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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.

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    $\begingroup$ The two locutions simply do not make a distinction with different "levels" of infinity. They simply states that both sets (of real and of integers) are not finite. $\endgroup$ – Mauro ALLEGRANZA Sep 11 '18 at 14:57
  • $\begingroup$ "There are at least two numbers", does that tell you anything about how many numbers are there? $\endgroup$ – Asaf Karagila Sep 11 '18 at 15:04
  • $\begingroup$ @AsafKaragila Yes? It tells me there are at least two. $\endgroup$ – Theoretical Economist Sep 11 '18 at 15:44
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    $\begingroup$ @TheoreticalEconomist: Exactly. $\endgroup$ – Asaf Karagila Sep 11 '18 at 15:47
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    $\begingroup$ @MauroALLEGRANZA And, correspondingly, "Finite" doesn't make a distinction between the different "levels" of "non-infinity". $\endgroup$ – David Richerby Sep 11 '18 at 19:01
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"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.

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  • $\begingroup$ For example, if we say, infinitely many group homomorphisms can be defined from the set of integers to itself, is it okay? $\endgroup$ – Shatabdi Sinha Sep 11 '18 at 14:59
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    $\begingroup$ @ShatabdiSinha Sure. In fact you can also say the set is countably infinite since you are just specifying an image for $1$ for each homomorphism, right? $\endgroup$ – rschwieb Sep 11 '18 at 15:01
  • $\begingroup$ Wow, what a poor choice of terminology, saying "countably many" meaing "at most countable". I didn't know people really used these words like this, thanks for the info. $\endgroup$ – Serge Seredenko Sep 11 '18 at 21:26
  • $\begingroup$ @SergeSeredenko It seems really like insisting that “countable=countable infinite” is a worse choice, because then you either have to say finite sets are uncountable, or you have to define uncountable sets as “infinite and not countable.” These two alternatives seem much less natural than using the inclusive definition of “countable.” But if you only rarely needed uncountable sets in a text, it would be convenient to have a convention that countable sets be infinite. $\endgroup$ – rschwieb Sep 12 '18 at 0:30
  • $\begingroup$ I am sorry, I was wrong. I based my judgement on the fact that "countable set" means infinite, so "countable set" and "countably many" would mean different things. But now I see that countable set may be finite as well. I just did not expect that because in my language nobody uses the word "countable" to mean a finite set. So, thanks for the info once more. $\endgroup$ – Serge Seredenko Sep 12 '18 at 1:28
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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.

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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.

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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.

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    $\begingroup$ Uncountable just means "not countable", just like "infinite" means "not finite". So by definition, a set cannot have a higher cardinality than "uncountable". $\endgroup$ – Asaf Karagila Sep 11 '18 at 20:39

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