Is Every Real Number The Limit Of A Sequence Of Irrational Numbers, Constructively?

Mar 10, 2025 by ADMIN 84 views

**Is Every Real Number the Limit of a Sequence of Irrational Numbers, Constructively?**

Introduction

In the realm of real analysis, the concept of sequences and limits plays a crucial role in understanding the properties of real numbers. A fundamental question that arises in this context is whether every real number can be expressed as the limit of a sequence of irrational numbers. This question has been extensively studied in the context of classical mathematics, where the law of the excluded middle and the axiom of choice are often employed. However, in constructive mathematics, which eschews the use of these principles, the situation is more nuanced.

Background

In constructive mathematics, the real numbers are typically defined as Dedekind cuts, which are subsets of the rational numbers that satisfy certain properties. A Dedekind cut is a partition of the rational numbers into two non-empty sets, the left set and the right set, such that every element of the left set is less than every element of the right set. The real number corresponding to a Dedekind cut is defined as the set of all rational numbers that belong to the left set.

The Axiom of Choice and the Law of the Excluded Middle

The axiom of choice and the law of the excluded middle are two fundamental principles in classical mathematics that are often used to prove the existence of certain mathematical objects. The axiom of choice states that for any family of non-empty sets, there exists a choice function that assigns to each set an element from that set. The law of the excluded middle states that for any proposition, either the proposition is true or its negation is true.

In constructive mathematics, these principles are not available, and instead, a more cautious approach is taken. This means that mathematical objects are not assumed to exist unless they can be explicitly constructed.

Constructing a Sequence of Irrational Numbers

Let $x \in \mathbb{R}$ be a real number. We want to construct a sequence of irrational numbers that converges to $x$ . To do this, we need to find a way to approximate $x$ with increasing precision using irrational numbers.

One possible approach is to use the decimal expansion of $x$ to construct a sequence of rational numbers that converges to $x$ . However, this approach is not constructive, as it relies on the law of the excluded middle to prove the existence of the decimal expansion.

A Constructive Approach

A more constructive approach is to use the fact that every real number has a binary expansion. This means that every real number can be written as a sum of powers of 2, where the coefficients are either 0 or 1.

Using this fact, we can construct a sequence of irrational numbers that converges to $x$ as follows:

Start with an arbitrary irrational number $a_0$ .
For each $n \in \mathbb{N}$ , define $a_n$ to be the irrational number that is obtained by adding $2^{-n}$ to $a_{n-1}$ if the $n$ -th digit of the binary expansion of $x$ is 1, and subtracting $2^{-n}$ from $a_{n-1}$ if the $n$ -th digit of the binary expansion of $x$ is 0.
The sequence $a_n$ converges to $x$ .

The Irrationality of the Sequence

To show that the sequence $a_n$ is irrational, we need to show that it does not contain any rational numbers. Suppose, for the sake of contradiction, that $a_n$ is rational for some $n$ . Then, we can write $a_n = p/q$ for some integers $p$ and $q$ .

However, this would imply that the $n$ -th digit of the binary expansion of $x$ is either 0 or 1, which is a contradiction. Therefore, the sequence $a_n$ is irrational.

Conclusion

In this article, we have shown that every real number can be constructed as the limit of a sequence of irrational numbers, without using the law of the excluded middle and the axiom of choice. This result is significant, as it provides a constructive proof of the existence of real numbers.

Open Questions

There are several open questions related to this result. For example, it is not clear whether every real number can be constructed as the limit of a sequence of rational numbers. Additionally, it is not clear whether the sequence $a_n$ can be constructed without using the binary expansion of $x$ .

Future Work

Future work in this area could involve investigating the properties of the sequence $a_n$ in more detail. For example, it would be interesting to study the distribution of the digits of the binary expansion of $x$ in the sequence $a_n$ .

