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?

  1. 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?

  2. We then create the language of mathematical logic.

  3. 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..

  • 2
    $\begingroup$ Correct? That depends on who you asked. I'm sure that there are prominent mathematicians on pretty much every opinion on the spectrum of "This is very wrong" to "That's the right way". And several others which are not on that spectrum at all. $\endgroup$ – Asaf Karagila Jun 24 '15 at 20:26
  • $\begingroup$ @AsafKaragila Thank you. I guess another way would be to ask if this in some way explains why it is allowed to use elements of set theory in creating mathematical logic, before set theory is created. I understand there are many ways of doing this, and that we probably can view the foundation of mathematics in many different ways. Put another way maybe: Is this an "ok" explanation to explain why books on mathematical logic are allowed to use the terms/objects/words in section 1., before set-theory is created? $\endgroup$ – user119615 Jun 24 '15 at 20:55
  • $\begingroup$ Yes. This is how set theory is used to investigate models of set theory. We formalize the theory internally to a mathematical theory that will provide us with formal and nice tools to simplify some of the work which will be tedious and boring otherwise. $\endgroup$ – Asaf Karagila Jun 24 '15 at 20:58
  • $\begingroup$ @AsafKaragila When you say "models of set theory", can this mean all of mathematics, that this is what you do to investigate mathematics? And with forming the theory internally, do you mean that we use the theory as help to create a formal langugage(logic), and then we formalize the set-theory with the help of logical language? Sorry for beeing a little slow. $\endgroup$ – user119615 Jun 24 '15 at 22:02
  • $\begingroup$ Your recipe seems to be good. But what are you going to do with power-sets? You haven't mentioned them at all. If you accept existence of the exponentials $A^B$, you accept existence of power-sets as well. $\endgroup$ – Ilya Vlasov Jan 27 '19 at 12:34

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