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I'm reading a book on Mathematical Logic (on my own) and from the beginning there are terms such as "functions" and "relations", but the only definitions of these words that I know are in terms of sets, but the purpose of mathematical logic is (among others) to be able to construct an axiomatic theory of sets. How is that not a circular construction ?

Mone
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    Mathematical Logic is a branch of mathematics, just like Group Theory. One uses the ordinary tools of mathematics to study axiom systems, models, etc. – André Nicolas Feb 18 '14 at 05:26
  • @AndréNicolas don't know exactly how that is supposed to answer my question, but thanks! – Mone Feb 18 '14 at 05:39
  • Not quite understand a trouble. Yeah, function is defined in terms of sets. Naturally, function is a set. But all set theory uses the concept of set. – sas Feb 18 '14 at 05:53
  • the problem is that at that stage (beginning of a book) we haven't developed a language to talk about sets yet. So (seemly) we are talking about sets but we "don't know" what are sets, I don't see how that is not circular. – Mone Feb 18 '14 at 05:55
  • This is circular, don't worry. Is there a second part in your book about 'set theory'? Don't waste your time on this problem: it will be explained by your author when he develops set theory 'from logic'...All you will get in this early stage is fancy talk on "metatheory" etc., just wait. – Blah Feb 18 '14 at 07:27
  • @Blah thank you, now I feel that the most important thing in approaching a subject like this one is to know "where" we should "waste time". – Mone Feb 18 '14 at 08:00
  • I don't like answer that sound as : "you are too young to understand ...". The issue of foundations in mathematics is really interesting and is at the "core" of XX century mathematical logic. If you are interested to it, you can read a couple of books (ask if you need). But in science (not only in math) and in human knowledge (I think) you must work with the indubitable fact that no absolute foundation is possible. – Mauro ALLEGRANZA Feb 19 '14 at 10:46
  • You can read this post and also this one – Mauro ALLEGRANZA Feb 19 '14 at 11:20
  • @MauroALLEGRANZA Thank you! I was following Kleene's "Mathematical Logic", but it seems to be pretty slow to me, now I'm following Srivastava's "A Course on Mathematical Logic" and Ebbinghaus's "Mathematical Logic", these two are the ones I'm most comfortable with. I study (Commutative) Algebra, and now and then we discuss the question of formalization, and that leads to Hilbert and then to Godel's theorem... – Mone Feb 19 '14 at 13:41
  • @MauroALLEGRANZA So I've decided I want to understand precisely what Godel's theorems say, but I've discovered a whole new universe. What books would you recommend ? – Mone Feb 19 '14 at 13:43
  • For Math Log, Kleene is not bad. But you can read Ian Chiswell & Wilfrid Hodges, Mathematical Logic (2007), or Dirk van Dalen, Logic and Structure (5th ed - 2013); the first one is more "slow", the second one covers a lot. About Godel's theorem, you must read: Peter Smith, An Introduction to Gödel's Theorems (2007 - there is also a new enhanced edition). About "foundations", I like the historical setting : Morris Kline, Mathematics: The Loss of Certainty (1982) and Marcus Giaquinto, The Search for Certainty: a philosophical account of foundations of mathematics (2002). – Mauro ALLEGRANZA Feb 19 '14 at 13:56
  • About philosophy of mathematics you can find a lot of articles online in Stanford Encyclopeida of Philosophy and of course read original papers re-edited in Paul Benacerraf & Hilary Putnam (editors), Philosophy of Mathematics: Selected Readings (2nd ed - 1984). – Mauro ALLEGRANZA Feb 19 '14 at 13:59

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The "construction" is circular, the reasoning ... not necessarily.

When you write a book about the syntax of (e.g.) english language, you use the language itself.

This "procedure" works because you have already learnt how to speak and read.

In mathematics you use the language of set (but also arithmetic : is very difficult to speak of "objects" without being able to count them ...) to set up your theory.

The same in mathematical logic that is a branch of mathematics: you need set language for describing basic objects like symbols (we need a set of primitive ones), formula (a string, i.e.a set of symbols), derivation (a sequence, i.e.a set of formulas), etc.

The "trick" is the interplay between the (mathematical) language you are "speaking of" (the english language subject to the study in your syntax book) and the (mathematical) language you are "speaking with" (the english language with which your syntax book is written).

The first we call it : object language.

The second we call it : metalanguage.

Mauro ALLEGRANZA
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  • Ok, at this stage I'm very confused but I think someplace ahead all of this will start to make sense, I just have to get used to this new way of thinking about math, thank you. – Mone Feb 18 '14 at 17:28
  • @Mone - the issue, according to my point of view, is that there is no absolute starting point : see Skolem's comment on set theory in his paper (Thoralf Skolem, Axiomatized set theory (1922)) reprinted in van Heijenoort (editor), From Frege To Gödel: A Source Book in Mathematical Logic, 1879-1931 (1967). – Mauro ALLEGRANZA Feb 18 '14 at 17:42