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I understand that $\mathbb{R}$ can be defined axiomatically in a number of different ways, including as an extension from more basic number systems (e.g., $\mathbb{N}$, $\mathbb{Z}$, or $\mathbb{Q}$) using equivalence classes or Dedekind cuts. However, my question here assumes that $\mathbb{R}$ is defined as a complete ordered field directly from the following axioms:

  1. Field Axioms (defining addition and multiplication with algebraic field properties -- in particular including a multiplicative constant $1\in\mathbb{R}$)
  2. Order Axioms ($\mathbb{R}$ is totally ordered)
  3. Completeness Axiom (the order is Dedekind-complete)

And now to the question: Given these axioms, how does one prove the existence of a Peano system $P\subseteq\mathbb{R}$ with $1_{\mathbb{R}}=1_P$ (i.e., I want to prove that $\mathbb{N}\subseteq\mathbb{R}$)?

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    $\begingroup$ How do you define ${\bf N}$? $\endgroup$ – tomasz Jan 24 '16 at 22:02
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    $\begingroup$ What are the axioms of a Peano system? Do they involve only a successor function, or are the axioms about $<, +, \cdot$, for instance? $\endgroup$ – David Jan 24 '16 at 22:05
  • $\begingroup$ @David, I would take the Peano axioms to simply be: P1: There exists $1_P\in P$ (or $0_P\in P$); P2: There exists a singulary operation $S$ on $P$; P3: $1_P$ is not a successor of any element of $P$ (i.e., $\forall p(S(p)\neq 1)$); P4: Different elements of $P$ have different successors (i.e., $\forall q\forall r(q\neq r\rightarrow S(q)\neq S(r))$); P5: Induction Axiom: Any subset of $P$ containing $1_P$ and closed under the successor operation must be identical with $P$. $\endgroup$ – Stephen K. Jan 25 '16 at 0:20
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With $S\colon \Bbb R\to\Bbb R$, $x\mapsto x+1_{\Bbb R}$ let $\Bbb N$ be the intersection of all subsets of $\Bbb R$ that contain $1_{\Bbb R}$ and are closed under $S$.

That is: With $e=1_{\Bbb R}$ (or $e=0_{\Bbb R}$ depending on taste) the definition $$\Phi(A)\stackrel {\text{def}}\iff e\in A\land \forall x\in A\colon S(x)\in A$$ we let $$\Bbb N\stackrel {\text{def}}=\bigcap\{\,A\in\mathcal P(\Bbb R)\mid \Phi(A)\,\}.$$ (This definition makes sense because such subsets $A$ do exist, for example we trivially have $\Phi(\Bbb R)$). One verifies that $\Phi(\Bbb N)$, so that $(\Bbb N,e,S)$ is a viable candidate: $\Bbb N$ is a set, $e\in\Bbb N$ is an element, and $S\colon \Bbb N\to\Bbb N$ is a map. The axioms are readily verified:

  • Assume $e=S(a)$ for some $a\in\Bbb N$. From $\Phi(\{\,x\in\Bbb R\mid x\ge e\,\})$ (which follows from $1_{\Bbb R}\ge 0_{\Bbb R}$) we conclude that $a\ge e$, hence $e=a+1_{\Bbb R}\ge e+1_{\Bbb R}$, contradiction
  • Assume $S(a)=S(b)$. Then $a+1_{\Bbb R} = b+1_{\Bbb R}$, hence $a=b$.
  • Let $A\subset\Bbb N$ be a set with $e\in A$ and $\forall x\in A\colon S(x)\in A$. In other words, $\Phi(A)$. Then by definition $\Bbb N\subseteq A$, hence $A=\Bbb N$.

Note that we didn't use completeness (so we find $\Bbb N$ also inside $\Bbb Q$) or Archimedean-ness of the order. Ordered field is enough.

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  • $\begingroup$ +1, even though you forgot $0$ :) $\endgroup$ – Carsten S Jan 24 '16 at 22:27
  • $\begingroup$ @Hagen: thank you for your response. From your definition, I understand how Peano axioms (defined in a comment above) P1, P2 are satisfied. I am afraid I do not see how P3 is satisfied, because wouldn't $0_{\mathbb{R}}$ be part of your defined $\mathbb{N}$? I do see how P4 could be satisfied though, using the Field Axioms of $\mathbb{R}$. $\endgroup$ – Stephen K. Jan 25 '16 at 0:28
  • $\begingroup$ You can prove by induction (i.e. P5) that $n \geq 1$ for any $n \in \mathbb{N}$. $\endgroup$ – David Jan 25 '16 at 0:44
  • $\begingroup$ @CarstenS That's because the truth value of $0\in\Bbb N$ is author-dependant $\endgroup$ – Hagen von Eitzen Jan 25 '16 at 7:13
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    $\begingroup$ @StephenK.You are right. While it is "obvious" that this set "is" $\Bbb N$, the properties need to be verified. And they won't hold for example in a finite field. But the verifications are not very hard $\endgroup$ – Hagen von Eitzen Jan 25 '16 at 7:16
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Every totally ordered field has characteristic $0$, hence contains an isomorphic copy of $\mathbb{Q}$ (which we can just identify with $\mathbb{Q}$). I guess you are comfortable with the fact that $\mathbb{N} \subset \mathbb{Q}$?

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    $\begingroup$ How do you prove that such a field contains ${\bf Q}$ without first (or simultaneously) proving that it contains ${\bf N}$? $\endgroup$ – tomasz Jan 24 '16 at 22:44
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Hint:

Along the lines of Hagen's answer, construct function $S$ such that:

$$\forall a\in R: S(a)=a+1$$

Construct $N$ such that:

$$\forall a:[a\in N \iff a\in R \land \forall b\subset R:[0 \in b \land \forall c\in b: [S(c)\in b] \implies a\in b]]$$

Then prove:

  1. $0\in N$
  2. $\forall a\in N: S(a) \in N$
  3. $\forall a,b \in N:[S(a)=S(b)\implies a=b]$
  4. $\forall a\in N: S(a)\neq 0$
  5. $\forall P\subset N: [0\in P \land \forall b\in P: [S(b) \in P] \implies P=N]$
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