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Tao gives the five axioms without mentioning sets and their elements. So I got confused about the two fundamental concepts the zero number $0$, and the increment operation $++$. I include two of the five axioms for reference.

Axiom 2.1

$0$ is a natural number.

Axiom 2.2

If $n$ is a natural number, then $n{+\!+}$ is also a natural number.

My first question is what is $0$. It seems we did not define it first and just gave a symbol. Is it unique or what it stands for, I have little knowledge of it.

My second question is what is increment operation. Since we omit the mention of sets, we can not define function. How can we know $0++$ is unique and gives another symbol $1$ to it. It seems very confusing to me.

From the link below, I got to know there is something related to second order logic, but even as a fourth math student I have little knowledge of it. Should second order logic be learnt first to understand the contents? The book is called Analysis I and is for freshmen.

Hagen von Eitzen (https://math.stackexchange.com/users/39174/hagen-von-eitzen), Successor in Peano Axioms, URL (version: 2020-11-25): https://math.stackexchange.com/q/3922951 Thanks for your help.

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  • $\begingroup$ The only thing we know is that $0$ is a natural number. That is all we need to know. Axioms don't define things, they state what we are assuming about them. Here, what we know is $0$ is a natural number, and if $n$ is a natural number, then $n++$ is a natural number. We don't know anything about the operation but what is in the axiom. There are no sets or definitions. $\endgroup$
    – Thomas Andrews
    Jul 20 at 14:26
  • $\begingroup$ Axioms are not definitions. To the contrary, axioms are statements of properties of certain objects which remain undefined. There are various ways you could restate Axiom 2.1, but none of them would be a definition per se; for instance you could say something a bit more elaborate like "There is a natural number denoted $0$"; or you could something more terse like "Zero exists". $\endgroup$
    – Lee Mosher
    Jul 20 at 14:46

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The concept of zero, only in regards to Peano's axioms, is not defined as anything other than a natural number. You build the other four axioms based on the fact that zero is a natural number. That is the nature of axioms - we take them as tautologies.

As for your second question: even if the increment operation satisfies the concept of a successor, I'd much rather reformulate the axiom as following:

Every natural number $n$ has a successor, which we denote as $n'$.

In this definition of the axiom, we don't need to define an increment operation, we only define the successor of the natural number $n$ as $n'$.

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  • $\begingroup$ Since we use a symbol to denote the successor, should we say 'Every natural number $n$ has and only has one successor, which we denote as $n'$'. $\endgroup$
    – Andrew_Ren
    Jul 20 at 14:48

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