# Defining natural numbers without $0$ or $1$.

Let's define Peano's axioms having $2$ as the first number:

1. $\newcommand\Nt{\mathbb N''}2\in\Nt$.
2. $\newcommand\next{\mathop{\mathrm{next}}}\forall n\in\Nt:\next n\in\Nt$ (or $\next:\Nt\to\Nt$).
3. $\forall n,m\in\Nt:\next n=\next m\implies n=m$.
4. $\forall n\in\Nt:\next n\ne2$.
5. $\forall A\subseteq\Nt:(2\in A,\wedge,\forall n\in A:\next n\in A)\implies A=\Nt$.

We define two operators and an order relationship $\langle\Nt,+,\cdot,<\rangle$ as follow.

Order:

1. $<\subset\Nt\times\Nt$.
2. $\forall n,m\in\Nt:n<m,\veebar,m<n,\veebar,n=m$.
3. $\forall n\in\Nt:n<\next n$.
4. $\forall n,m,p\in\Nt:n<m,\wedge,m<p\implies n<p$.

1. $+:\Nt\times\Nt\to\Nt$.
2. $\forall n\in\Nt:n+2=\next\next n$.
3. $\forall n,m\in\Nt:n+\next m=\next(n+m)$.

Multiplication:

1. $\cdot:\Nt\times\Nt\to\Nt$.
2. $\forall n\in\Nt:n\cdot2=n+n$.
3. $\forall n,m\in\Nt:n\cdot\next m=(n\cdot m)+n$.

We are defining $\Nt$ as the natural numbers with $2$ as the first element.

Some properties of $\Nt$:

1. A prime number is a number which has no proper divisors.
2. Two numbers $n,m$ are co-primes if they have no $\gcd$.
3. The fundamental theorem of arithmetics does not have to exclude $1$ or $0$ but it gets weird by not being able to define $p_k^1$: either $p_k$ is no part of the prime expansion, is part, or is part with an exponent.

Which theorems, lemmas or definitions of arithmetics and number theory would become the most cumbersome to formulate in $\Nt$ as opposed to $\mathbb N$ (with or without $0$)?

Which theorems, lemmas or definitions of arithmetics and number theory would become easier to formulate in $\Nt$?

There's no difference in expressive power. The only people who would ever notice even the slightest difference between the language and theory of $\mathbb{N}$ and $\mathbb{N}''$ are the handful of people who like to study the first few steps of how to derive the theory from the specific axioms and definitions. Everybody else just uses the integers, the rational numbers, Turing machines, and everything else you construct out of Peano's axioms in exactly the same way.
It may help to notice that the algebra $\langle \mathbb{N}'', +, \cdot \rangle$ is isomorphic to the algebra $\langle \mathbb{N}, +'', \cdot'' \rangle$ defined by
• $a +'' b = a + b + 2$
• $a \cdot'' b = (a+2) \cdot (b+2) - 2 = a \cdot b + 2 \cdot (a+b + 1)$