Cardinality as "size of a set" [closed]
Solution 1:
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.