# What is equivalent of Peano axioms for Real numbers?

I know Peano arithmeic defines natural numbers and their behaviour. I am interested in whether there exists equivalent for real numbers, or even for rational numbers. And whether there is a consensus on one preferred system of axioms for that sets too (I assume Peano arithmetic is most widely accepted).

• I'm not sure what you mean by 'equivalent'. But, each natural number has a successor which is a natural number and unique and "the next" natural number. Each real number and rational number does not have a similar successor. So, if "equivalent" means that we have similar axioms to the successor axioms of Peano arithmetic, then the answer is no; there does not exist an equivalent for real numbers, nor for rational numbers. Commented Jul 23 at 2:41

There is a first-order axiomatization of $$\mathbb{R}$$ given by the theory of real closed fields. There are several equivalent ways to describe this theory, one of which is the following:

• $$\mathbb{R}$$ is an ordered field, and
• every polynomial $$P(t) \in \mathbb{R}[t]$$ satisfies the intermediate value theorem.

This theory is strong enough to prove every true first-order statement about Euclidean geometry, as well as the fundamental theorem of algebra (that the algebraic closure of $$\mathbb{R}$$ is $$\mathbb{R}[i]/(i^2 + 1)$$). However, it is very far from characterizing $$\mathbb{R}$$ uniquely: there are many other real closed fields, for example the real algebraic numbers $$\mathbb{R} \cap \overline{\mathbb{Q}}$$. On the one hand this is interesting because it means any statement about $$\mathbb{R}$$ you prove using only the theory of real closed fields is true in any other real closed field, but on the other hand this means the theory of real closed fields is not capable of expressing the fundamental property of $$\mathbb{R}$$ needed for analysis, namely the least upper bound property, also known as Dedekind-completeness, which is not first-order. In the context of thinking about the Peano axioms this is the analogue for the real numbers of induction.

Instead there is a second-order axiomatization of $$\mathbb{R}$$ given by the statement that $$\mathbb{R}$$ is the unique Dedekind-complete ordered field; this is the analogue of the second-order induction axiom characterizing $$\mathbb{N}$$. This means, for example, that we can prove that different constructions of the real numbers (such as via Cauchy sequences or Dedekind cuts) are actually constructing the same object by proving that they construct Dedekind-complete ordered fields.

• +1, but I want to point out that RCF does satisfy definable completeness: Every bounded definable set has a least upper bound. This is analogous to the situation with PA, which proves induction for all definable properties. In both cases, we have to move to second-order logic, replacing definable completeness with second-order completeness, and the induction schema with second-order induction, in order to get categoricity. I think you know all this, but the analogy between PA and RCF is closer than your answer makes it sound. Commented Jul 22 at 18:37
• Of course, the real difference between PA and RCF is that the definable sets in PA are far more complex than those in RCF. So in this sense, first-order induction in PA may seem to have "more power" than definable completeness in RCF. Commented Jul 22 at 18:39
• @Alex: yes, that's what I had in mind. Definable sets in RCF are just finite unions of intervals or something like that, right? (Possibly open or closed or infinite at either end.) Commented Jul 22 at 18:50
• Exactly (also counting isolated points as closed intervals $[a,a]$). Of course, we can give arbitrarily complex definitions (e.g. one can talk about coordinates of singular points of real algebraic varieties, etc), but it's a consequence of quantifier elimination that these all have a simple form. Commented Jul 22 at 19:13