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I made this question several months ago. I guess that recently, I have come to the realization that there are admissible things sayable with a certain theory and also that this is very interesting! So what do I need to know in order to know what can be said with Presburguer arithmetic, for example? My first guess is logic and I am reading some books about it, it seems the obvious step. But are there more things one needs to know for this purpose?

In the mentioned question, they said that certain number-theoretic concepts can't be formalized with Presburguer aritmethic. Wikipedia's article even states that "Presburger arithmetic cannot formalize concepts such as divisibility or prime number." But what can be said about other realms such as analysis, algebra, etc? I am interested by this type of questions.

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Red Banana
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    Are you finally asking the vast question: what is in the reach or out (=beyond) out of the reach of a given axiomatic system ? – Jean Marie Jan 20 '17 at 21:07
  • @JeanMarie Yes, I guess. I understand that I may be asking a very complicated thing just as dav11 told me about. So I'd like to know a bit about the roads of this field of study. – Red Banana Jan 21 '17 at 02:50
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    Besides Enderton's book, which I don't think is freely available online, there are other equally useful resources that I list at http://math.stackexchange.com/a/1684208. In particular if you already have some mathematical background you probably should start with Simpson's notes. And see also the other links in my profile. – user21820 Jan 21 '17 at 04:41

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To answer your question: Logic would be a good place to start. And there are many good books you can use. Usually any textbook that talks about the incompleteness theorem would have some sort of treatment of Persburger. I would recommend "A mathematical introduction to logic" by Enderton. It has a good treatment of the Persburger arithmetic.

To answer your second question is hard: It depends on how you formalize the problem. Persburger arithmetic is what is known as first order theory; and what you can say, reduces to the things you can say in first order using the language you have at hand. The way we study Real Analysis in an intro to real analysis class, on the other hand, can be thought of as higher order (what we usually do is think of the reals as a set and use the language of set theory; where set theory is first order). If the comments within the parenthesis seems confusing; don't worry too much about it for now. It's definitely not the easiest thing to grasp just going in and is probably much better understood after you gain an appreciation for what you can and cannot say in first order.