Most likely this subject has been covered many times here, still I fail to grasp this.
I can't understand how do we know that the successor of $1$ is $2$ based on Peano's axioms, given that we start from $1$ (for convenience, since the book I am reading starts from $1$)? It is all good to the point where we demonstrate that there is a unique function such that
- $f(x, 1) = x'$
- $f(x, y') = (f(x, y))'$
However, it all falls apart for me, when somehow magically $f(x, y')$ becomes $x+y'$ or particularly $f(x, 1) = x+1$, hence $x+1 = x'$. My problem is that, as we have never been told by axioms what successors really are, just defining the function $f$ through some plus-sign notation doesn't make it clear how to perform (calculate, evaluate) that operation. In my eyes, it is the same as if it was written like $f(x, 1) = x ? 1 = x'$ (yes, the operation is a question mark). So, how does $x ? 1$ imply that the successor of $1$ should be $2$? Where does it come from that we assign $2$ to the successor of $1$, $3$ to the successor of $2$, etc. besides the intuition that it should be this way?
P.S. It reminds me of a square root function $r(x)=\sqrt{x}$ with fancy notation $\sqrt{x}$, definition of which doesn't really tell us how to calculate one but rather defines a property of the result that whatever it is, its square should be equal to the $x$. Similar feeling about $x+1$ is also present here:
- either I have to know where it comes from that the successor of $1$ is $2$, then $x+1$ becomes a mere conventional notation for the successor, on top of which more general operation $x+y$ is defined
- or how to actually perform $x+1$ to get $2$, when $x=1$, which is not obvious to me from the axioms. Although I have an intuition how to define $x+1$ through set-theoretic notation, where natural numbers are defined through empty set ($0 = \varnothing,1 = \{\varnothing, \{\varnothing\}\}, \dots$), I am interested how it derives from Peano's axioms, not from the set theory.
Best Answer
The general concept of an axiom system does not tell you what the objects are that it refers to, it tells you only properties of those objects and of the interactions between them. The axioms do not prescribe what things actually are, they merely describe things.
So, for example, the Peano Axioms do not tell you what $2$ is. Also, they do not tell you what $+$ is. Nor do the tell you how to go about actually performing $x+1$.
In fact that's the whole point of the Peano Axioms. They are not authoritative. Their sole purpose is to give one a tool for building the natural numbers on a simpler foundation.
Our experience of the natural numbers includes many named objects like $1, 2, 3$ and $485096$, and complicated binary operations such as $+$ and $\times$, and complicated binary relations such as $<$, and long lists of useful identities such as $a \times (b+c) = a \times b + a \times c$. The point of the Peano Axioms is that they give us a much less complicated system of objects and properties to accept, from which we, using our powers of logic, can construct the much more complicated system of objects and properties which comprise the theory of natural numbers. Once you accept this simpler foundation (based on your intuition, or on whatever basis you choose to accept), you can then then apply the Peano Axioms to define $2$ and $3$ and so on.
Have you accepted the successor function? Great, now define $2=1'$. Next, define $3=2'$ and $4=3'$ and $5=4'$ and $6=5'$ and $7=8'$ and $9=8'$. That was boring, onwards.
Have you accepted the axioms of the successor function? Great, now define addition, and prove it is commutative and associative, then define multiplication and prove is it commutative and associative and that multiplication distributes over addition, and so on.
What if you don't want to accept the Peano Axioms? What if you really, really want someone to tell you what $2$ is and $3$ is and $+$ is and so on?
Well, there are alternatives.
For example, many modern textbooks of analysis will start instead with axioms for the real numbers, and from those will construct the natural numbers satisfying Peano's Axioms; Fitzpatrick's Advanced Calculus textbook is an example, which I used in my Advanced Calculus class.
For another example, in modern set theory one starts with the ZFC axioms, and then using Von Neumann ordinals one constructs the natural numbers satisfying Peano's Axioms.
However, in both cases it's just a matter of pushing the ball backwards: you have to accept some axioms as your starting point: Peano's axioms; the axioms of the real numbers; the ZFC set theory axioms; or something.