thanksarildno said:Hi, roger:
If you think HallsofIvy's answer was a bit difficult, think in the following manner:
1) Ideally speaking, EVERY concept or symbol we want to use, must enter our maths by a DEFINITION.
That is, we MUST KNOW WHAT WE TALK ABOUT, before deducing consequences from our quantities/structures.
( It might be that we reach a situation in which some concepts, or some relations between such concepts are so fundamental that we are unable to define these in terms of other, deeper concepts, but that is not the issue here.)
2) Hence, before we can use the symbol "2" we must define it somehow, and basically, that is done by stating that "2" is the symbol for the quantity we get when adding 1 to 1 (this does, to some extent, assume we have clarified what we mean by "1" and "adding" (and "being equal to"))
That is, we INTRODUCE the symbol "2" through the equation
2=1+1
That is, we say that this particular equation is true, by definition of the symbol "2".
If you proceed deeper along these lines, the formalism sketched by HallsofIvy is what mathematicians have ended up with.
It's just an axiom, the reason it is there because otherwise we cannot tell whether zero has a sucessor or not. (it's worth noting that just if not even more frequently we start with the number 1 rather than zero i.e. we don't include zero as a natural number).roger said:thanks
Would you please explain why zero is not a successor of any number ?
Because they do not fufil Peano's axioms.Why are the negative integers not natural numbers ?
Yes, they are 'countably infinite', infact the defintion of a countably infinite set is one for which there exists a bijection (a bijection can be thought of as a map that takes a number from one set and assigns it another set in a manner that uses all the numbers of both sets) between it and the natural numbers.Are natural numbers countable even though it could contain infinite elements ?
Note that, of course, they haven't actually proven 1 + 1 = 2, since they haven't actually successfully defined addition yet. ;)http://en.wikipedia.org/wiki/Image:Principia_Mathematica_theorem_54-43.png" [Broken]
You see? :-)
Do you have statistics to back up that claim? Certainly some people find it counterintuitive, but that's not enough to claim 'most'.It's just a mathematical fact that most people find deeply counterintuitive.