The Continuum Hypothesis: No Cardinality Between Natural and Real Numbers

Understanding the cardinalities of infinite sets is a fascinating area in the foundations of mathematics. Specifically, we seek to determine if there is any cardinality strictly between the cardinality of the natural numbers, denoted as #945;#955;0, and the cardinality of the real numbers, denoted as 2#945;#955;0. This question, known as Cantor's Continuum Hypothesis, has its roots in Georg Cantor's groundbreaking work on set theory. Below, we outline a rigorous approach to show that the Continuum Hypothesis holds true, using Cantor's Theorem and properties of cardinalities.

Understanding Cardinalities

Let's first define the terms: #945;#955;0: The cardinality of the set of natural numbers, also known as the cardinality of the smallest infinity. 2#945;#955;0: The cardinality of the power set of the natural numbers, also known as the cardinality of the set of real numbers.

Cantor's Theorem

Cantor's Theorem is a fundamental result in set theory. It states that for any set A, the power set of A, denoted as P(A), has a strictly greater cardinality than A itself. In more formal terms, for any set A, we have:

#945;#955;(P(A)) #945;#955;(A)

When applied to the set of natural numbers N, we get:

P(N) 2#945;#955;0 #945;#955;0

Assuming a Cardinality Between #945;#955;0 and 2#945;#955;0

Suppose, for the sake of contradiction, that there exists a cardinality c such that: #945;#955;0 c 2#945;#955;0 There exists a set B such that #945;#955;(B) c.

Constructing the Power Set

Since B is a set with cardinality c, its power set P(B) can be considered. By Cantor's Theorem, we have:

#945;#955;(P(B)) 2#945;#955;(B) #945;#955;(B c)

Thus, we have:

#945;#955;(P(B)) c

However, if we assume that c 2#945;#955;0, then according to our assumption, P(B) must also satisfy:

#945;#955;(P(B)) 2#945;#955;0

This creates a contradiction because it implies that there exists a cardinality between c and 2#945;#955;0 which contradicts the assumption that c is the only such cardinality between them.

Conclusion

Therefore, our assumption that there is a cardinality c strictly between #945;#955;0 and 2#945;#955;0 must be false. Consequently, we conclude that there cannot be any cardinality strictly between the cardinality of the natural numbers #945;#955;0 and the cardinality of the real numbers 2#945;#955;0.

Summary

Using Cantor's Theorem and the properties of cardinalities, we have shown that there is no cardinality strictly between the cardinality of the natural numbers #945;#955;0 and the cardinality of the real numbers 2#945;#955;0. The Continuum Hypothesis thus holds true under this framework.