# Logic + Definability (R,<)

1. Oct 25, 2007

### moo5003

1. The problem statement, all variables and given/known data

What subsets of the real line R are definable in (R;<)? What subsets of the plan R x R are definable in (R:<)?

3. The attempt at a solution

R and the empty set are the only definable subsets of (R;<) since:

x to x+1
Is an automorphism and changes all subsets except for R and the empty set, therefore those subsets are the only possible definable subsets.

R(x) := All x ~(x<x)

ie: All real numbers hold this property

Empty Set (x) := All x (x<x)

ie: Nothing holds this property.

Question: When answering the second part of this question for RxR. I'm not completley sure how you can say (a,b) < (c,d). My answer which I'm a little unsure of right now is that you can define (R,a) and (R,a) for some fixed a. (As well as the empty set). Any help would be appreciated.

2. Oct 25, 2007

### HallsofIvy

Perhaps it would help if you defined "definable"! What is the definition of "definable set" you are using?

You can't say (a,b)< (c,d). That's why you problem says "(R: <)". The order relation is still on the real numbers.

3. Oct 25, 2007

### AKG

x to x+1 doesn't change the set Z.