Book on foundational reasoning of standard arithmetics "curriculum"

Question

I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it.
The closest example I can think about is the way Bertrand Russell has some books presenting such foundations in mathematics; but not in such a strong level of formality preferable.
Is there something like this available? Does anyone have something to suggest?

Answer 1

I believe Frege's The Foundations of Arithmetic is considered to be a classic in the subject.

Answer 2

There are several ways to construct a fully functional axiomatic theory of mathematics.

I would recommend you to start your journey with Set Theory, which has been the classical framework for math since early in the past century. 'Naive Set Theory', by Halmos, is a short but very complete introduction to the topic.

You may also want to dabble a bit in metalogic: the completeness and soundness of formal logic and the incompleteness of minimal arithmetic. For that I can recommend the second half of 'Computability and Logic', by Boolos et al.

Both books need no prerequisites, and will help you progress to more advanced topics quickly.