When it is said that there are infitely many points on a line, does it imply anything about the cardinality of set of all points on a line? I think the answer to the question depends on the definition of a line and what is meant by the infinitely many points on it, so apologies if there are ambiguities in my question (but I'd appreciate if you can choose an unambiguous case and answer that!).
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One of the most standard definitions of what it means for a set to be infinite is:
A set is infinite if there exists a bijective function (1-1 and onto) witha o proper subset of the set.
Now, if you say that there are infite member in a set the only thing you can conlude is that this set isn't finite or in other words that it's cardinality isn't a finite number.
For example, if you "start" constructing the real line (I am now describing one of many such constractions) , first you create the natural numbers then the integers and then the rationals. At this point a line contains infinite numbers but only countably infinite (which means that there is a bijective between the naturals and the reals). If you now construct also the reals then again your line has infinite points but uncountably infinite, meaning that there doesn't exists any bijection between the naturals and the reals.
So, in conclusion, if we only have the data tha a set is infinite we can't conclude anything else.
On
Usually it is assumed that when we say "line" we mean something where we can choose a starting point and a unit, and then the points on the line correspond one-to-one to the real numbers.
Under this standard interpretation, the cardinality of the set of points on a line is exactly $|\mathbb R|$, which also known as $\mathfrak c$. This is known to be the same as the cardinality of the set of all possible subsets of $\mathbb N$ -- that is, $2^{\aleph_0}$ or $\beth_1$ -- but it has no other simpler name.
However, there are also models of Euclidean geometry where each line (and indeed the entire plane) contains only countably many points. The general view is that such models are in some sense "cheating" in that they arise by ignoring points that "ought" to be in the plane simply because we can. But some people find that view simpler anyway.
There seem to be two distinct questions here: the title question, and the body question "When it is said that there are infinitely many points on a line, does it imply anything about the cardinality of set of all points on a line?" These have different answers. The answer to the latter is: "It implies that the cardinality is infinite." Saying that a set is infinite is just a statement about the set's cardinality, in the same way that saying that a set has five elements is a statement about the set's cardinality.
So with regard to that, I think you may be over-thinking what "there are infinitely many ..." means - it just means exactly that, that the set of [things] is infinite, or has infinite cardinality.
Meanwhile, the title question has a reasonably exact answer. Different sets, infinite or otherwise, can be compared by considering injections; we say $A$ has cardinality at least that of $B$ if there is an injection from $B$ to $A$. (There are various alternatives we can use here, which are equivalent if we assume the axiom of choice; in general, some care needs to be taken.) Two sets have the same cardinality if there are "injections both ways;" by the Cantor-Bernstein theorem, this is the same as the existence of a bijection (note that this does not require choice!).
Using this language, we can show that each of the following sets have the same cardinality:
The set of points on a line, as usually understood (e.g. a line in Euclidean space); that is, the cardinality of $\mathbb{R}$.
The set of points on a plane: the cardinality of $\mathbb{R}^2$.
The set of infinite sequences of zeroes and ones.
The set of infinite sequences of natural numbers.
The set of sets of natural numbers.
The cardinality of all these sets is denoted "$2^{\aleph_0}$." Another important fact is that these sets are uncountable.
There are, interestingly, some basic facts about this cardinality which are not known, and in fact known to not be establishable from the usual axioms of set theory. Chief among these is whether there is any cardinality "strictly between" that of $\mathbb{N}$ and $\mathbb{R}$ (which are different since the latter is uncountable). This question is known as the continuum hypothesis, and it is known that the usual ZFC axioms of set theory cannot resolve the question one way or another. But that's going fairly far afield, so I'll stop there.