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...
Homework Statement
Suppose that T and F are both axiomatizable, complete, consistent theories. Is T\cap F axiomatizable?
Homework Equations
A theory T is a set of sentences such that if yo can deduce a sentence a from T, then a is in T.
I have already proved that T\cap F is a theory...
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...
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...
Homework Statement
Prove that if A and B are two sets of well-formed formulas (logical statements, abv. wff) such that A union B is not satisfiable, then there exists a wff k such that A tautologically implies k and B tautologically implies not k.
Homework Equations
This question is in...
Homework Statement
Show that if A is a tautology, then so is *(A). A is a well formed formula, * is a function that replaces all sentence symbols A_1, A_2, etc. with formulas B_1, B_2, etc. , respectively
Homework Equations
* is defined recursively, starting with the fact that if A_n is...
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...