That’s the basic idea.
You’re misunderstanding what I mean by “model”. The natural numbers are not meant as a model of something else. The natural numbers simply are, and what they are is what is described by the Peano axioms. Any collection of things and relations that obeys those axioms is a model of the natural numbers, not the other way around. Whether the real world contains any models of the natural numbers is a physics or philosophy question, and can’t properly be addressed by mathematics.
OK, I’m thoroughly confused now! 
Isn’t a model (in this context) a symbolic representation of a physical reality? I mean, we model a lot of things these days - weather, stock market performance, engineering projects, etc. We set up a symbol with symbolic properties that we believe reflect a thing in the real world, we constrain it to the rules we believe are operative in the real world, and then we do something to it and watch what the thing does/becomes, right? I mean, that’s MY understanding of modelling. But from what you’re saying above, you mean something completely different!
For example, if I have five apples and I gain another three, I don’t have to count them all to know that I have eight apples in total. I can use the natural numbers model, which in human experience has always corresponded to reality, to add five and three and get eight. It sounds to me (and maybe this is because I’m an idiot) that you are saying that the eight apples are a model for the natural numbers. I’m confoosed!
Mathochist is using the word model in the context of model theory. Don’t worry about being confused–it’s not something that most undergrad math students would run into.
Isn’t a ring something you wear on your finger? I’ve explained a number of times now what I mean by “model”. Why do you insist that words have the same technical meaning as non-technical meaning?
I’m saying that if you have a collection of apples such that
[ul]
[li] one of them is called “zero”[/li][li] there is a rule that given one apple tells you which apple is its “successor”[/li][li] the apple “zero” isn’t the successor of any other apple[/li][li] two apples have the same successor if and only if they are the same apple[/li][li] if a statement about the apples is true for “zero” and holding true for one apple implies it holds true for its successor, then it is true for all apples[/li][/ul]
then the collection of apples and various rules for manipulating the apples provides a model for the natural numbers.
In that case, any statement which can be derived purely from those five axioms is a property of the structure of natural numbers, and thus holds for any model of the natural numbers. In particular, if we define addition by iterated succession the apple “one” “plus” the apple “one” is the apple “two”. If the apple “one” “plus” the apple “one” isn’t the apple “two”, then one of those axioms must fail for the apple system, and the system isn’t a model for the natural numbers.
That “1+1=2” holds in the natural numbers has been proven a number of times in this thread by arguing from the Peano axioms. Whether there are any models like the proposed apple system in the real world that we can apply this fact to is beyond the scope of mathematics.
I don’t blame you for being confused. The word “model” is used in math in two very different ways. One of them is the one you’ve described. But in the context of axiomatic systems, “model” means something different (in fact, almost completely opposite). Basically, if you have a set of axioms, any particular “thing” that satisfies that set of axioms, is a model. Or, as Ian Stewart explains it in Concepts of Modern Mathematics,
Gee, Mathochist, maybe it’s because I’m an idiot and was asking for clarification, just as I said in my last post. You may have written about this topic in this thread, but I am sufficiently (and admittedly) ignorant that I couldn’t even tell that. I’m sorry that I so terribly inconvenienced you by not realizing that Modeling was a technical term for something that is almost the exact reverse of what is meant by the word (even technically) in fields other than pure Math.
ultrafilter, I followed your Wikipedia link, and now my brain hurts. Such as it is, which apparently isn’t much. I was hoping to maybe get a little grasp on higher math, since I had tried back in '76/'77 and found it hopeless then. But apparently whatever quality of mind it takes to grasp this kind of stuff, I don’t have it. I sure as heck respect and admire those of you who do! It’s just a whole different way of thinking that I can’t quite wrap my mind around. I thought I was good at what I called “math,” until I reached set theory and realized that what I was good at was actually arithmetic.
I’d *still * like to know what sverresverre had in mind with respect to “proof.” I mean, either way, you come back down to human-invented, completely abstract axioms, don’t you? You can prove internal consistency given those axioms, but is that what sverresverre wants?
I don’t think sverresverre knew what sverresverre wanted. The structure/model distinction isn’t really apparent to the casual observer, and doesn’t even show up in the “mathematical philosophy” debates still mired in the 19th century. That is, I don’t think at the time of the OP it had even occurred that there was any difference between “1+1=2” in terms of abstract natural numbers and in terms of concrete things.
Well, it doesn’t look like we’ll know for sure, because he (she?) seems to have disappeared from this discussion. A pity. I just can’t imagine putting proof of 1 + 1 = 2 (which concept to me is meaningless, since at my level it’s true by definition) on a par with proof of God, which must be an external reality. I was hoping he could make it clearer to me what he was looking for. Oh well.