This question already has an answer here:
- When does the set enter set theory? 6 answers
I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations.
Now is the "=" taken as undefined? I have seen it been defined in terms of the identity relation.
But in order to talk about functions and relations you need set theory. However, set theory seems to be a part of mathematical logic.
Does this mean that (naive) set theory comes before sentential and predicate logic? Is (naive)set-theory at the absolute bottom, where we can define relations and functions and the eqality relation. And then comes sentential logic, and then predicate logic?
I am a little confused because when I took an introductory course, we had a little logic before set-theory. But now I see in another book on introduction to proofs that set-theory is in a chapter before logic. So what is at the bottom/start of mathematics, logic or set theory?, or is it circular at the bottom?
Can this be how it is at the bottom?
naive set-theory $\rightarrow$ sentential logic $\rightarrow $ predicate logic $\rightarrow$ axiomatic set-theory(ZFC) $\rightarrow$ mathematics
(But the problem with this explanation is that it seems that some naive-set theory proofs use logic...)
(The arrows are of course not "logical" arrows.)
simple explanation of the problem:
a book on logic uses at the start: functions, relations, sets, ordered pairs, "="
a book on set theory uses at the start: logical deductions like this: "$B \subseteq A$", means every element in B is in A, so if $C \subseteq B, B \subseteq A$, a proof can be "since every element in C is in B, and every element in B is in A, every element of C is in A: $C \subseteq A$". But this is first order logic? ($(c \rightarrow b \wedge b \rightarrow a)\rightarrow (c\rightarrow a)$).
Hence, both started from each other?