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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?

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  • $\begingroup$ Try googling "classic set theory for guided independent study." $\endgroup$ Feb 3 '16 at 19:11
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    $\begingroup$ Foundations of Analysis by Landau might be what you're looking for. If you want a very formal presentation, but less formal than Russell's, you could look at Bourbaki's Set Theory. However, be warned that the book is still very difficult. $\endgroup$
    – David
    Feb 3 '16 at 19:16
  • $\begingroup$ That's “Bertrand Russell”. $\endgroup$
    – MJD
    Feb 3 '16 at 19:27
  • $\begingroup$ @David:What exactly is the target of Landau's book? $\endgroup$
    – Jim
    Feb 3 '16 at 21:12
  • $\begingroup$ @Jim: you made a request for references and you've been given one. Landau's book is beautifully written and very well organised: why are you asking us to read it for you? $\endgroup$
    – Rob Arthan
    Feb 3 '16 at 22:13
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I believe Frege's The Foundations of Arithmetic is considered to be a classic in the subject.

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  • $\begingroup$ But isn't this book only focusing on the idea of what is a number? $\endgroup$
    – Jim
    Feb 3 '16 at 19:50
  • $\begingroup$ @Jim That is included in the book for sure, but I don't know whether it is the focus of the book. You should maybe specify what kind of book you require more than "little formality" and "closeness to Russell". $\endgroup$
    – Alp Uzman
    Feb 3 '16 at 21:39
  • $\begingroup$ I am interested more in topics like: logarithms, ratios, series, fractions, basic operations i.e. all the basic topics of classic arithmetic. When I mean less formality I mean that I would prefer to not be too abstract in a way that is not practical. If something exists that is good but is formal I am fine with that if there is nothing else. $\endgroup$
    – Jim
    Feb 3 '16 at 22:34
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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.

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  • $\begingroup$ But how is set theory related to the type of book I am asking about? May be I am missing something here? $\endgroup$
    – Jim
    Feb 3 '16 at 19:53
  • $\begingroup$ Well, you see. All formal systems of math which talk about simple things like arithmetic and the like are grounded upon an axiomatic system: a set of propositions which we take for true and upon which we build the rest of the mathematics. Set Theory exposes one such axiomatic system, and proceeds to show how its relatively simple axioms can be used to talk about complex math such as arithmetic, etc. $\endgroup$ Feb 3 '16 at 20:13

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