Inductive defintion for real numbers I was learning about inductive definitions and an example was given for the natural numbers :
 
\[  0 \in N \wedge n \in N\implies n+1 \in N  \] 


I found a definition for all integers and all rational numbers (please correct me if I'm wrong) :




  
  
  


 
\[  0 \in Z \wedge n \in Z\implies n+1 \in Z \wedge n-1 \in Z  \] 
\[ and\]
\[ 0 \in Q \wedge n \in Q \implies n+1\in Q \wedge n-1 \in Q \wedge n \ne 0 \implies \forall x \in Q : \dfrac{x}{n} \in Q \]


I could however not come up with any inductive definition for all real numbers or all imaginary numbers. Does anyone have knowledge of such inductive definition? Thanks in advance.
 A: There is a serious "size" issue here: $\mathbb{R}$ (and related structures like $\mathbb{C}$) is uncountable, and so there is no way to inductively generate it using a "small" starting set and a "small" number of inductive clauses.
Instead, we need to bring second-order logic - that is, quantification over sets - into the picture in a serious way. (Technically an inductive definition is already somewhat second-order, but its second-order-ness is "tamer.") One way to do this is to first use an inductive definition to build a "scaffold," and then describe how $\mathbb{R}$ "covers" that scaffold. For instance:

We can build $\mathbb{Q}$ according to an inductive procedure, and then define $\mathbb{R}$ via Dedekind cuts.

Another option is to discard any semblance of induction and go for a "top-down" definition:

Up to unique isomorphism, there is exactly one ordered field with the least upper bound property. We call any such field $\mathbb{R}$ (since any two examples are uniquely isomorphic, there isn't a serious issue here re: non-uniqueness of referent).

But ultimately, there's no way around this: defining $\mathbb{R}$ will necessarily take more work than defining $\mathbb{N}$. (See also this earlier thread treating a closely-related question.)
