# Set of natural numbers and Peano axioms

Under standard Peano axioms (below, from Wikipedia), what implies how the set of natural numbers actually looks like, e.g. that 1 = S(0), 2 = S(1), 3 = S(2), etc.?

Why not for example 2 = S(0), 4 = S(2), 6 = S(4), with no odd numbers or some other variation of it?

Or is the above also a part of the definition of natural numbers and I'm missing some axioms here?

1. 0 is a natural number.
2. Every natural number has a successor which is also a natural number.
3. 0 is not the successor of any natural number.
4. If the successor of x equals the successor of y, then x equals y.
5. The axiom of induction: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.
• My response is speculative and may easily be wrong. I surmise that thousands of years ago, math people said, what if we have the number 1, and then construct other numbers by 1+1, 1+1+1, ... Commented Feb 2, 2021 at 1:52
• @user2661923 I get that from intuitive perspective, but I'm wondering why isn't it a part of the formal definition (but maybe I'm wrong and it indeed is). Commented Feb 2, 2021 at 2:03
• I emphasize - my response is (highly) speculative, and based on the idea of : if I were alive back then, and numbers didn't exist, how would I conjure them? Commented Feb 2, 2021 at 2:04
• Well, I could say $s(0) = BOOMCHUNKIFUNKWAAWA$ if I wanted to..... If we say $s(0) = 2$ then we can't say $2 = s(0) + s(0)$. Your axioms define what the natural numbers are but they don't tell us anything about what squiggles of ink we choose to use to represent them. If we want to set $s(0)=$ (upright fishhook with a flat long base) nothings going to be much different then $s(0)=$ (straight line up and down with a narrow short horizontal base and tiny tick at the top). ... As stated your question doesn't actually mean anything. Commented Feb 2, 2021 at 3:56
• ...As stated your question doesn't actually mean anything....unless you have some intuitive idea what $\color{green}{\large2}$ means opposed to what $\color{purple}{\small1}$ means. So: When you say $\color{green}{\large2}$ and $\color{purple}{\small 1}$ what do you think those symbols mean? To my mind using Peano as our constructive basis then $\color{purple}{\small 1}$ means nothing more or less than $\color{purple}{\small 1}:=s(0)$ and $\color{green}{\large2}$ means nothing more or less than $\color{green}{\large2}=\color{purple}{\small 1}+\color{purple}{\small 1}=s(s(0))$. Commented Feb 2, 2021 at 4:10

I think you are missing the point of the Peano Axioms. It postulates that there is a set $$\,\mathbb{N}\,$$ which is by convention called the set of natural numbers. We are given that zero is a natural number and is by convention denoted by $$\,0.\,$$ In order to avoid confusion and emphasize its nature, perhaps it would be better to use a distinct notation such as "$$0$$". The successor of zero is a natural number by convention denoted by "$$1$$". This conventional defining property of "$$1$$" as the successor of "$$0$$" in conjunction with other definitions and the Peano Axioms leads to all of its properties. Similarly for all of the other natural numbers. Each natural number is defined to be the successor the the previous number. In other words, Only the number zero is initially given and all of the rest of the natural numbers are determined as the repeated successors of zero. For example,
$$\text{"}1\text{"} :=S(\text{"}0\text{"}),\; \text{"}2\text{"}:=S(S(\text{"}0\text{"})),\,\dots .\,$$
Thus, if you wish, you can use the set of even numbers as a model of the natural numbers and then define $$\,2=S(0),\,$$ $$4=S(2),\,$$ $$6=S(4),\,$$ $$\dots.\,$$ In this model the number denoted by $$\,2\,$$ is the successor of zero but this does not change its properties in the model of the natural numbers. For example, in this model, we have $$\,2\times 2=2\,$$ using the Peano definition of $$\,\times\,$$ (multiplication) of natural numbers. This is because $$\,2\,$$ models the natural number "$$1$$" and has the same properties in the model as "$$1$$" has.