Precise meaning of infinitely many

Question

I have a rather lame question here. I need a clarification with the definition of "infinitely many". I have come across statements like:

There are infinitely many reals.

I know that reals are non-denumerable (uncountably infinite).

Again we have:

There are infinitely many integers.

I also know that integers are denumerable (countably infinite).

So my question is what do we actually infer from "infinitely many" about the countability or the uncountability?

Also kindly correct me if I am wrong somewhere.

Answer 1

"Infinitely many" is just the negation of "finitely many."

You can't infer anything about countability from "infinitely many." There are infinitely many rationals and there are infinitely many real numbers.

To be more specific, you'd have to say something about "countable" in there somewhere.

"Countably many" is usually taken to mean "either finitely many or countably-infinitely many." Then you have "uncountably many," the negation of that.

Answer 2

When we use infinitely many, we mean the set under consideration is not finite. It does not address the question that whether the set is countable or not. We can see it as opposed to finitely many which means the set is finite.

Answer 3

A very concise and dry definition of infinity: a set $S$ is infinite if and only if there is an injection from $\mathbb{N}$ to $S$.

A set $S$ is countably infinite if there is a bijection from $\mathbb{N}$ to $S$. From this definition, you can see that the set $\mathbb{R}$ is infinite, but not countably infinite. Also, you can see that all countably infinite sets are also infinite sets.

Answer 4

A set S is finite if there exists an injection from S to a subset of N having a maximum element. A set is infinite if and only if it is not finite. A set is also infinite if and only if it can be put into a bijection with a proper subset of itself.

S is countable if there exists a bijection to the integers, uncountable otherwise. Is interesting to note there are multiple possible cardinalities of uncountable sets. The cardinality of a power set of a set always has a cardinality greater than the original set. So any of these "sizes" of infinity is possible, and perhaps others, especially given some assumption on the Continuum Hypothesis.