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On the first page of Hardy, Wright'a An Introduction to Theory of Numbers, they write:

It is plain that
 b|a.c|b -> c|a,
 b|a -> bc|ac
   if c != 0, and
 c|a. c|b -> c|ma+nb
for all integral m and n. 

where -> means "implies", a|b means "a divides b", . means "and" and we are talking of non-negative integers.

Now while the statements are intuitive it makes me wonder about the basis/proof of most basic laws in arithmetic. For example, above statements can be proved to be valid on the basis of the basic laws of arithmetic (commutative/associative laws etc). But then, aren't those laws just assumptions then?

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1 Answer 1

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Yes.

Nothing can be proved from nothing, so you have to start with 'something' to get anywhere. For $\mathbb{N}$, this 'something' is usually the Peano Axioms, which we then extend to $\mathbb{Z}$ via groups.

But we have no way of proving the Peano Axioms or anything like that, these are taken to be 'true' since they seem to give us objects that work in the same way that we want numbers to work.

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  • $\begingroup$ Thank you for your answer! $\endgroup$
    – Lavya
    Commented Aug 12, 2013 at 10:15
  • $\begingroup$ when I picked the above book (Hardy, Wright), I was hoping it'd talk of such basics first, but it seems to start with prime numbers etc. Do you know of some other book that starts with a discussion of basic axioms and builds up on that..? Thanks! $\endgroup$
    – Lavya
    Commented Aug 12, 2013 at 10:52
  • $\begingroup$ Halmos' "Naive Set Theory" is pretty good for this purpose. $\endgroup$ Commented Aug 12, 2013 at 11:26
  • $\begingroup$ @Lavya If you're interested in the actual axioms, you can either learn some naive set theory or read up on the axioms via Wikipedia or the like. If you want to learn how you use axioms to prove theorems etc, you want to learn some mathematical logic. We used 'Leary: A friendly introduction to mathematical logic', but I don't know if there exists better books in this genre. $\endgroup$
    – Greebo
    Commented Aug 12, 2013 at 12:50
  • $\begingroup$ @Lavya "Foundations of Analysis" by Landau $\endgroup$
    – awkward
    Commented Aug 12, 2013 at 23:46

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