Are There More Sets or Groups?

In an effort to procrastinate from revising for my exams next month, I pondered the question of whether or not there are more sets than groups. The intuitive answer is that there should be more sets than groups, since a group is a set with added structure. I think it is possible to show that in fact the “number” of groups is greater than or equal to the “number” of sets, using a little bit of Model Theory.

In Model Theory, the language of groups has similarity type \langle -; 2,1;1\rangle, and has an alphabet consisting of Predicate Symbols: “=”, Function Symbols: “\cdot“, “^{-1}“, and Constant Symbol: “e“. We say a group is any structure with the above similarity type that models the following axioms:

  1. Associativity: \forall xyz ( x \cdot (y \cdot z) = (x \cdot y) \cdot z) ,
  2. Identity: \forall x (x \cdot e = x \land e \cdot x = x),
  3. Inverse: \forall x ( x \cdot x^{-1} = e \land x^{-1} \cdot x = e).

Now, let’s say that we have a group G and a set X and there exists a bijection f: G \longrightarrow X.  We can induce the structure of G onto X by defining a \cdot_X b \iff f^{-1}(a) \cdot f^{-1} (b) and defining inverses of elements in X the same way, by the inverse of the pre-image of the element. This shows that given a group with cardinality \kappa, and set with cardinality \kappa gives rise to a group.

As the axioms of group theory are consistent, we now apply both Upwards and Downwards Skolem-Löwenheim theorems to deduce that there is a group of every cardinality. Combining this with above, we see that every set can be given a group structure. We can also note that if there is a bijection between two sets, then we can form another bijection simply by swapping the output of two inputs. This means that every set might give rise to multiple non-isomorphic groups, implying that the “number” of groups is greater than or equal to the “number” of sets.

We can apply the same reasoning to any algebraic structure, so this could equally be about Rings or Fields. I feel like there should be an equality between them, however I am not sure how to show that yet. I think I would need a deeper understanding of axiomatic set theory to prove such a thing. One way would be to show that my argument of producing more groups from a set actually only gives rise to isomorphic groups.

 

James

 

My Model Theory knowledge is based off of van Dalen’s Logic and Structure, which is my reference for this post.

 

 

 

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