question about the induction axiom and principle of mathematical induction

hi,
i'm doing an extra credit assignment for my math class on mathematical
induction and in my research of it i've come across both an induction
axiom and a principle of mathematical induction. Other than the axiom
being often written in terms of sets and the principle in terms of
P(n), is there any difference between the two?

this leads me to a more general quesiton of is there a difference
between an axiom and a principle, or do they just kind of build on top
of one another?

from what i've gathered, the induction axiom states that:
let S be a subset of positive integers satisfying:
1. 1 is an element of S, and
2. if k is an element of S, then (k+1) is an element of S
Then S = positive integers

and the principle of induction states that:
we have a proposition P(n) concerning two positive integers, one being
n and a (which is a fixed integer often seen as being 1 or 0). And that
P(n) is true when n > a. Suppose that...
1. P(a) is true, and
2. we let k, being an arbitraty number less than or equal to a and
that for every k which is a positive integer, P(k) implies P(k+1).
Then P(n) is true for all positive integers n.

now, to me being a 17 yr old math student, these two things seem to
state the same thing just in two difference notations. am i missing
something here, or does the principle just build upon the axiom?

thanks :)

.

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