# First-Order Languages and Circular Reasoning

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 ?

• 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 5:26
• @AndréNicolas don't know exactly how that is supposed to answer my question, but thanks! – Mone Feb 18 '14 at 5:39
• 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 5:53
• 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 5:55
• 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 7:27

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.

• 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