# Set Theory vs. Logic

## Main Question or Discussion Point

It makes me wonder what should be studied first - whether the basics of axiomatic set theory or mathematical logic? Although I initially that logic should be studied first, set theory second, now something makes me think that it should be vice-versa. The reason for this shift is that - when studying logic - we use various concepts that are introduced in set theory - numbers (&mathematical induction), sequence (definition of proof*) etc. What do you think?

*formal proof is usually defined as follows: "a formal proof in propositional logic is a finite sequence of statements ..."

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It's the summer vacation that causes so much confusion in my head :-) I have a lot of time to think and then I end up thinking about these obscure theories that prior to using numbers in metalanguage (used for describing our object language - first-order logic), we have to define them somehow - for example using sets.

But that is, obviously, not possible. So we take a part of mathematics - let's call it e.g. "informal mathematics" - with some basic notions (natural numbers, mathematical induction - both taken as intuitively granted) and with their help define the framework (mathematical logic, axiomatic set theory, etc.) for exact definition of them and other - more complicated - mathematical concepts.

Do you find this reasoning correct?

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MathematicalPhysicist
Gold Member
at least universities think like you, i.e the preliminary for logic in mathematics courses is set theory or discrete mathematics which include only an introduction to set theory.
i personally think that they are interlinked, every set theory class starts its first day in introducing basic logic connectors {v,&,->,<->,~}, with which you define the operation: union intersection and so forth.

i think the best way, is to learn mathematical logic with set theory, they complement each other, which rarely youll find an expert in set theory who doesnt have some expertise in logic, and vice versa.

loseyourname
Staff Emeritus
Gold Member
I personally think that one may as well learn the basics of sentential (or propositional) and predicate logic first, since it has the widest range of application. After that, further study tends to specialize, to set theory, mathematical induction, and the derivation of theorems in algebra and topology and all that good stuff in mathematics; to modal logic, deontic logic, inductive logic (which has nothing to do with mathematical induction), and counterfactuals in philosophy (and possibly in linguistics); to Boolean algebra and fuzzy logic in computer science and systems engineering.

Of course, I have no background in education to be able to say this. It stands to reason that, generalizing my opinion, we should study the basics of universal grammar and phonetics before studying any specific second languages, but that obviously is not the way languages are taught, although it is the way they used to be taught when people that thought like me were running the universities. Immersion and natural language acquisition are all the rage today, though.

loseyourname
Staff Emeritus
Gold Member
To be honest, I really have no knowledge of fuzzy sets/logic and no opinion on the matter, but I was under the impression it had some applications to electronic hardware problems; train switching on subways or something.

matt grime