Existence of Sets with Cardinality Between Integers and Real Numbers

Introduction

Is there a set of numbers whose cardinality is greater than that of the integers but less than the cardinality of the real numbers? The question seems paradoxical at first, but through the lens of set theory and the concept of cardinality, we can explore this intriguing mathematical phenomenon.

Understanding Cardinality

Cardinality in set theory refers to the size of a set. Two sets have the same cardinality if there exists a one-to-one correspondence (bijection) between them. Different sets can have different cardinalities, and some sets are considered larger than others in terms of cardinality.

Cardinality of the Integers

The set of integers, denoted as (mathbb{Z}), has a cardinality denoted by (aleph_0) (aleph-null). This is the smallest infinite cardinality, representing a countably infinite set. You can list all integers in a sequence, making them countable.

Cardinality of the Real Numbers

The set of real numbers, denoted as (mathbb{R}), has a cardinality denoted by (2^{aleph_0}), or the cardinality of the continuum. This is strictly greater than (aleph_0). The real numbers include all rational and irrational numbers, making them uncountably infinite.

Intermediate Cardinalities

Let's explore some sets that have cardinalities between that of the integers and the real numbers.

Countable Subsets of Real Numbers

A classic example is the set of all countable subsets of the real numbers. The cardinality of this set of all countable subsets of (mathbb{R}) is (2^{aleph_0}). However, the set of all finite subsets of (mathbb{R}) is still countable and has cardinality (aleph_0). This shows that the set of countable subsets has a cardinality greater than (aleph_0) but less than (2^{aleph_0}).

Sequences of Rational Numbers

Another interesting set is the set of all sequences of rational numbers. Each sequence can be seen as a function from (mathbb{N}) to (mathbb{Q}). This set has a cardinality greater than (aleph_0) but less than (2^{aleph_0}). While it is uncountably infinite, it does not reach the cardinality of the continuum. This is because the set of all sequences of rational numbers is a proper subset of the power set of (mathbb{N}).

The Interval (0, 1)

The set of all real numbers between 0 and 1, denoted as ((0, 1)), has the same cardinality as the entire set of real numbers, (2^{aleph_0}).

The Continuum Hypothesis

The question of whether such sets exist is closely related to the Continuum Hypothesis (CH). The Continuum Hypothesis states that there is no set whose cardinality is strictly between that of the integers and the real numbers. However, this hypothesis cannot be proven or disproven within the standard axioms of set theory (ZFC).

In models where the Continuum Hypothesis holds (such as the constructible universe (V L)), no such set exists. In models where the Continuum Hypothesis is negated, the first uncountable ordinal can serve as an example of a set whose size is strictly between that of the natural numbers and the real numbers.

Conclusion

In summary, while the Continuum Hypothesis remains independent of standard set theory axioms, there are indeed sets with cardinalities that fit between the integers and the real numbers. The sets of countable subsets of real numbers and certain constructions involving sequences are examples of such sets.