Cardinality as "size of a set" Given the discussions on Refuting the Anti-Cantor Cranks in which I left a comment that I suspected would not get any attention, I decided to start a new thread instead.
Now, I can't read the mind of "anti-Cantorians" so I don't really know what their objections are. But I get the impression that most reasonable objections aren't based upon the validity of the proofs that $\mathbb{R}$ has higher cardinality than $\mathbb{N}$. That seems pretty undisputable.
But a viewpoint I personally haven't been able to shake is the notion of cardinality as capable of describing "sizes" of sets. Now, different people may have different views on which criteria this notion of size absolutely needs to adhere to and which we can discard, but I think it's certainly thinkable to require that any reasonable notion of size needs to satisfy $B \subset A \Rightarrow \text{size}(B) < \text{size}(A).$
Starting from the assumption that $B \subset A \Rightarrow \text{size}(B) < \text{size}(A)$, then the statement "if two sets can be put into one-to-one correspondence with each other, then they are of the same size" leads to a contradiction and can therefore not be true, leading us to a chicken-and-egg axiom argument with no right answer.
Which makes me wonder. Could the interpretation of cardinality as the "size of a set" simply be seen as a mneumonic? Am I wrong in assuming all mathematical results relating to cardinalities of sets will be as relevant without thinking of it as a "size"? 
What does the interpretation really buy us and what would we lose by leaving the notion of the size of an infinite set undefined?
EDIT: The very nice discussion in Relative sizes of sets of integers and rationals revisited - how do I make sense of this? is probably sufficient for this question, but I'll reiterate my position since it apparently wasn't as clearly formulated as I thought given the down-votes.
We want a nice, intuitive notion of the "size" (or better yet, "numerosity") of a set. So we make a list of properties that we know hold for finite sets that ideally we would want to hold for infinite sets. Two of these:
1) If you can take all the elements of set $A$ and place each element next to a unique member of set $B$, then $A$ and $B$ are "of the same size".
2) If you take a set $A$ and proceed by removing some elements from it, then you will have a set smaller in size than you started out with.
Taking the first one as an axiom leads to the concept of "cardinality", causing property 2 to lead to a contradiction, while taking the second one as an axiom leads to the first one causing a contradiction. Clearly we can't have both of them.
From some answers and comments, you get the impression that expecting the second property to hold is a "fallacy", sprung through some naive expectation that properties of finite sets should automatically carry over to infinite sets. But it seems to me that the same could be said for the first property. The only reason to prefer using #1 as a definition seems to be "more interesting mathematics spring from it", which is clearly an excellent reason - but it doesn't make it a more natural candidate for capturing the notion of size than #2.
EDIT #2: There was an excellent point brought up in the question I linked above which is the formulation of various closure operations on sets - for example the convex hull of a $A$ is the smallest convex set containing $A$. So letting $A = \{0,1\}$, what's its convex hull? Well, $[0,1]$ certainly can't be the unique correct answer, since $\mathbb{R}$ is of the same size! Not that this invalidates the cardinality concept, but it shows that maybe we should be careful with equating cardinality with size.
 A: Why any reasonable notion of size should satisfy $A \subsetneq B \implies size(A) < size(B)$? 
Mass is a good notion of "size", but we can have a bar of chocolate surrounded by vacuum and it does not respect your properties (ignoring any physics "mumbo-jumbo"). Mathematically, measure (in the context of measure theory) is also a good notion of "size", and it does not respect your property either.
Not only that, but cardinality is not a measurement of "size". It measures "correspondencicity". This is true even for finite sets. When we count them (for example, with fingers), we are corresponding each finger with the elements of the finite set. For instance, if you take $10$ balls of steel, and put them scattered in a closed room , the "size" that they are enclosing is very big. If you cluster them in the table in that room, not so much. However, I can correspond to each element of the previous arrangement an element of the new arrangement in a bijective manner. This is the intuition from finite sets. 

As a sidenote, based on the comments, I feel this personal input may be useful:
 In my understanding, intuition is when you work, see or deal with something on a regular basis and has developed an acquaintance with the subject, sufficient enough for you to be able to infer something without a clear logical concatenation. This seems to aggree with the entry on the wiktionary for intuition:

Noun, intuition ‎(plural intuitions)
  
  
*
  
*Immediate cognition without the use of conscious rational processes.
  
*A perceptive insight gained by the use of this
  faculty.

and also to the "colloquial usage" section: "Intuition, as a gut feeling based on experience, (...)"
Therefore, a person who has dealed with finite sets has no intuition with infinite sets. Period. What he has is naivety:

Naivety (...) is the state of being naïve, that is to say, having or showing a lack of experience, understanding or sophistication, often in a context where one neglects pragmatism in favor of moral idealism.

or

Adjective, naive
Lacking worldly experience, wisdom, or judgement; unsophisticated.
(of art) Produced in a simple, childlike style, deliberately rejecting sophisticated techniques.

And one of the utmost goals of Mathematics is to get rid of naivety.
