Comparing sets of possible values Say $x$ is a real number, can one argue that the set of possible values for $x$ when $x < 2$ is less than when $x < 4$? Is this mathematically true?
One could argue that the set of possible values in the latter case contains that of the former case, plus other values. But, we are comparing infinite values.
 A: You've hit on the fact that 'size' is a difficult notion when it comes to infinite sets.
Mathematicians have tried to come up with mathematical definitions to compare the 'sizes' of infinite sets, and arguably one of the most 'well-behaved' definitions is the one of cardinality: we can say that two sets have the same 'cardinality' if and only if their elements can be put into a one-to-one correspondence. 
This one-to-one correspondence certainly seems to capture an intuitive aspect of 'size': if we can reach into two bags and pull out one element from each and throw them away, and if continuing to do this will exhaust the bags at the same time, then the bags are of equal size. Now, we can't exhaust infinite sized bags this way, but a one-to-one-correspondence certainly captures this very idea. 
Moreover, the relation of two sets having the same cardinality has lots of nice properties corresponding to our intuitive notion of 'size: it's reflexive, symmetric, and transitive, and it has some more important and again intuitive properties as well. So, when mathematicians talk about 'size', they are typically talking about this notion of 'cardinality'.
But, this still leaves open the question as to whether this notion of 'cardinality' is really the best way to think about 'size', let alone that this some how is the same as 'size'. Indeed, the two sets in your example turn out to have the same cardinality, even though the one is a proper subset of the other, which is rather counterintuitive, and one could certainly make an argument that any set that is a proper subset of some other set is 'smaller' than that set.
In the end, though, mathematicians will simply work with their definitions. Thus, they will say that the two sets in your example have the same cardinality ... and thereby simply sidestep the notion of 'size' or 'smaller than' ... even though they will often use that language informally when talking about cardinality.
A: It depends on your notion of size of a subset. If we partially order subsets of the real number using the subset relation then yes in a sense i.e. $\{x\in\mathbb{R}\mid x<2\}\subset \{x\in\mathbb{R}\mid x<4\}$. 
If we compare the sets in terms of cardinality, then no. They have the same cardinality. Let $A=\{x\in\mathbb{R}\mid x<2\}$ and $B=\{x\in\mathbb{R}\mid x<4\}$. The map $f\colon A\to B$ given by $f(x)=x+2$ is a bijection. 
If we compare using Lebesgue measure, then no. Both sets have measure equal to $\infty$.
A: mathematicaly a real line R is isomorphic with the interval $(0,1)$. I mean they have the same cardinality. can you imagine that?? 
