Motivation:

It recently occurred to me that we may construct the real numbers in such a way that they inhabit an infinite-dimensional vector space. In developing this construction, I was partly motivated by the elegance of this construction from the perspective of theoretical computer science which has developed the notion of types and type hierarchies.

What I find particularly elegant is that this construction shows that you may use the prime numbers to construct the integers, then the rationals, then the real numbers. These are all derived types relative to the prime numbers. Furthermore, I show that this allows a recursive definition of the real numbers that is unique assuming that we optimise for the rate of convergence of the subsequences which would satisfy the principle of minimum energy.

I will add that such a construction is consistent with the implications of the Physical Church-Turing thesis [4].

Construction:

Given the set of prime numbers $$\mathbb{P}$$, it is known that the elements of $$\log \mathbb{P}$$ are dimensionally independent $$p_{i \neq j}, p_j \in \mathbb{P}, \log p_{i \neq j} \perp \log p_j$$ in the sense that:

$$\exists \alpha_i \in \mathbb{Z}, \sum_{i=1}^\infty \alpha_i \log p_i = \log p_j \iff \alpha_j = 1 \land \alpha_{i \neq j} = 0$$

Furthermore, if we define the infinite-dimensional vector space:

$$\text{span}(\log \mathbb{P}) = \Big\{\sum_{i=1}^\infty \alpha_i \log p_i < \infty \lvert p_i \in \mathbb{P} , \alpha_i \in \mathbb{Z}\Big\}$$

and if we define $$\log \mathbb{Q}_+ = \{\log q |q \in \mathbb{Q}_+\}$$ we may deduce from the unique prime factorisation of the integers that:

$$\log \mathbb{Q}_+ = \text{span}(\log \mathbb{P})$$

Now, it recently occurred to me that using the Riemann rearrangement theorem we may deduce that:

$$\forall \alpha \in \mathbb{R} \exists q_i \in \mathbb{Q}, \alpha = \sum_{i=1}^\infty q_i \log p_i$$

I find that this construction illuminates the algebraic structure of the real numbers. They appear to inhabit a kind of Hilbert space. This also suggests that both logarithms and prime numbers were computational primitives at the origin of time. The prime numbers were a fundamental data type and the logarithm was a fundamental operation.

Unique prime factorisation of the positive real numbers:

Since $$\forall x \in \mathbb{R}, e^x \geq 0$$ we may deduce that every positive real number has a prime factorisation $$\forall \alpha \in \mathbb{R}_+ \exists q_i \in \mathbb{Q}$$:

$$\alpha = \lim_{N \to \infty} \alpha_{N} = \lim_{N \to \infty} \prod_{i=1}^N p_i^{q_i}$$

and we may show that this factorisation is unique under the constraint that the reals are defined recursively in terms of subsequences $$\alpha_{N}$$ that have an optimal rate of convergence, which would satisfy the principle of minimum energy.

Indeed, if we define $$\alpha_N$$ recursively we have:

$$\alpha_{N+1} = p_{N+1}^{q_{N+1}} \cdot \alpha_{N}$$

and choosing the $$\alpha_{N}$$ that have an optimal rate of convergence is equivalent to choosing $$q_N$$ in such a way that:

$$q_{N+1}= \min_{q_{N+1} \in \mathbb{Q}} \lVert p_{N+1}^{q_{N+1}} \cdot \alpha_N - \alpha \rVert$$

so we have a unique construction of the real numbers via the Principle of Minimum Energy.

Discussion:

The definition of $$\log(\cdot)$$ appears to presuppose the existence of the real numbers. However, $$\forall p \in \mathbb{P}, \log p$$ is not meant to be evaluated as we are mainly exploiting the property in (1) to construct an infinite-dimensional vector space. This issue is carefully addressed in theoretical computer science, specifically the Lambda calculus, by treating data types, function definitions and function evaluation separately.

References:

1. Michael Nielsen. Interesting problems: The Church-Turing-Deutsch Principle. 2004. https://michaelnielsen.org/blog/interesting-problems-the-church-turing-deutsch-principle/
2. David Deutsch. Quantum theory, the Church–Turing principle and the universal quantum computer. 1985.
3. Eugene Wigner. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. 1960.
4. Aidan Rocke (https://cstheory.stackexchange.com/users/47594/aidan-rocke), Understanding the Physical Church-Turing thesis and its implications, URL (version: 2021-02-20): https://cstheory.stackexchange.com/q/48450