The question is whether a first-order definable subset ( E \subseteq G^n ) (where ( G ) is the directed union of finite groups ( G_i )) necessarily has a density, defined as the limit of ( |E \cap G_i^n| / |G_i^n| ) as ( i \to \infty ).

Key observations:

1. First-order formulas have finite quantifier depth: Any first-order formula ( \phi ) defining ( E ) has a bounded number of ...

No. Even for a very simple first‑order (indeed, positive existential) definable set, the density need not exist and can oscillate wildly along the chain.

Here is a concrete counterexample.

• Let G be the (locally finite) direct sum of countably many copies of a cyclic group of order 4: G = ⊕m≥1 C4. In this group, the set of squares is exactly 2G (coordinatewise squaring). In particular, an e...

No.
There is a directed system of finite groups and a parameter–free first–order formula ϕ(x) such that the sequence

    |{g∈G i :  G⊨ϕ(g)}| / |G i |

does not converge. We exhibit an explicit example.

1. The groups in the chain

Put
G 0 = A 5 , G 1 = S 5 .

For n ≥ 1 define

        G 2n   = A 2n+5 ,            G 2n+1  = S 2n+5 .  

In ot...

This is a fascinating question about the interaction between model theory, group theory, and asymptotic density. Let me analyze the key aspects and provide some insights.

Understanding the Problem

The question asks whether first-order definable sets in a directed union of finite groups necessarily have well-defined densities. This is a significant generalization of the simpler case where sets...

Density of First-Order Definable Sets in Directed Unions of Finite Groups

This question asks whether the density of a first-order definable subset in a directed union of finite groups necessarily exists.

Understanding the Setup

We have an inductive family of finite groups:

The directed union is def...

At the moment nobody knows the correct general answer – the problem is open already for the much more restricted situation considered by J. Wiltshire–Gordon and no technique is known that would settle it for all first–order formulas.

1. “First–order definable’’ includes as a very special case the sets that are defined by a single group–word equation
     w(g1,…,gn)=1.
   (These are the set...

First of all let me rephrase the situation in my own words. You have an increasing chain of finite groups

G₀ ↪ G₁ ↪ G₂ ↪ ⋯

and you set G = ⋃ᵢ Gᵢ. A subset E ⊆ Gⁿ is called first‐order‐definable if there is a fixed group‐language formula φ(x₁,…,xₙ) (with its quantifiers ranging over all of G) so that

E = { a∈Gⁿ : G⊨φ(a) }.

One would like to define the “asymptotic density” of E by

...

This is an excellent and subtle question. The distinction you draw between the truth of a formula in the limit group versus in the finite approximations is crucial.

The answer to your question is no, the density of a first-order definable set does not necessarily exist.

A counterexample can be found for the atomic case, which, as you note, is a special case of your question. Your p...

<think> We are considering a directed union of finite groups: G_0 <= G_1 <= ... <= G_i <= ... and G = union_{i} G_i. We are interested in the density of a first-order definable set E in G^n, defined by d(E) = lim_{i->infty} |E ∩ G_i^n| / |G_i^n|, if the limit exists.

The question is: if E is first-order definable in the language of groups, does the density d(E) necessarily exist?

First, recall t...