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Continuum Hypothesis

The "Continuum Hypothesis" states that the cardinality
of the set of real numbers is two to the power of the
cardinality of the set of integers.

The cardinal number for a set is the size of the set.

There are infinitely many integers, denoted by Ω  omega,
or more modern א zero  Aleph-Naught.

There are infinitely many reals, denoted by 
2^א zero  = א one  Aleph-One.

The following is called a "proof by picture" that is typically
not accepted by strict mathematicians, yet, this is about
numerical computation.

We will write the real numbers from zero to about one,
as binary fractions. This is a subset of the real numbers
because each fraction may have every integer as its
integer part. We do not even bother to count that when
determining the cardinality of the real numbers.

We use a one-to-many  mapping from the integers
to the real fractions. The integer is at the top and
the real fraction runs vertically as binary.

At each step we take each real and make two reals by
appending a zero and one. The number of bits in the
fraction is the mapping from the integer.

Step 1: integer   1  there are 21 reals with 1 fractional bit
        real    0.0
                0.1

Step 2: integer    2 there are 22 reals with 2 fractional bits
        real    0.00
                0.01
                0.10
                0.11

Step 3: integer     3 there are 23 reals with 3 fractional bits
        real    0.000
                0.001
                0.010
                0.011
                0.100
                0.101
                0.110
                0.111

Step 4: integer      4 there are 24 reals with 4 fractional bits
        real    0.0000
                0.0001
                0.0010
                0.0011
                0.0100
                0.0101
                0.0110
                0.0111
                0.1000
                0.1001
                0.1010
                0.1011
                0.1100
                0.1101
                0.1110
                0.1111

The steps continue for all of the integers.
Thus: there are 2n real fractions in the range zero to one
for n integers.

Thus: there are two to the power Omega real fractions.
With a little hand waving, proving by picture that:

There are 2Ω real numbers, א Aleph Null, the cardinality of
the real numbers.

The hand waving is that n 2n becomes 2n as 
n becomes infinity.

qed.

Function programming is different
some information on Functional Programming

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