What Does It Mean to Be a Number? (The Peano Axioms) | Infinite Series
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as an accessible introduction to the Peano axioms when asking or answering questions about how the natural numbers are defined and how arithmetic is built from them.
Recommended by 1 people across 1 different months, 2021 to 2022.
What it is
A PBS Infinite Series episode hosted by Gabe Perez-Giz that asks whether numbers can be described without already assuming the idea of number. It introduces the Peano axioms, built on a zero element and a successor function, and shows how they let you construct the natural numbers without mentioning counting or arithmetic.
What people said
Recommends user1060801 · 2022-05 · Mathematics Stack Exchange · on “Absolute definition of 'True' and 'False' in Mathematical Logic”
@AlexKruckman in this video: [this video] , they have explained how "Numbers" could be defined by using concepts like set theory, function and equality (the Peano Axioms). Similarly, I expected a definition for "True" and "False", which is built by using some foundamental concepts.
Cites user716881 · 2021-10 · Mathematics Stack Exchange · on “Peano axioms vs set construction of natural numbers”
When I first looked at the construction of natural numbers the Peano axioms were shown as a way to do this, without the need of anything else, in the way described in this video [this video] and Analysis I by Terence Tao. The Peano axioms were described as agnostic on the subject of what $0$ or the successor function was. Then I saw that you could use $0=\emptyset$ and $S(n) = \{n\}$ to get the Zermelo ordinals and $S(n) = n \cup \{n\} $ to get the von Neumann ordinals. However, then I saw that you can use the ZF axioms, like the axiom of infinity, and VN ordinals to construct the natural numbers as shown here [LINK] and here [LINK] . Here the natural numbers were constructed without the Peano Ax
Cites Chang Yao Liu · 2022-08 · Mathematics Stack Exchange · on “Peano addition commutativity proof by induction”
I hope you are already familiar with the five Peano axioms [this video] , from which point we use the standard numerals $0,1,2,3,\dots$ as some shorthand notation for each element of our set $\mathbb{N}$ . The successor function from axiom #2, $S$ , simply adds $1$ in the familiar fashion e.g. $S(2)=3$ . From this we can define addition by the following rules: $$ \begin{align} \\\text{For all } x,y\in\mathbb{N}, \\ \\x\oplus0&=x \\x\oplus S(y)&=S(x\oplus y) \end{align} $$ I’m not using the standard addition sign because we do not yet know if it is both associative and commutative. I can prove associativity by induction. Suppose there is some $k$ where $(a\oplus b)\oplus k = a\oplus (b\oplus k)$ f
Cites MonkeyWithDarts · 2022-11 · Mathematics Stack Exchange · on “Applying Peano Axioms to Subsets of Natural Numbers”
Concise Question Is it possible to apply the Peano axioms to subsets of the natural numbers and thus define arithmetic operations on those subsets strictly in terms of those subsets? If so, is it also possible to create a correspondence between those arithmetic operations and the natural numbers' operations? Wordy Question My knowledge of these subjects is largely informal, so I welcome corrections of the errors I am about to make. My informal understanding of Peano axioms is similar to my software engineering experience: If I can define a suitable initial value and successor function (satisfying the contract that is the Peano axioms), then I can meaningfully talk about arithmetic on the res