[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] We do not need any natural numbers, we just need an ordered field, then construct the rational numbers from the ordered field. We can do it impredicatively, by taking the intersection of all ordered subfields of the ordered field. Similarly, if we have an arbitrary ordered integral domain, the we can construct the integers as the intersection of all ordered integral subdomains of the ordered field. In particular, the integers are the intersection of all ordered integral subdomains of the rational numbers. ---- Big question, do the natural numbers have zero in them? We can sidestep this entire discussion by beginning with the integers, which can be constructed via universal properties or the induction principle of integers, and then constructing the partition of the integers into the positive and negative integers and zero. Then we also have the non-negative, non-positive, and non-zero integers. * [[Christian Sattler]], *Natural numbers from integers* ([pdf](https://www.cse.chalmers.se/~sattler/docs/naturals.pdf)) ---- There is a hierarchy of ordered field structures Rational numbers -> Archimedean ordered fields -> Archimedean Euclidean fields -> Archimedean analytic Euclidean fields -> Cauchy complete Archimedean ordered fields -> the terminal Archimedean ordered field ---- We need Archimedean ordered fields which has all analytic functions... ## Entire functions What is an entire function? * Wikipedia, [Entire function](https://en.wikipedia.org/wiki/Entire_function) Given real number $a$, an entire function is a function $f:\mathbb{R} \to \mathbb{R}$ with a sequence $c:\mathbb{N} \to \mathbb{R}$ such that for all real numbers $x$, $f(x)$ is the limit of the power series $$\sum_{i = 0}^{\infty} c(n) \frac{(x - a)^n}{n!}$$ What is the limit of the power series? It is the limit of the sequence of partial sums $g:\mathbb{R} \times \mathbb{N} \to \mathbb{R}$ defined as $$g(x, m) = \sum_{i = 0}^{m} c(n) \frac{(x - a)^n}{n!}$$ Now we need to express it in $\epsilon$-$\delta$ terms. However, in order to do that we additionally need to assume a modulus of convergence for the series. $$\prod_{x:\mathbb{R}} \prod_{\epsilon:\mathbb{Q}_+} \sum_{N:\mathbb{N}} \prod_{m:\mathbb{N}} \prod_{m:\mathbb{N}} (m \geq N) \times (n \geq N) \to (-\epsilon \lt g(x, m) - g(x, n) \lt \epsilon)$$ $$\sum_{M:\mathbb{R} \times \mathbb{Q}_+ \to \mathbb{N}} \prod_{x:\mathbb{R}} \prod_{\epsilon:\mathbb{Q}_+} \prod_{m:\mathbb{N}} \prod_{m:\mathbb{N}} (m \geq M(x, \epsilon)) \times (n \geq M(x, \epsilon)) \to (-\epsilon \lt g(x, m) - g(x, n) \lt \epsilon)$$ Then to say that $f$ is the limit of the sequence we have $$\prod_{x:\mathbb{R}} \prod_{\epsilon:\mathbb{Q}_+} \sum_{N:\mathbb{N}} \prod_{m:\mathbb{N}} (m \geq N) \to (-\epsilon \lt g(x, m) - f(x) \lt \epsilon)$$ We talk about convergent power series. Every convergent power series has a limit function on the real numbers. ## Analytic functions What is an analytic function? * Wikipedia, [Analytic function](https://en.wikipedia.org/wiki/Analytic_function) Given real number $a$ and positive real number $\epsilon$, let $(a - \epsilon, a + \epsilon) \subseteq \mathbb{R}$ denote the open interval with endpoints $a$ and $b$. An analytic function is a function $f:(a - \epsilon, a + \epsilon) \to \mathbb{R}$ with a sequence $c:\mathbb{N} \to \mathbb{R}$ such that for all real numbers $x$ such that $a - \epsilon \lt x \lt a + \epsilon$, $f(x)$ is the limit of the power series $$\sum_{i = 0}^{\infty} c(n) \frac{(x - a)^n}{n!}$$ What is the limit of the power series? It is the limit of the sequence of partial sums $f:(a - \epsilon, a + \epsilon) \times \mathbb{N} \to \mathbb{R}$ defined as $$f(x, m) = \sum_{i = 0}^{m} c(n) \frac{(x - a)^n}{n!}$$ Now we need to express it in $\epsilon$-$\delta$ terms. However, in order to do that we additionally need to assume a modulus of convergence for the series. ## Analytic continuation We just add to the Archimedean ordered field the axiom that it has all analytic functions. We need analytic continuations to continue the functions to the unbounded situation. * Wikipedia, [Analytic continuation](https://en.wikipedia.org/wiki/Analytic_continuation) which is necessary for defining the commonly used analytic functions, such as the exponential, logarithm, trigonometric functions and their inverses. category: redirected to nlab