moo5003
Oct25-07, 05:08 PM
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.
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.