Recent content by ky2345

Homework Statement Prove that the theory of dense linear orderings with no endpoints is not categorical in the cardinality of the reals. Homework Equations A theory is categorical in the cardinality of the reals (denoted c) if every c-model is ismorphic. Isomorphic means that there...

I'm pretty sure all I need is an automorphism that stretches everything about the line y=x.

Oh and thanks for being patient- I guess I wasn't confused about what I thought I was confused about!

Ok, I think the automorphism definition is really what I was confused about. So, a function f((x,y))=(x+c,y+c) would be an example of an automorphism, but f((x,y))=(x+c,y+d) where c and d are not equal would not be an automorphism. Now, I think I've figured out some of the definable sets. I'll...

I am using the definition of definable that I mentioned in my first post. Then, a formula with n free variables defines the set of n-tuples that make the formula true in the given structure. So, in RxR for example, we are looking at ordered pairs. An example of a set being definable can be seen...

Oh, and also I have equality, =

My language includes all five formula building operations, so or, and, implies, iff, and not. There are no constants. The two place predicate symbol < is the only predicate symbol I have. I have a definition of definable, and am having difficultly applying what it means for a subset of RxR to...

the variables can vary over all of R, and < is the only relation. So, automorphisms need to preserve <. I am viewing R as an ordered set. I'm not sure what you mean by "nonlogical symbols"

Homework Statement What subsets of the real line R are definable in (R,<)? What subsets of the plane RxR are definable in (R,<)? Homework Equations A subset is definable if there is a formula in first order logic that is true only of the elements of that subset. For example, in the...

If I apply a rotation, and I know that (cos(t), sin(t)) is the arclength parameterization for a circle, then can I just use those formulas to prove that the acceleration is tangent to the sphere? How do I apply a rotation, should I just say I'm applying a rotation and viewing P as the xy plane...

Homework Statement When a plane intersects a sphere at more than two points, it is a circle (given). Let x^2+y^2+z^2=1 be a sphere S, and P be a plane that intersects S to make a circle (called C). Let q:[a,b] -> R^3 be a unit speed parameterization whose trace is C. Prove that the second...