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I have been wondered the definition of cardinality and number of elements. One mathematician told me that one can't said that the cardinality or size of the set $\{1\}$ is one, it should be said that the number of elements in the set is one. I guess that his opinion is that there is no term size in mathematics. Is these true? On the other hand, if we have an infinite set like $\mathbb Z$, can we say that the number of elements of $\mathbb Z$ is infinite or is the term "number of elements" used only in finite sets?

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I have no idea what the mathematician was thinking. The cardinality of $\{1\}$ is indeed $1$. Of course it’s also true that $1$ is the number of elements in the set. I would probably not use the expression number of elements when the set is infinite, but I’d not complain if someone said ‘$\Bbb R$ has $2^\omega$ elements’ instead of ‘the cardinality of $\Bbb R$ is $2^\omega$’. –  Brian M. Scott Jun 15 '12 at 10:51
    
For finite sets there is absolutely no problem; a set has cardinality or size $n$ if it is in bijection with $\{ 1, 2, ... n \}$, and it has cardinality or size $0$ if it is empty. (The empty set is unique!) It is silly to prevent yourself from being able to say this. –  Qiaochu Yuan Jun 15 '12 at 12:39

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Converting my comment to an answer:

I have no idea what the mathematician was thinking: the cardinality of the set $\{1\}$ is indeed $1$, and that’s a perfectly normal way to express the fact. Of course it’s also true that $1$ is the number of elements in the set.

I would probably not myself use the expression number of elements when speaking of an infinite set, but ‘$\Bbb R$ has $2^\omega$ elements’ is a perfectly fine synonym of ‘the cardinality of $\Bbb R$ is $2^ω$’.

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Okay. I guess the mathematician wanted that I won't be his graduate student as he asked me to modify my bachelor thesis so heavily. That was one point where I made a "mistake" and got a bad grade. And is the word "size" a synonym of the cardinality? –  mathenthusiast Jun 15 '12 at 15:52
    
@mathenthusiast: I wouldn’t say so: it’s more general, as it can also refer to length, area, volume, etc. –  Brian M. Scott Jun 15 '12 at 21:38

The question you need to ask yourself is "what is the meaning of the notion number?"

Natural numbers can be used to measure size, length, etc. while rational numbers measure ratio between two integers, real numbers measure length... what do complex numbers measure?

In mathematics we allow ourselves to define new ways of measuring things, and usually we refer to the values of these measures as numbers. Indeed there is no real difference between the length of a $100$ meters running track, and a $100$ meters long hot dog...

When measuring the size of a set we came up with a clever way to discuss infinite sets. This way is "cardinality", and the cardinal of a set represents in a good sense the "number of elements in the set". So it has a perfectly good meaning to say that a set has $\aleph_0$ many elements, or that a set is of size $\aleph_1$.

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