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As the title stated , what is the meaning of infinitely many ? When we say a set contains infinitely many elements, does this mean we cannot finish counting all the elements in the set ? Does infinitely many same as $\forall$ ?

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up vote 7 down vote accepted

No, $\forall$ means that for all elements. Infinitely many just means there are infinitely many elements with a certain property.

For example $A=\{p\in\mathbb N\mid p\text{ is a prime and } p\equiv 1\pmod 4\}$ contains infinitely many prime numbers, but it is not true that all the prime numbers are in $A$, or that all the natural numbers are in $A$. Therefore the property defining $A$ holds for infinitely many elements, but hardly for all of them.

Another example would be $\mathbb R$ with the $\leq$ relation and $0$. For infinitely many numbers it holds that $0\leq x$, but hardly $\forall x.0\leq x$.

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$\forall$ is just a shorthand way of saying "for all". It can be used for infinite or finite sets.

For example:

$$\forall x \in \{ 1, 2, 3\},\ x > 0.$$

"Infinitely many" means that there are not finitely many. In other words, "infinitely many" means that there does not exist some real integer $n$ such that you can describe the objects with a set of cardinality (size) $n$.

There are different types of infinitely many: countably many, which means the infinite set can be counted with a bijection to the integers, and uncountably many, which means the infinite set cannot be counted with a bijection to the integers.

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Infinitely many means that you can find a subset and a bijection with $\mathbb{N}$, that is, you have an infinite elements.

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Aren't there infinitely many real numbers? – Rahul Feb 12 '13 at 17:34
@ℝⁿ. Well, sometimes there are only countably many of them if you ask certain cranks. :-) (Also, change that username back already!) – Asaf Karagila Feb 12 '13 at 17:59
Yes, but this is another infinite 'greater than' the first one. You know, that notion of $\aleph$. – Sigur Feb 12 '13 at 18:00
@Asaf: "Display name may only be changed once every 30 days; you may change again on Feb 28 at 0:06" My hands are tied! – Rahul Feb 12 '13 at 18:52

You have to understand this from basic definitions.

Firstly, a set A is said to be infinite if it is "not finite" OR equivalently, there exists a bijection from A to a strict subset of itself. Additionally, we don't say a set contains infinitely many elements, instead we say it is infinite. We use the phrase "infinitely many" in the following way for example:

"A has infinitely many elements satisfying property P"

This in NOT synonymous as $\forall$ as the other answers have answered.

So what does finite mean? A set A is finite if $\exists K \in \mathbb{N}$ such that there is a bijection from A to $\{1,2,..K \}$

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