this doesn't make sense to me, smart people? (infinity)

[daviddem]

This seems to be the key thing, especially after checking out some of your earlier links on cardinality and such. I can happily accept the proofs in the videos if we're talking about concepts that are specific to infinity, but saying that this means (for example) there are more numbers between blah than there are blah, doesn't seem to be a simplification or a translation to layman's terms, it just seems to be incorrect.

Yes cardinality and numerosity have been invented to cope with infinite sets. However it is also important to notice that they are almost completely natural extensions of the conventional notion of size to infinite sets, even if they require a bit of mind twisting to start with.

It goes like this:

- if we only ever deal with finite sets, we can just compare them by saying that one has a larger size than another if it has more elements than the other. You can easily say that set 1 is bigger than set 2, or that two sets are the same size just by counting the elements in each set.

- enter the infinite sets and mathematicians were scratching their head because they recognized that the notion of size does not apply per se to infinite sets

- then some clever guy took a long hard look at finite sets again, and noticed that:
1) two finite sets are the same size <-> there exists at least one bijective function between the two sets
2) finite set A has strictly greater size than finite set B <-> there exists at least one injective function from B to A and there exists no bijective function between A and B

The point is that, for finite sets, by 1) and 2), talking about set size is completely equivalent to talking about the existence or non-existence of bijective/injective functions between the sets. You could completely dump and forget the very notion of size, and only deal with the existence of injective/bijective functions between the sets, and you would not loose anything at all.

- the next step was to notice that, while you cannot talk about "size" per se for infinite sets, you can still very well talk about injective or bijective functions from/to/between inifinite sets. That is why mathematicians decided to start using this criteria instead of "size": because it applies equally well to finite and infinite sets. And finally all they had to do was to replace the notion of "size" which applies only to finite sets, with the notion of "cardinality" which applies equally well to finite and infinite sets, by replacing 1) and 2) with:
1) two sets (finite or infinite) have the same cardinality <-> there exists at least one bijective function between the two sets
2) set A has strictly greater cardinality than set B <-> there exists at least one injective function from B to A and there exists no bijective function between A and B