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I am not a mathematician. So I am not even sure if what I am asking is logically coherent. But I do have some application-based curiosity that I would like to enlighten myself about. I will first pose, to the best of my ability, what I think I am trying to figure out, and then explain why I am asking this question (i.e. what made me think the answer to the following question can be useful for me):

Is there a branch of mathematics that assumes an upper limit for the size/number of objects that it studies? In this hypothetical branch, we do not assume that numbers can be infinitely large, so e.g. any hypothesis about natural numbers would be true as long as it is true for all numbers less that our upper limit. This branch can be of two forms:

  1. We use the same math that we already use (e.g. ZFC), but we only ask the types of questions in it that concern limited objects. For instance instead of asking whether Goldbach conjecture holds, we ask whether Goldbach conjecture holds for all numbers less than an upper limit N (let's call it Limited Goldbach Conjecture). In this example, while this does not change the difficulty of actually solving this conjecture (since N is not known), it does change our confidence in our ability to solve it, because while the former is not known to be decidable at all, the latter is definitely decidable because we know that N is finite. (unless we also have a limit on the size of the solution, in which case we are asking if we can solve Limited Goldbach Conjecture in less than a certain number of steps, M, which can be a function of N. Regardless, we are still in the realm of normal math).
  2. Another kind of math in which infinity is not a thing at all. In this axiomatic system, Goldbach conjecture would be a syntactically or semantically meaningless problem, but Limited Goldbach Conjecture would not be. I am not sure whether a useful non-trivial form of such a system is possible to create at all or not, and if possible, whether it would be equivalent to the option #1 described above.

Here's why I am asking this. Assuming we live in a universe with a finite number of particles, if I am trying to use a formalization to explain the universe , one possible way to do it is to incorporate that number of particles (N) into my formalization, so that, practically (option #1) or systematically (option #2), I don't care about anything that requires more number of particles. This might have practical consequences. For example, I think such a limit in our formalization will change the kinds of problems that are decidable, and the type of computing framework that is required for solving them. For example, any look-up table that requires more memory than our number of particles cannot exist. Or any turing machine that has a program tape of more than what the universe can offer would not exist. So some problems that are decidable would become undecidable in this formalization. But also, some problems that are undecidable might become decidable, because the input size is going to be limited in our formalization... Or maybe not, but somebody has to figure this out! Hence why I thought maybe such a branch might exist.

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  • $\begingroup$ Please feel free to add tags that might be relevant. $\endgroup$
    – Feri
    Nov 4, 2023 at 1:46
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    $\begingroup$ Although this question was migrated from MO, the OP may be interested in another MO question concerning efforts to found ultrafinitism. $\endgroup$
    – J.G.
    Nov 4, 2023 at 12:31

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There are two schools of mathematics that reject infinity as it is presented in classical mathematics.

  1. Finitism. Finitism is a philosophy of mathematics that takes the Zermelo-Frankel axioms as the foundation of mathematics, except rejecting the axiom of infinity, either replacing it with its negation or another axiom that implies its negation. However, this shouldn't be seen as a straight rejection of infinity, because finitists do believe in potential infinity. What they take issue with is there being a set of infinite cardinality, which implies the existence of an actual infinity which they judge as logically incoherent.

  2. A more extreme school is ultrafinitism, which rejects even the idea of a potential infinity. They believe that the universe is strictly finite and so there exists a greatest conceivable integer $N$ which can never be approached, but is the upper bound of all quantities in mathematics. There are strong philosophical arguments that ultrafinitism is logically incoherent, and that logic does require at the very least a notion of potential infinity. The only serious proponents of this school I know of is Dr. Doron Zeilberger and as @user4894 added, the late Edward Nelson.

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