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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.
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  • $\begingroup$ 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, ... $\endgroup$ Commented Feb 2, 2021 at 1:52
  • $\begingroup$ @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). $\endgroup$
    – Treex
    Commented Feb 2, 2021 at 2:03
  • $\begingroup$ 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? $\endgroup$ Commented Feb 2, 2021 at 2:04
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    $\begingroup$ 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. $\endgroup$
    – fleablood
    Commented Feb 2, 2021 at 3:56
  • $\begingroup$ ...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 $\color{green}{\large2}=\color{purple}{\small 1}+\color{purple}{\small 1}=s(s(0))$. $\endgroup$
    – fleablood
    Commented Feb 2, 2021 at 4:10

1 Answer 1

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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,

$$ \text{"}1\text{"} :=S(\text{"}0\text{"}),\; \text{"}2\text{"}:=S(S(\text{"}0\text{"})),\,\dots .\,$$

The actual identity of the other natural numbers is not important. What is of great importance is that they are all definite successors of zero. You are allowed to use any given set as the set of natural numbers as long as one element is singled out as the zero element and all the rest of the elements are the successors of zero.

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.

This is an example of the abstract nature of modern axiomatic mathematics. The natural numbers are not defined by what they are, but by what they do. All models of the natural numbers based on the Peano Axioms are equivalent in the sense that they all have the same properties in the model. The Wikipedia article Peano axioms has a lot of details, but the fundamental idea is that it is an axiomatic model of the natural numbers. Previously generations of mathematicians took them as given somewhat as in the quote "God made the integers, all else is the work of man". The natural numbers have not changed, but our view of their nature has.

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  • $\begingroup$ "According to Peano postulates the defining property of 1 is that it is the successor of 0." Where is this postulated in the Peano axioms? It seems this should be an additional axiom. Peano axioms state that 0 is member of the set of natural numbers but never explicitly state that 1 is the successor of 0. $\endgroup$ Commented Dec 24, 2022 at 16:17
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    $\begingroup$ @CharbelBejjani Thanks for your comment! Point taken. I will have to restate my answer accordingly. $\endgroup$
    – Somos
    Commented Dec 24, 2022 at 16:58

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