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Thus, to define the natural numbers, we will use two fundamental concepts: the zero number 0, and the increment operation. (Digression to computer languages...).

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We thus see we should have the following axioms concerning 0 and the increment operation ++

Axiom 2.1: 0 is a natural number

Axiom 2.2: if n is a natural number, then n++ is also a natural number

Axiom 2.3: 0 is not the successor of any natural number; i.e., we have $n++ \neq 0 $ of every natural number n

Axiom 2.4: Different natural numbers must have different successor; i.e., if n, m are natural numbers and $ n \neq m$ , then $n++ \neq m++$. Equivalently, if $n++ = m++$ , then we must have $n=m$

Axiom 2.5: Principle of mathematical induction

pg 17,18,19

If you read carefully, the recipe for constructing $\mathbb{N}$ began with an object and some sort of recurrence operation defined on it. The choice of the axioms makes the recurrence act on the subject such that the whole natural number system is generated.

My question is, is there a general theory about such recurrence operation? For example, could there be other interesting number systems developed beginning at $0$ and choosing different axioms conducting the behavior of $++$?

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  • $\begingroup$ There is nothing special about the fact that it is a 'recurrence operation', n++ brings with it the same meaning as s(n) where s is defined as mapping n to n+1. If you are interested in fun recurrences, such as the Mandelbrot generating f(z) = z^2 + c, search for recurrence relations in general. $\endgroup$
    – egglog
    Commented Jan 18, 2022 at 17:29
  • $\begingroup$ I don't understand, why not just begin with mappings then? Since mappings are defined on sets and sets have seperate axiom set than N, it should have been conceptually simpler to do it $\endgroup$
    – Cantor Dust Drachen
    Commented Jan 18, 2022 at 17:30
  • $\begingroup$ Sets are just one way to talk about math. Not everything is a set. We can have lots of things in the universe that match these axioms. Sets is just one of them. $\endgroup$
    – Thomas Andrews
    Commented Jul 20, 2023 at 14:30
  • $\begingroup$ The whole point of the Peano axioms is to find a limited set of assumptions for the natural numbers. Set theory has a lot of assumptions in addition - the axioms of set theory. Set theory logically includes all of the Peano axioms, but it adds a lot more. $\endgroup$
    – Thomas Andrews
    Commented Jul 20, 2023 at 14:32
  • $\begingroup$ Kronecker wrote: "Natural numbers were created by God, everything else is the work of men.” If so, we want axioms for this most basic object, because, in the end, we cannot define it. What exactly is zero? Six? We cannot say - that is a matter for philosophy, not mathematics. $\endgroup$
    – Thomas Andrews
    Commented Jul 20, 2023 at 14:47

1 Answer 1

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My question is, is there a general theory about such recurrence operation?

Yes. The entire subfield of dynamical systems studies iterated maps.

For example, could there be other interesting number systems developed beginning at $0$ and choosing different axioms conducting the behavior of ${++}$?

Yes, our description of the integers is known to be incomplete: there are mathematical statements whose truth depends on the choice of ${++}$. Worse, the existence of such statements cannot be eliminated simply by adding more axioms.

But these statements are unusual, and tend not to arise in the study of the integers (Matiyasevitch's result notwithstanding). If they were more common, we'd add more axioms until the behavior of the questions we ask settled again.

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  • $\begingroup$ What do you mean when you say "our description of integers" .. are you saying integers are some physical thing which exists without our description? $\endgroup$
    – Cantor Dust Drachen
    Commented Jan 18, 2022 at 18:19
  • $\begingroup$ @Buraian: No; I don't think "the integers" has no meaning except that which we give it. By mathematical consensus, it means anything satisfying the (linked) Peano axioms. $\endgroup$
    – Jacob Manaker
    Commented Jan 19, 2022 at 3:21
  • $\begingroup$ (Also worth mentioning: this answer is kinda glib. The Peano axioms are incomplete in first-order logic, but not in second-order; in that sense, there is only one reasonable definition of ++.) $\endgroup$
    – Jacob Manaker
    Commented Jan 19, 2022 at 3:22
  • $\begingroup$ (Yet a third comment: if you're willing to drop some axioms on ++, in addition to add, then modular arithmetic becames a very powerful and important system arising from a successor operation. I might turn this into a separate answer.) $\endgroup$
    – Jacob Manaker
    Commented Jan 19, 2022 at 3:23
  • $\begingroup$ @JacobManaker Second order Peano is also incomplete, in that it has a lot of different models, and there are statements true in some models and false in others. $\endgroup$
    – Thomas Andrews
    Commented Jul 20, 2023 at 14:38

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