s) and B(s) are true.  
• v(A or B) prob sA(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
  a0  or it is not the case that a0 is
  determined by the probability of s a(s)0 or it
  is not the case that a(s) 0. This set is O, so
  the probability is 1 and this statement is
  P-true.  

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The Value of the Test Statement

• Now let's look at the TEST STATEMENT RESTATED
  using negation and or
• Either a0, b0,  or it is not the case that
  ab0.
• 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 O, if a(s)b(s)0, then either
  a(s)0 or b(s)0 is true. So the set under
  consideration is O, and the probability is 1. So
  the test statement is P-true.

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Some Hand Waiving

• 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.

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Constructing a model for the real numbers where
the CH fails.

• 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 Bf(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 O 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 O is a sample space for a
  probability measure and any of the projection
  functions are random reals.

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The Counterexample to the CHA Large Set of
Random Real Numbers

• 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.

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References and Reading

• 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

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• The End ?