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Clearly it's not set theory, because set theory is built on first order logic, which is in a sense more fundamental as it comes first (you need first order logic to even state the axioms of ZFC)

And then comes the philosophical nightmare that I have been wrestling with for weeks. What comes before first order logic.

The question started when I began my quest on studying logic. It bothered me to a great extent that the books I read always assumed a knowledge of sets, natural numbers, functions, relations and in one of them, even cardinality! And to put the cherry on the cake, a few chapters later the books would go on to describe ZFC and building set theory, as if they had not been using sets casually for the past few chapters.

Here are the answers I have received so far

  1. First-order logic does not require the use of set theory
  2. There are certain things that you have to accept as primitives and part of an accepted metalanguage, which are needed as a starting point to lay the foundations of mathematics
  3. It's turtles all the way down

Regarding the first one, I have literally seen books on logic use terms like a language is a set of symbols or even "sets of well formed formulas", particularly something like $\{\phi_1,\phi_2,\dots,\phi_n\}\vDash \psi$ so I really don't know where the first point is coming from.

Regarding the second point, this makes much more sense, except that it seems no one can tell me what are the things we take as primitive and part of "common sense". Do we start off with a "common sense" understanding of sets and then build logic? Is it the same for natural numbers, functions, countability etc.? Is that to say that the ultimate foundation of all of mathematics is Naive Set Theory and Naive Number Theory? That just sounds so... unsatisfying. How can math be consistent, if our metalanguage of English is not (we can form contradictions like "This statement is false"). Ok basically my main question for this part is what are the things that we take as accepted and "common understanding" before we build mathematics. Like precisely what are they. Sets? Natural Numbers? (Is the domain of discourse in first order logic also a "common sense" term?)

The third one really grinds my gears because it just brushes the whole issue aside and doesn't explain it at all.

To sum up my question, as a foundation for mathematics, there are some things that we humans have to take as understood for us to draw a baseline. What precisely are these things and does a naive understanding of them in a metalanguage mean that mathematics is not as precise as it appears to be. Also as an aside, when we prove things about first order logic, do we do it in our imprecise metalanguage? I have seen logic proofs that use induction and I'm so confused because Number Theory is a much higher level thing than logic.

Saad Haider
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    I answered a similar question here, maybe you'll find that useful. – Oscar Cunningham Dec 10 '19 at 21:32
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    Related: https://math.stackexchange.com/q/1334678/291100 – Nap D. Lover Dec 10 '19 at 22:23
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    Also: https://math.stackexchange.com/q/121128/291100 – Nap D. Lover Dec 10 '19 at 22:24
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    “Using set theory” and “using the concept of a set” are two drastically different things. – spaceisdarkgreen Dec 10 '19 at 22:47
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    Like I’m pretty sure my python interpreter is thinking about arithmetic much more than about large cardinals when I tell it to take the intersection of two sets I give it. – spaceisdarkgreen Dec 10 '19 at 22:59
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    "there are some things that we humans have to take as understood for us to draw a baseline." Of course, YES: we humans have to use natural language and the rule-following mechanism built-in into language to reason; this is the "natural foundation" for logic: formal and mathematical. And we have to use natural language and the counting procedures learnt very early at school as a "natural foundation" for mathematics. – Mauro ALLEGRANZA Dec 11 '19 at 09:25
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    Why should there be an "ultimate foundations"? – Asaf Karagila Dec 11 '19 at 13:58
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    Does this answer your question? Does mathematics become circular at the bottom? What is at the bottom of mathematics? – user21820 Dec 11 '19 at 16:50
  • Related: Founding mathematics from a set of axioms – Mark S. Dec 11 '19 at 23:29
  • @AsafKaragila Otherwise how do we know what we are talking about? If I say 1 + 1 = 2, I'd have to know what is 1, 2, + and =. to understand this I would need to understand functions, equality and logic etc. So these are the foundations I would need to talk about 1+1=2. But then I would need to know what I mean when I say functions, equality and logic. If this goes down an infinite regress, it means I have no idea what I'm talking about when I say 1+1=2 – Saad Haider Dec 12 '19 at 19:04
  • @user21820 thanks for sharing that. I have seen that before and that post actually just confused me even more. The top answer states that 1. you don't need set theory for first order logic which I adressed in my question and 2. it first says set theory needs first order logic and then a few lines later says that set theory is stronger than first order logic which makes no sense to me. – Saad Haider Dec 12 '19 at 19:07
  • @MarkS. that's actually really helpful thanks! – Saad Haider Dec 12 '19 at 19:09
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    How are you able to understand me now? Or comprehend my existence, or your own? Or anything? It's not that mathematics has this problem, it's knowledge in general. At some point, you have to put your faith in some common understanding and build from there. Unfortunately this common understanding is neither mathematical nor quantifiable and fully describable. – Asaf Karagila Dec 12 '19 at 19:09
  • @AsafKaragila So I guess that is our "ultimate foundation" then. Not gonna lie I'm really bummed out that math isn't 100% objective and devoid of human biases. And I'm more bummed out by the fact that we can't even fully describe these foundational notions. It's just so... imprecise. And precision of language is one of the things I love so much about mathematics – Saad Haider Dec 12 '19 at 19:14
  • @spaceisdarkgreen so when we talk about sets in logic and proofs in logic, we are not talking about the same object that is described by ZFC? – Saad Haider Dec 12 '19 at 19:15
  • @Saad The same objects, perhaps (a philosopher could write a book on what that even means), but we don’t need even a tiny fraction of the amount of strength that ZFC has (e.g. we arguably don’t even need any infinite sets). The mechanics of formal proofs in first order logic is formalizable in arithmetic, as one learns in the study of Godel’s theorem. (On the other hand, model theory / semantics requires some set theory.) – spaceisdarkgreen Dec 12 '19 at 20:29
  • @SaadHaider: I linked you specifically to my answer, not the currently top-voted one, which in my view is too vague to be helpful to you. Read my answer first, and if you have further inquiry about any specific point in it then ask again. Also read this post, which touches on the same issue spaceisdarkgreen mentioned, namely that formal deduction can be mechanically handled with essentially 100% objective precision, but semantic interpretation will need more meta-logical assumptions. – user21820 Dec 13 '19 at 17:26
  • In particular, basic results in logic can be proven in weak meta-systems such as ACA, so at least it should be very clear that point (1) in your post is more or less correct. As for point (2), different mathematicians will have different notions of what is "mathematics". Some of them think that every ZFC theorem is true (whatever that means), so for them the ultimate foundation had better be as strong as ZFC. As shown clearly by Godel, you cannot bootstrap from a weaker to a stronger system, so you are going to have to decide what you accept and what you don't, more or less from the start. – user21820 Dec 13 '19 at 17:31

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