Is Russell's proof of addition with Peano's 5. Axiom valid?

Question

This is a follow up question to my previous question: Why define addition with successor?

In this one I'd like to ask about Russell's use of Peano's 5. Axiom to prove his definition of addition:

Suppose we wish to define the sum of two numbers. Taking any number $m$, we define $m+0$ as $m$, and $m+(n+1)$ as the successor of $m+n$. In virtue of (5) this gives a definition of the sum of $m$ and $n$, whatever number $n$ may be.

I follow T. Tao's definition of the 5. Axiom as stated here: Fifth Peano axiom — Properties of the natural numbers

In Russell's case the property $P$ is the identity property of zero: $P(0) \equiv 0 + m = m$

Here Russell is not testing the relation P as true or false for zero but he defines the identity property of zero by using addition symbol "+".

But $P(n)$ and $P(n+1)$ are not true because neither $n$ nor $n+1$ has the identity property.

Since $P(n)$ and $P(n+1)$ are not true, the conditions for the 5. Axiom are not satisfied.

What's wrong?

Answer 1

We have to start from the definition of sum in terms of $0$ and sucecssor:

$m+0=m$

and:

$m+s(n)=s(m+n)$,

and then we have to use induction to prove that the definition "works", i.e. "for every natural number $m$ there is natural number $n$ satisfying the definition and that this number is unique".

The relevant property $P(m)$ will be: $\forall y \exists z(m+y=z)$.

First step, to prove it for $0$, i.e. to prove that $\exists z(m+0=z)$. The result is immediate from the definition.

The next step, according to Russell's Axiom 5 (induction), will be: assume that the above holds for $m$ and prove that it holds also for $s(m)$.

See Addition of natural numbers for details. The same in Ch.2.2 Addition of Tao's Analysis I (page 24-on).