References

[1] Bishop, E., and D. Bridges. Constructive Analysis. Springer-Verlag, 1985.
[2] Bridges, D. Constructive Mathematics: A Foundation for Computational Mathematics. Chapman and Hall/CRC, 1999.
[3] Myhill, J. "Some Properties of Intuitionistic Zermelo-Fraenkel Set Theory." In Logic Colloquium '73, edited by H. E. Rose and J. C. Shepherdson, 97-106. North-Holland, 1975.

Appendix

The following is a proof of the fact that every real number has a binary expansion.

Theorem

Every real number has a binary expansion.

Proof

Let $x \in \mathbb{R}$ be a real number. We want to show that $x$ has a binary expansion.

Since $x$ is a real number, it can be written as a Dedekind cut. Let $L$ be the left set of the Dedekind cut corresponding to $x$ .

We can define a function $f: \mathbb{N} \to \mathbb{N}$ by setting $f(n) = \min \{k \in \mathbb{N}: k \in L \text{ and } k > 2^n\}$ .

We claim that the sequence $f(n)$ converges to a real number $y \in \mathbb{R}$ . To see this, let $y$ be the real number corresponding to the Dedekind cut $\{k \in \mathbb{N}: k \in L \text{ and } k > 2^n \text{ for some } n \in \mathbb{N}\}$ .

Then, for each $n \in \mathbb{N}$ , we have $f(n) \in L$ and $f(n) > 2^n$ . Therefore, the sequence $f(n)$ converges to $y$ .

Since $y$ is a real number, it has a binary expansion. Therefore, $x$ also has a binary expansion.

QED

Introduction

In our previous article, we explored the question of whether every real number can be constructed as the limit of a sequence of irrational numbers, without using the law of the excluded middle and the axiom of choice. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the significance of this result?

A: This result is significant because it provides a constructive proof of the existence of real numbers. In constructive mathematics, the existence of a mathematical object is not assumed unless it can be explicitly constructed. Therefore, this result shows that real numbers can be constructed without relying on the law of the excluded middle and the axiom of choice.

Q: How does this result relate to the axiom of choice?

A: The axiom of choice is a fundamental principle in classical mathematics that states that for any family of non-empty sets, there exists a choice function that assigns to each set an element from that set. In constructive mathematics, the axiom of choice is not available, and instead, a more cautious approach is taken. This result shows that real numbers can be constructed without relying on the axiom of choice.

Q: Can every real number be constructed as the limit of a sequence of rational numbers?

A: This is an open question in constructive mathematics. While we have shown that every real number can be constructed as the limit of a sequence of irrational numbers, it is not clear whether every real number can be constructed as the limit of a sequence of rational numbers.

Q: How does this result relate to the binary expansion of real numbers?

A: The binary expansion of a real number is a way of representing the number as a sum of powers of 2, where the coefficients are either 0 or 1. This result shows that every real number has a binary expansion, which is a fundamental property of real numbers.

Q: Can this result be generalized to other mathematical structures?

A: This result is specific to the real numbers and does not generalize to other mathematical structures. However, it is possible that similar results could be obtained for other mathematical structures, such as the complex numbers or the quaternions.

Q: What are the implications of this result for constructive mathematics?

A: This result has significant implications for constructive mathematics, as it provides a constructive proof of the existence of real numbers. This result shows that constructive mathematics is a viable alternative to classical mathematics, and that it can be used to prove the existence of mathematical objects in a constructive way.

Q: Can this result be used to prove other mathematical theorems?

A: This result can be used to prove other mathematical theorems, such as the intermediate value theorem or the Bolzano-Weierstrass theorem. However, the specific implications of this result for other mathematical theorems are still an open question.

Q: What are the limitations of this result?

A: This result has several limitations. For example, it only applies to real numbers and does not generalize to other mathematical structures. Additionally, it relies on the binary expansion of real numbers, which is a fundamental property of real numbers.

Q: Can this result be used to prove the existence of other mathematical objects?

A: This result can be used to prove the existence of other mathematical objects, such as the real numbers themselves. However, the specific implications of this result for other mathematical objects are still an open question.