I recently asked a question about the fact that logic and set theory seems circular. link
I got a lot of good and thoughtful answers, that probably explain everything, but I must admit I did not understand them all (not the fault of any of the people who answered). So I am wondering if someone can check the "recipe" I have below, and if this is a sufficient or ok way to view the construction of mathematics?
Before creating mathematical logic we just have to know what some something means. We assume we know these things:
1.1 We know what a string is, and we know what symbols is. We also know what ordered sequences of symbols is, even though we have not formalized what it is.
1.2 We can talk about function symbols and relation symbols, and we know that there are rules when writing these things. But we have not yet made a formal definition of functions and symbols.
1.3 We know what the equality symbol is, and we create rules on how to use it in our logic language. But we do not have any more precise definition of what it is, other than a symbol, and our own intuitive meaning.
1.4 We have an idea about the natural numbers, and we are allowed to use these as symbols, even though we have not created them.
1.5 We can use the induction principle on proving things about our logical language. We assume that if something holds for "the base"-object, and if something should hold for an arbitrary object then it also also for its "successor", then it must hold for all the objects starting with the base-object and all the successors. We just assume that this holds, and that we can use it?
We then create the language of mathematical logic.
We create set-theory:
3.1 Here we formalize ordered pairs in terms of sets.
3.2 We formalize what a function and a relation is in terms of sets (cartesian-product etc.).
3.3 We formalize equality as a particular relation.
3.4 We define and create the natural numbers in terms of sets.
3.5 We create an axiom of mathematical induction in terms of the natural numbers? In other words, we assume that induction holds, just as we did before we created the logical language?
And hence we do not have any circularity in using set theory before it was created?
Is this a correct way to view the construction of mathematics? If it is wrong I would love answers, but I would very much love answers in a list form, where it is explicitly stated what we do from the start and where we end up etc..
You are mixing up two things which are similar only by pure coincidence. The meta-language, and the study of mathematical logic.
In mathematics, generally, we use the meta-language, to formalize and model something, either from some platonic idealistic world or from our universe. This is the language you describe as the things we assume exist, the very primal things like finite strings and so on. We don't actually assume their existence, these are our tools for building new mathematical objects.
The study of mathematical logic, on the other hand, is taking small slices of this meta-language, and using the very tools created by the meta-language, to study these slices and gain insight, the same way we do with the movement of planets. (Remember Gödel, we can't really take everything in one go!).