Infinity and structures

Question

Do you know any case (example) where an "infinite" object with a structure (say, an infinite group) cannot be extended (in the sense of adding elements) in any way without it no longer having the structure? A case where extending the infinite object in any possible way necessarily "breaks" the structure?

Thank you

Answer 1

No proper extension of the natural numbers satisfies the full second order Peano Axioms.

No model of Hilbert's axiomatic geometry of the plane can be properly extended to a model of that geometry.

Note that by the Lowenheim-Skolem Theorem, an example of the type you seek cannot be specified in a purely first-order way.

Answer 2

$\Bbb{R}$ cannot be extended as an Archimedean ordered field.

Answer 3

The integers cannot be extended to a larger Abelian group which is a free group.

The algebraic closure of the rationals cannot be extended into a field $F$ which is an algebraic extension of $\Bbb Q$.

Answer 4

The Frobenius Theorem states that $\mathbb{H}$, the quaternions, cannot be embedded into a higher dimensional normed division algebra over $\mathbb{R}$.