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

[kiwiMark]

Forget talking about "bigger", "larger", "more", "size", etc when talking about infinite sets because these concepts don't really make sense for infinite sets.

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.

[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

[surviva316]

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.

I don't know (and can't be bothered to do a ton of wikipedia research at the moment on) the exact specifics of all of the mathematic definitions of "more than" and "bigger" and "larger" and all of that, but it doesn't seem all that preposterous to me that some infinities can be larger than others, at least by every practical definition we have of the word larger.

Infinites aren't endless. They're not boundless at all. If I represent a ray (let's say Ray A) on a piece of paper with its starting point on the left and the arrow to the right, then that ray has allllll sorts of boundaries that I can point to. The starting point on the left is the most obvious boundary. There's also everything above and below the line. Everything in front of and in back of the plane of the paper on which you wrote it is outside of the bounds of this infinity. All of the time before you drew the ray and all of the time after it's been erased (I realize it's only a representation of a ray, but I'm sure that rays in the real world aren't eternal).

There is simply one very specific way in which it is endless; it just keeps going to the right ad infinitum. If we were to draw a different ray (Ray B) to the direct left of Ray A, that after an inch of paper space, that overlaps with Ray A and continues in the same exact manner to the right ad infinitum, then Ray B is literally Ray A + 1 inch of paper space. Since Ray A is infinite, I know that we can't actually add that together in a way that makes the expression Ray A + 1 make any sense, but that's still exactly what Ray B is. There are several sections of spacetime you can look at that include Ray A, and all of them would include at least as much of Ray B. There are some sections of spacetime you can look at that include only Ray B. Then, finally and most obviously, there are an endless amount of sections of spacetime that you could look at that include neither. There are no instances whatsoever (though I'm only looking at finite sections of space, which I know rong will take issue with) where Ray A occupies more than Ray B. It's impossible; it doesn't exist. Its bounds don't allow for such a scenario to be possible, even though there exist some that include more of Ray B than Ray A (I'm guessing this is exactly what davidem is talking about with the injective/bijective functions).

And that's just another ray that's just an inch "longer." There's also the line that's endless both to the left and to the right that includes all of Ray A, plus infinite more. Then, there's the plane that contains Ray A, which is infinite more than that line which is infinite more. Then, there's whatever the 3-d equivalent of a plane is, that includes the plane that includes the line that includes Ray A. Then there's an eternally extant 3-d equivalent of the plane. Etc.

If we talk about this last thing, the "subset" of infinite time and infinite space in all directions, then ALMOST EVERY MOTHERFUCKING THING EVER is an example of something that is in one subset and not in the other. The only points that aren't examples of that, are points that both include the TRULY endless thing, AND points on Ray A, which is admittedly (in the grand context of all things that ever were, are, will be, could be and couldn't be) an extremely small percentage of things, though infinite they are. I mean, I know that it's infinite, so you can't divide by it to get the exact probability, but surely we're willing to admit that if we had an RNG spit out a point for anything ever, then most of the time, it would not be a point on that ray.

So I don't know whether it's literally impossible to ever "account" (bad word, but it's the best one I can think of) for the relative size of these sets' inclusion of points, or if it's just our counting-based math that makes it impossible because we're dealing with uncountable entities, but I can't see a way to where one isn't more inclusive simple because they're not finite. Sure, to say that we can't know what the probability is that a random point within the subset of {Everything that was + Everything is + Everything that will be + Everything that could be + Everything that couldn't be} would include {Ray A} seems tenable; but to say that we can't be certain that that probability would be very low seems a bit crazy.