I have an equivalent definition of finite and countable sets that seems 'less' based on set theory than the usual definitions. I am wondering if anyone has encountered these definitions anywhere, esp. in the pre-set theoretic era.
First of all, set theory provides the definition of countable set (bijection to $\mathbb{N}$) and finite set (bijection to $\{0, 1, \dots, k\}$ for some $k\in \mathbb{N}$. One can also define Dedekind-finite, which is equivalent to 'finite' assuming the Axiom of Choice.
Secondly, the following definitions (restricted to subsets of the reals for convenience) are equivalent to the set-theoretical ones over a strong enough logical system.
A set $A\subset \mathbb{R}$ is finite if there is $N\in \mathbb{N}$ such that any finite sequence $(x_0, \dots, x_N)$ of real numbers, there is some $i\leq N$ such that $x_i\not \in A$.
*) A set $A\subset \mathbb{R}$ is countable if there is a sequence $(A_n)_{n\in \mathbb{N}}$ of finite (as in the previous paragraph) sets $A_n$ such that $A=\cup_{n\in \mathbb{N}}A_n$.
**) Note that the definition of 'finite' is not circular as finite sequences of reals are readily coded as real numbers.
I repeat my question: has anyone encountered *) or **), esp. in the pre-set theoretic era?
