# The definition of 'countable' and 'finite' set

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?

• I do not think so... The above definitions rely heavily on set-theoretic symbols and concepts. Apr 6, 2022 at 5:57