The finite analogue of the axiom of choice is true, and it seems highly intuitive that it would be true for the infinite case. It is, however, undecidable. When explaining this to myself or to others, what I typically note is that our intuitions about finite sets don't necessarily hold over for infinite sets, and we should try to discard them. But this itself is an intuition that I'm looking to fortify. I thus ask, what are examples of theorems that hold true for finite sets, and to most people would seem to hold true for infinite sets, but don't. I.e., I'm looking for something that's provably false for infinite sets rather than merely undecidable.
[Math] Properties that are true for finite sets but are (non-trivially) false for infinite sets
elementary-set-theory
Best Answer
Take a set of rational numbers $A$ such that $$ \sum_{a\in A}=x<\infty. $$ If $A$ is finite, then $x$ is rational, which is not necessarily true if $A$ is infinite. This doesn't have as much to do with sets really, but shows how being closed under finitely many operations does not determine behavior under arbitrarily many operations. You could play this same game with intersections of closed sets.
You could also look at the power set of a set. The power set of $A$ is finite if and only if $A$ is finite, and is in fact uncountably infinite if $A$ is infinite.
Another option deals with functions from a set to itself. Let $f$ be a function from $A$ to itself. If $f$ is injective, then it is automatically bijective and hence invertible if $A$ is finite. However, this does not necessarily hold if $A$ is infinite.
One last example: any proper subset of a finite set has strictly less cardinality than the original set. However, this fails in the infinite case, as can be seen by looking at the set of integers and the subset of even integers.