Question: What subsets of R x R are definable in (R:<)?

[moo5003]

Homework Statement

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

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.

 

[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.

 

[AKG]

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