Im at that stage in my mathematical understanding where I kinda understand what set theory is and what first order logic is but dont really understand how they fit together to create Mathematics. I assume that the ZF system uses first order logic to create the foundations of mathematics and in the grand scheme of things, set theory is dependent on logic for its existence whereas logic or any formal system can exist on its own. Is this the correct view?
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$\begingroup$ Neither mathematical logic nor set theory "create Mathematics" ... Math "was there" centuries before math log and set theory. The "foundational" point of view was linked to the historical "foundational schools" : Logicism, Formalism, Intuitionism and must be understood in that setting. A "simple" point of view is that math log and set theory provide the language and the "tools" to represent the existsing math theories in a "uniform" way: but this does not means that math "dependent on logic [and set th] for its existence". $\endgroup$– Mauro ALLEGRANZAAug 4, 2015 at 7:57
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3 Answers
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The received wisdom is that pretty well any mathematical statement can in principle be formulated and hence formalised in ZF. I think this is rather an overstatement, but let's bear with it. So in some sense, logic can be viewed as providing a formal foundation for mathematics. However, to do logic rigorously, you need to be able to define and reason about syntax, so a certain amount of mathematics is required to underpin logic. Many people take the view that the finitistic mathematics you need to reason about syntax is sufficiently certain that the apparent circularity is harmless. (Personally, I am agnostic on nearly all such issues.)
answered Aug 4, 2015 at 16:46
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1$\begingroup$ I too had doubts about the circularity you point out. So it is a still unsettled area is it?? $\endgroup$ Aug 4, 2015 at 17:09
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$\begingroup$ I don't think philosophical debate about mathematics will ever be settled (but fortunately I'm not a philosopher). $\endgroup$ Aug 4, 2015 at 17:25
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Not the best answer, but I'll try with a short answer. Some of the set theory books, you will find it says for example $A\subseteq A\cup B$, where $A$ and $B$ are both sets. If we want to check if it's true, we have to prove that the statement $(x\in A)\implies (x\in A\cup B)$ is true. Note that this is equivalent to $(x\in A)\implies [(x\in A)\vee (x\in B)]$, and it's actually true. The reason why it's true is because the statement $p\implies (p\vee q)$ is a tautology which can be read more in Mathematical logic. There is a nice table that shows different tautologies in Rules of Inference. The rule we used here is called "Addition".
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From my understanding, you can use set theory to model the objects studied in mathematics. Then major mathematical theorems can be proven from these models. Take for example, ZFC. If you accept the axioms, then you can use the language of first-order logic to deduce theorems. In Enderton's book on set theory (Elements of Set Theory), he gives a very easy to understand construction of the real numbers from these basic axioms. He first constructs the natural numbers, then defines ordered pairs and equivalence classes using sets. This allows him to construct integers, then rationals, and finally real numbers. Now we can prove theorems about real numbers and eventually theorems in calculus. So in this sense, first-order logic describes the rules of the language in which the axioms of set theory are written, and how rules of inference can be applied to these axioms to create theorems. And since we can use sets to model a lot of things in mathematics, we can say that they can be used as a foundation for mathematics.
