An Algebraic Proof of the Continuum Hypothesis

A Proof of the Continuum Hypothesis by Analysis of the Function f(x) = C on the Interval (a, b)

The continuum hypothesis was advanced by Georg Cantor in 1878. It is a hypothesis based on the possibility that infinite sets come in different sizes. Cantor proposed the hypothesis after proving that there is no one to one correspondence between the set of real numbers R and the set of natural numbers N. His conclusion that the real numbers is a larger infinity than the natural numbers was based in part on a constructive proof called the diagonal method.

The diagonal method shows that when trying to list the natural numbers and the real numbers side by side, one for one, that it is possible to construct a real number that won’t appear in the list; for which there is no corresponding natural number. He reasoned that the set of real numbers must be larger than the set of natural numbers even though both sets are infinite. Another way of looking at his results is to assert that it is not possible to list the real numbers using brute force.

Problems arise with the diagonal method due to Cantor’s reliance on:

1.                 The employment of only infinitely long real numbers,
2.                 The use of decimal approximation of real numbers of finite length, and
3.                 The requirement that the real number list be constructed in a very specific manner.

Cantor’s constructive proof is just too contrived. It’s like a trading system that’s been optimized in order to perform well when back tested but can’t generate the same returns in real-time trading.

Kurt Gödel (1940) showed that the CH cannot be disproved from ZF (Zermelo-Fraenkel set theory), even if the axiom of choice (AC) is added. Paul Cohen (1963, 1964) showed that the CH cannot be proven from the ZFC axioms. Since the HC cannot be proved or disproved within ZFC, it is independent with regard to ZFC.

If the HC is independent of the axioms of set theory is there another method available that can be employed to settle the question of its truth or falsity? Using a simple algebraic function, we can show that it is possible to map the natural numbers onto an infinite subset of the real numbers in a very straight forward manner; without resorting to the tricks and restrictions of the diagonal method.

Given – 

1. The set of natural numbers N. Let n stand for an element of N so that

N, {n ∊ N | 1 ≤ n}

2. The set of real numbers R. Let r stand for an element of R so that

R, {r ϵ R | a < r < b}

where a and b are real numbers.

3. The continuous function

f(x) = C where C is any constant and a < x < b.

4. The Evaluate Function Operator – Let E f(x) d_i be taken to mean “evaluate the function f(x) over the domain d where i is an index of the number of iterations of E and i = (1, 2 …).

Equivalencies:

1. The domain d of f(x) is the set R, {r ϵ R | a < r < b} that is d = R.

2. The index i is the set N, {n ∊ N | 1 ≤ n} that is i = N

Since the function f(x) = C is continuous over the interval (a, b) the domain d of f(x) = C must, by definition, contain all real numbers between a and b. This fact is central to the proof of the following theorem.

Theorem 1: The domain d of the continuous function f(x) = C is countable over the interval (a, b).

Proof: Invoking the evaluate function operator on f(x) = C we have

E f(x) = C d_i over (a, b) where i = (1, 2 …)

We can plot the data points as an array showing the values of i, x and f(x), depicted below.

i	1	2	3	…	n	…
	↓	↓	↓	…	↓	…
x	r1	r2	r3	…	rn	…
f(x)	C	C	C	…	C	…

The process of plotting data points can go on indefinitely. We will never run out of elements of the domain to plot nor will there be any missing elements, since the domain by virtue of the fact that f(x) = C is continuous over the interval (a, b), contains all real numbers between a and b. Conversely no matter how long the process goes on, we will never run out of values of i. These facts lead to the conclusion that there is exactly one i for every r and exactly one r for every i in the array. We have thus constructed a one to one correspondence between i and d and we can then write the correspondence as a bijective function,

f : i → d

Having shown that  f : i → d exists we can assert that the cardinal numbers of i and d are equal and the sets are the same size.

Theorem 1 can now be used to prove Theorem 2.

Theorem 2: The set of natural numbers

N, {n ∊ N | 1 ≤ n}

and the set of real numbers

R, {r ϵ R | a < r < b}

have the same cardinality and, as a result. are the same size.

Proof: Theorem 1 proves that  f : i → d exits and equivalencies 1 and 2 establish that i = N and d = R. Substituting N for i and R for d in the bijective function

f : i → d

we have

f : N → R

which completes the proof.

Theorem 3: The Continuum Hypothesis: there can be no infinite set with a cardinality strictly between that of the set of natural numbers and the set of real numbers.

Proof: The proof is a straight forward extension of Theorem 2. It has been shown that the cardinal numbers of N and R are the same. It is self-evident that it is impossible for any infinite set to have a cardinal number in between two infinite sets of the same cardinality. We have shown that N and R are of the same cardinality and therefore it is not possible for any infinite set to have a cardinal number between them. This completes the proof and confirms the truth of the hypothesis.

Ron Ragusa
May 27, 2018