Vtgyjfy

In what follows we shall assume the conventional interpretation of
the symbols $'=','ne'$ and the other logical symbols used to formulate
definitions. Lowercase Latin letters $a,b,dots,z$
will designate number variables. The notation $mathcal{P}left[nright]$
represents a proposition regarding a number variable $n$. By a number
we mean an element of the set $mathbb{N}$ defined by the following
axioms:

I. $1inmathbb{N}.$

II. $underset{a}{forall}exists a^{prime}inmathbb{N}.$

III. $a^{prime}=b^{prime}implies a=b.$

IV. $underset{a}{forall}a^{prime}ne1.$

V. $left(mathcal{P}left[1right]landunderset{n}{forall}left(mathcal{P}left[nright]impliesmathcal{P}left[n^{prime}right]right)right)impliesunderset{n}{forall}mathcal{P}left[nright].$

The object $a^{prime}$ is called the successor of $a$. That is my rendering of the axioms attributed to Peano.

We now define the set of functions

$$
Phi=left{ varphi_{a}:mathbb{N}tomathbb{N}backepsilonvarphi_{a}left[x^{prime}right]equivvarphi_{a}left[xright]^{prime}right} .
$$

For notational convenience we also define

$$
left[_+right]:mathbb{N}_{1}toPhi
$$

so that

$$
left[a+_right]equivvarphi_{a},
$$

to be interpreted as
$$
a+b:mathbb{N}timesmathbb{N}tomathbb{N}text{ where }a+bequivvarphi_{a}left[bright].
$$

From the above definitions we may prove the associative and commutative
properties of addition. By defining the symbols $'<'$ and $'>'$
as meaning

$$
a<bimpliesunderset{c}{exists}a+c=btext{ and }a>bequiv b<a,
$$

we may define (or show) the well ordering of $mathbb{N}.$

If we now define the notation $bar{a}$ to mean $a<bar{a},$ we
have
$$
negunderset{x}{exists}x+bar{a}=a.
$$

That is, $mathbb{N}$ contains no negative numbers. We also have
the condition
$$
underset{a,b}{forall}a+bne a.
$$

Which says $0notinmathbb{N},$ or that there is no additive identity
element in our set of numbers.

If we define numbers to be the set $mathbb{N}_{0}$ by replacing
the symbol $'1'$ with the symbol $'0'$, without bestowing additional
properties to $'0'$, we arrive at exactly the same algebraic structure
as we have for $mathbb{N}.$

Let us assume we agree that by the natural numbers we intend
the most fundamental complete set of numbers of interest to a mathematician.
We shall further assume that the only reasonable choices are between
$mathbb{mathbb{N}}$ and $mathbb{N}_{0}$ as defined above. To
argue that it is preferable to choose $mathbb{N}_{0}$ rather than
$mathbb{N}$ presupposes that the element which is not a successor
has a different interpretation when symbolized as $'1'$ than when
symbolized $'0'.$

So my question is this: is the basic algebraic structure intended
by $mathbb{mathbb{N}}=left{ 1,2,dots,0right} $ distinct form
that intended by $mathbb{N}_{0}=left{ 0,1,2,dotsright} ?$

Clearly, $nmapsto n+1$ is an isomorphism between these two, hence both have the same logical structure.
If you prefer to always work with $Bbb N$, you may want to introduce he operation $+^*$, where $a+^* b$ is the unique $c$ with $c'=a+b$. Then $+^*$ is associative, abeliam amd does have a neutral element - an operation that sometimes comes in handy. If you need suc an operation on $Bbb N$, you can use $+^*$ - or work in the isomorphic $Bbb N_0$, where $+^*$ becomes ordinary addition. This makes $(Bbb N,+^*)$ isomorphic to $(Bbb N_0,+)$, but of course $(Bbb N,+)$ and $(Bbb N_0,+)$ are not isomorphic.