Each set of axioms has its own purposes, and neither is better than the other in all circumstances.
It is true that, when they are interpreted in set theory, the second-order Peano axioms are categorical. They are useful for characterizing the natural numbers once we have a notion of "set" to ...12
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Let $\left\{\mathbb{F}_{p^n}| n \in \mathbb{N}\right\}$ be a family of finite fields of characteristic $p$. Whenever $m$ divides $n$, we have the inclusion homomorphism $\mathbb{F}_{p^m} \to \mathbb{F}_{p^n}$. We also have the surjection $\mathbb{F}_{p^n} \to \mathbb{F}_{p^m}$ given by the norm m...
In mathematics, a direct limit (also called inductive limit) is a colimit of a "directed family of objects".
== Formal definition ==
=== Algebraic objects ===
In this section objects are understood to be sets with a given algebraic structure such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. With this in mind, homomorphisms are understood in the corresponding setting (group homomorphisms, etc.).
Let
⟨
I
,
≤
⟩
{\displaystyle \langle I,\leq \rangle }
be a directed set. Let
...
