This question is kind of a follow-up question to this. I am also using Terence Tao's book and I still struggle to understand why the fifth Peano axiom is valid.
Tao defines the fifth axiom in the following way.
Axiom 2.5 (Principle of mathematical induction). Let P(n) be
any property pertaining to a natural number n. Suppose that P(0)
is true, and suppose that whenever P(n) is true, P(n++) is also
true. Then P(n) is true for every natural number n.
He then continues to explain the axiom intuitively and prove informally that now we can create the natural numbers.
The informal intuition behind this axiom is the following. Suppose P(n) is such that P(0) is true, and such that whenever
P(n) is true, then P(n++) is true. Then since P(0) is true,
P(0++) = P(1) is true. Since P(1) is true, P(1++) = P(2) is
true. Repeating this indefinitely, we see that P(0), P(1), P(2),
P(3), etc. are all true – however this line of reasoning will never
let us conclude that P(0.5), for instance, is true. Thus Axiom
2.5 should not hold for number systems which contain “unnecessary” elements such as 0.5. (Indeed, one can give a “proof” of this
fact. Apply Axiom 2.5 to the property P(n) = n “is not a half-integer”, i.e., an integer plus 0.5. Then P(0) is true, and if P(n)
is true, then P(n++) is true. Thus Axiom 2.5 asserts that P(n)
is true for all natural numbers n, i.e., no natural number can be a
half-integer. In particular, 0.5 cannot be a natural number. This
“proof” is not quite genuine, because we have not defined such
notions as “integer”, “half-integer”, and “0.5” yet, but it should
give you some idea as to how the principle of induction is supposed
to prohibit any numbers other than the “true” natural numbers
from appearing in N.)
This is where he loses me.
To me, it looks like Tao just took the property we need to create the natural numbers "n 'is not a half-integer'" and therefore, of course this set must be the natural numbers.
However, as far as I can tell, the property is not defined, so with that train of thought, I could also start my induction at P(1) and state that every natural number is uneven. Since 1 is uneven, 1++ = 2 is uneven as well and so forth, which of course is false.
What am I missing? Why does the fifth axiom only allow for the natural numbers as we know them? And specifically: What does "property" really mean in this context so that I cannot prove anything? There must be a certain property that only non-decimal postive numbers have, but as far as I can tell, we cannot define such a property with the five peano axioms.
Best Answer
We have an official way of writing PA that only uses a little more than a dozen different characters. They include the successor function, plus, and times. The induction axiom then accounts for all statements with one free variable that you can write in this language. It allows you to derive all we generally think of as number theory.
Your example that all integers are odd fails because you can prove $P(1)$ but you cannot prove the induction step. In particular $P(1)$ does not imply $P(2) $
When the induction axiom is stated this way it does not really say that everything we are talking about is a natural number, where we informally define a natural as one that you get by applying successor to $0$ a finite number of times. It is stated this way because it is the best you can do in first order logic, which has very nice proof properties. Unfortunately, it does not let you rule out that there are more numbers than the naturals we know and love. It is possible there are what are called nonstandard numbers. I could define a statement with one free variable $Q(n)$ that means $n$ is either even or odd. That statement is provable by induction. I am allowed to conclude that all numbers are either even or odd, even the nonstandard ones (if they exist). It is a large and complex subject.