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Is it possible to construct a coherent axiomatic theory to ground the integers, rational and irrational without induction?

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    $\begingroup$ Could you be more specific? $\endgroup$ Sep 1, 2018 at 21:03
  • $\begingroup$ What do you mean by a "real number theory" do you mean able to number theory or do you mean able to do work with the real numbers? $\endgroup$
    – JoshuaZ
    Sep 1, 2018 at 21:13
  • $\begingroup$ What do you mean by the word "ground"? We can certainly write a list of axioms about natural, rational, and real numbers which does not include induction for the naturals; however, such a system will be very weak. So it depends on what you mean by "ground" - presumably you want these axioms to prove "basic facts" about the number systems in question, and the answer to your question will depend on what basic facts you want. $\endgroup$ Sep 1, 2018 at 21:16
  • $\begingroup$ "What do you mean by a "real number theory" do you mean able to number theory or do you mean able to do work with the real numbers?"I say aritmetic operations properties ( sums, products, divisibility but with real numbers) $\endgroup$ Sep 1, 2018 at 21:19
  • $\begingroup$ @Israel: Divisibility with real numbers is mostly trivial: $x$ is divisible by $y$ if and only if either (1) $y \neq 0$, or (2) $x = y = 0$. As an explicit example, $3$ is divisible by $2$, because $3 = 2 \cdot \frac{3}{2}$. Is this what you mean, or do you want to be able to talk about both the properties of integer arithmetic and real arithmetic? $\endgroup$
    – user14972
    Sep 1, 2018 at 21:46

2 Answers 2

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The Peano axioms are for constructing the natural numbers, not the real numbers.

As for answering your question, Robinson arithmetic is a theory of the natural numbers without induction, and because it lacks induction general statements over variables (e.g. x + y = y + x) are not provable. Induction is needed to prove statements involving arbitrary natural numbers represented by variables. Without induction, you are stuck with proving sentences that use only concrete values like 5 + 3 = 3 + 5. So no, there is no axiomatization of the natural numbers without induction that gives you the full expressive power of Peano arithemtic. This is justified by noticing that in order to prove that a proposition P holds for a variable, one must have a rule that has a consequence $\forall x P$... which is going to be induction if the rule is to be consistent with the property of the successor function.

However, as Andrés E. Caicedo pointed out, there may be some other rule (e.g. reflection principle) which serves as a replacement for induction and yet produces an axiomatic system equivalent to Peano arithmetic. I am not aware of any papers.

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    $\begingroup$ I don't think this addresses the question. The problem is whether it is possible to find an axiomatization for the natural numbers which does not include induction (or anything straightforwardly equivalent to it). $\endgroup$ Sep 1, 2018 at 21:10
  • $\begingroup$ I edited it. I said that there is no axiomatization of the natural numbers which is as powerful as the Peano axioms but lacks inductions. $\endgroup$
    – Fomalhaut
    Sep 1, 2018 at 21:12
  • $\begingroup$ Even your first point seems off. One usually starts with $\mathbb N$ (axiomatized using induction), builds $\mathbb Z$ using equivalence classes of pairs of naturals, builds $\mathbb Q$ using equivalence classes of certain pairs of integers, and builds $\mathbb R$ using $\mathbb Q$ via Cauchy sequences or Dedekind cuts or somesuch. But it is conceivable there is an alternate approach that avoids induction altogether. $\endgroup$ Sep 1, 2018 at 21:15
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    $\begingroup$ No one is saying what you think I am saying. There is an interesting question here you are not addressing. It is not something trivial as your comments and your answer seem to indicate. $\endgroup$ Sep 1, 2018 at 21:23
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    $\begingroup$ For instance, obviously a mild set-theoretic platform is needed in addition to the axiomatization of $\mathbb N$ to carry out the usual constructions. I assure you I am well aware of it. $\endgroup$ Sep 1, 2018 at 21:24
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The usual second-order axiomatization of the real numbers is that they are a "complete ordered field".

  • Fields axiomatize basic arithmetic and algebra
  • Ordered fields axiomatize the order relation and how it relates to arithmetic
  • Completeness describes the basic analytic features of the continuum. It implies (or is equivalent to) a number of basic results, such as the intermediate value theorem.

The analogous first-order theory is that of real closed fields.

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