Peano Axioms, it's possible construct a real number theory [closed]
Peano Axioms, it's possible construct a real number theory [closed]
Is it possible to construct a coherent axiomatic theory to ground the integers, rational and irrational without induction?
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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?
– JoshuaZ
Sep 1 at 21:13
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.
– Noah Schweber
Sep 1 at 21:16
"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)
– Israel Meireles Chrisostomo
Sep 1 at 21:19
@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 $3 = 2 cdot frac32$. Is this what you mean, or do you want to be able to talk about both the properties of integer arithmetic and real arithmetic?
– Hurkyl
Sep 1 at 21:46
2 Answers
2
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.
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).
– Andrés E. Caicedo
Sep 1 at 21:10
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.
– Tomislav Ostojich
Sep 1 at 21:12
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.
– Andrés E. Caicedo
Sep 1 at 21:15
Thanks for the edit, but you haven't really justified your claim at all.
– Andrés E. Caicedo
Sep 1 at 21:16
I'm sorry Andre, but the fact that down the line you can build the real numbers using the Peano axioms (and a whole bunch of other things) does not entail that the real numbers are axiomatized by the Peano axioms. I also added justification for why induction is needed to construct the natural numbers.
– Tomislav Ostojich
Sep 1 at 21:20
The usual second-order axiomatization of the real numbers is that they are a "complete ordered field".
The analogous first-order theory is that of real closed fields.
Could you be more specific?
– Sean Nemetz
Sep 1 at 21:03