Models for the real numbers
based on Probability Measure
Theory.
(work by Scott- Solovay)
A random real:
a measurable function from a probability
sample space, S, to the real numbers, R:
r: S-> R so that for any a < b, the probability of the set {s: a<r(s)<=b} is measureable.
Note: The total measure of S is 1, and S can have sets of measure 0.
In particular S can be the cartesian product of a large
number of copies of the interval [0,1].
{We'll decide how large later.}
Think about S= [0,1]x[0,1] as an example.
There are several random reals on S:
For example:
TEST STATEMENT: If ab= 0 then either a=0 or b=0.
Now if a and b
are random real numbers
then the fact that ab=0 doesn't imply that
a=0
or b=0.
Here is a specific example:
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 nor b = 0.
We'll say that a simple arithmetical/algebraic statement P(x) about a real number is true in this model for the random real r if the probability for{ s: P(r(s)) is true} is 1.
For example, the Dirichlet function f(x)=0 when x is rational and f(x)=1 when x is not rational is a random real for the sample space [0,1] and the statement that f = 1 is true in this model.
Even using this standard for truth, our test statement for the random reals to model the real numbers is not true.
What we need is an interpretation not only of the real
numbers, arithmetic, and equality, but a different interpretation for the
logical connectives and quantifiers used in the statements describing this
model of the real numbers.
We'll say that value of a statement L, v(L),
is the probability of the sample set {s:L(s)}.
We'll say that a statement is
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} which is 1.
So this statement is P-true.
Now let's look at the TEST STATEMENT RESTATED:
We consider prob{ s: a(s)=0, b(s)=0,
or not a(s)b(s) =0 is true.}
But for any s, if a(s)b(s)=0,
then either a(s)=0 or b(s)=0 is true.
So the set under consideration is S, and the probability
is 1.
So the test statement is P-true.
With more work extending the structures and logic, Scott and Solovay showed that the random real numbers for any particular probability measure space would provide a consistent model for the reals. [Assuming Set Theory is already consistent.]
Now the consistent model that is needed is one in which the continuum hypothesis fails in some way.
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 S= 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 S is a sample space for a probability measure and any of the projection functions are random reals.
Now take take the set T of random reals that correspond to the projections for the single element subsets of the reals.
The following can then be shown:
I. The set of random reals that correspond to the natural numbers in this model cannot count (be mapped onto) the set T.
II. 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 model.