Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch
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Tuesday, October 04, 2016
A Second Order Statement true in (R,+) but not (Q,+)
In my last post I asked
Is there a first order statement true in (R,+) but false in (Q,+)
Is there a second order statement true in (R,+) but false in (Q,+)
First order: No.
One reader said it followed from the Lowenheim-Skolem Theorem. I can believe this but don't see how.
One reader said the theory of (Q,+) is complete. True, but would like a reference.
I would prove it by a Duplicator-Spoiler argument.
Second order: Yes
Some readers thought I had < in my language. I do not so I think that examples using < do not work- unless there is someway to express < in my language.
Some readers thought that second order meant you could quantify over functions. My intent was just to be able to quantify over sets.
Some had the right answers. The one I had in mind was
There exists A,B such that A,B are subgroups with at least two elements that only intersect at 0.
Here is a writeup.
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The first order equivalence of (Q, +) and (R, +) follows from the Lowenheim-Skolem theorem for the following reason: (Q, +) and (R, +) are both torsion-free divisible abelian groups-- i.e. essentially vector spaces over Q. Thus when you use Lowenheim Skolem to blow (Q, +) up to a model of cardinality |R|, it will be a vector space over Q of the same dimension as (R, +).ReplyDelete
Clever! I didn't think of using subgroups at all...ReplyDelete
I think you can define < in second-order (R,+). Sort of. The idea is to consider the second order predicate Pos(X) = X is closed under + and 0 \not\in X and forall x \not= 0, exactly one of x and (-x) are in X. Note here that the maximality condition in the definition of Pos implies that it is closed under multiplication with positive rationals.ReplyDelete
Clearly this predicate is true of R^+, but it is also true of other subsets of R too. We can understand their structure. Let B be a basis for R over Q. The elements of B come with "natural" positive or negative assignments, but any predicate X satisfying Pos imposes its own assignment of positive or negative to the basis elements, and moreover, any assignment of positive or negative to the basis elements uniquely determines a set X that satisfies Pos(X).
We can now convert any closed formula \phi in second-order (R,+,<) to an equivalent formula in (R,+) via
forall (X : R -> Prop) (< : R -> R -> Prop), Pos(X) and (x < y iff exists z \in Pos, x+z = y), \phi.
Rod Downey and I had a paper on recursive ordered groups that relied being able to play this sort of game with the order relation, although it wasn't at all framed in this language.
A thought: Isn't it the case that a formula F(x_1,..,x_n) is true in (R,+) iff F(-x_1,..,-x_n) is true in (R,+)? And that holds for first and second order. But then there can be no F(x,y) which is true iff x is less than y (iff -x is greater than -y).Delete
Regarding the Löwenheim–Skolem theorem solution: by the LS theorem, there exists a structure (M, +) of cardinality continuum that is elementary equivalent to (Q, +). Note that (M, +) and (R, +) are linear spaces. Since |M| = |R|, (M, +) and (R, +) are isomorphic as linear spaces over Q; thus, (M, +) and (R, +) are isomorphic structures. By transitivity, (Q, +) and (R, +) are elementary equivalent.ReplyDelete
"...Some readers thought that second order meant you could quantify over functions. My intent was just to be able to quantify over sets..." ---> I suspected it, but in this case there is the *widely* used "monadic second order logic" ( only quantification over unary relations :-)ReplyDelete
... I mean it's more appropriate to use the term "monadic second order" (instead of "second order") in the note to define your problemDelete
In the argument using the Lowenheim-Skolem theorem one probably needs more care to argue that the blown up structure (M,+) is indeed elementary equivalent to (R,+). As vector spaces over Q they are not elementary equivalent as (Q,+) is of dimension 1 and (R,+) of infinite dimension.ReplyDelete
To the last anonymous: the point is that when you blow up (Q, +) to have cardinality |R| it will still be a vector space over Q (because it will still be a torsion-free divisible abelian group) and so it is *isomorphic* to (R, +) and elementary equivalent to (Q, +) because it was chosen as such (using the Lowenheim-Skolem theorem). Of course, as vector spaces Q and R are not isomorphic, but nobody claimed they were.ReplyDelete