So gimmel, does it make sense now?
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So gimmel, does it make sense now?
no
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.Quote:
Originally Posted by kiwiMark
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
I wasn't not taking it seriously; I was agreeing. The passage you quoted when you said "infinity isn't a number, it's a concept" said almost the exact same thing except in different words.Quote:
Originally Posted by ImSavy
The passage you quoted was me using a hypothetical to disprove that infinites are equivalent. I said later in the post that the whole exercise was irrelevant and impossible. The point was that given rong's premise (that infinites are an amount, all of which are equivalent to each other), then any amount that is divided by an equivalent is 1, which would disprove rong's own point.
Again, though, later in the post I went on to say that the premise is wrong anyway and all of the terms couldn't be compared because infinity isn't a number.
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.Quote:
Originally Posted by kiwiMark
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.
^ gangsta
Your ray example to explain there are infinities of different "sizes" than others isn't very good, because they are the same "size", because you can find a bijective function between the two rays.
In your example, let's define a reference scale in the direction of the rays such that ray A starts at 1 and ray B starts at 0. Then an example of bijective function from ray B to ray A is:
f: B->A: x->f(x)=x+1 where x is any point of ray B.
The inverse function is of course:
invf: A->B: y->invf(y)=y-1 where y is any point of ray A.
So these two rays have the same cardinality... they are the same kind of infinity. One is not "bigger" or "longer" than the other. They are the same "size".
This may be counterintuitive, but that is how it is. Take for example the sets of real numbers comprised in the intervals [0, 1] and [0, 2]. These two infinite sets also have the same cardinality (with the bijective function f(x)=2*x and its inverse invf(y)=y/2). Even though you can find an infinity of elements of [0, 2] that are not in [0, 1], and even though [0, 2] obviously overlaps more of the set of all real numbers than [0, 1] does, these two sets still have the same cardinality, the same "size": their infinities are the same mathematic animal.
Weird, uh? Especially when you think that the (finite) sets of natural numbers comprised in the intervals [0, 1] and [0, 2], do not have the same cardinality: [0, 1] has cardinality 2 and [0, 2] has cardinality 3...
Head ass-plodes yet?
So x is a whole, counting number. What's the probability that x is even? Surely it's 50%, right? How is this the case unless there are twice as many whole counting numbers as there are whole counting numbers that are even?
Maybe we're unable to use our mathematic system to calculate probabilities that involve infinite numbers, but I can't see how it's not more likely for any x that we know nothing about to be a counting number than it is that it's an even counting number.
It does make my head hurt a bit to think about.
You are confusing yourself again because you are talking about "how many" elements there are in infinite sets, or "twice as many" elements in an infinite set as in another. Drop that language, and you'll be all right. It doesn't make any more sense than saying that there is twice as much liquid in this empty glass as there is in that other empty glass. It's true though, because 2 times zero is still zero, just the same as two times infinity is still infinity. It's true, but it's just not a useful statement to make. Multiply zero by 5 and it's still the same zero, so why bother multiplying it? Multipy infinity by 10 and it's still the same infinity. Raise zero to the square, it's still the same zero. Raise infinity to the square and it's still the same infinity. Doing any of that to the infinite size of an infinite set does not change its cardinality.
The trick to get it is to forget entirely the conventional notion of size. Pretend you never learned what size was, or never knew how to count elements in a set. Replace it entirely with the notion of cardinality and the existence of bijective/injective functions between a set and some reference set. That is actually what you subconsciously do when you say there are 3 elements in the set {a,b,c}, what you really do when counting the elements is creating a bijection between this set and the reference set {1,2,3}. Work from this premise, and the rest will come naturally.
But it just seems like semantics. I'm fully willing to admit that I'm using the strict denotations of the words too loosely, but the concept still seems true. If, as you say, it's true but an impractical way of thinking of it from a calculation standpoint, then I don't see how this should dismiss practical applications of this perspective?
I mean, even if its incalculable or even mathematically silly, to consider how low of a percentage a ray's existence has in the world, it can be conceptually useful to know that on some level that it's low, and so that by some perspectives, an infinite can be relatively "small." There are ephemeral infinites. There are infinites that have no effect on our existence. There are infinites that allow room for contradictory infinites that do not ever interact and so that do not ever "cancel each other out."
Saying that even numbers make up only half of the number line seems like a perfectly sensible and practical perspective to take on the number line even if we can't find a mathematically sensible way to divide one infinite by another to get 2.
So in other words, if the math deems it no more "wrong" than saying that there is twice as much liquid in an empty glass as there is in another empty glass, then if it's practical to look at it that, then doesn't it make sense?
And then, of course, I might be wrong, and thinking that there are any more counting numbers than even numbers is just more proof that humans are inherently stupid.
It's not an impractical way of thinking about infinity from a calculation standpoint, it's an impractical way of thinking of infinity altogether.
Your reasoning seems to have lead you to think that the infinity of points on a straight line or the infinity of points in a plane or the infinity of points is some volume or all of space-time (assuming space-time is a continuum) are somehow different. They're not. All these sets of points are equinumerous to each other, and are equinumerous to the set of real numbers (if you want to decide that R is the reference, which is usually the convention). They are, however, more numerous than the set of natural numbers, or any other infinite set of discrete elements.
You are fooling yourself looking at the natural numbers as a "line". We are talking about sets and you should picture the natural numbers and the even natural numbers as distinct sets to think correctly about this. Picture them side by side, not as one included into the other.
What you want to express with your natural number line is that, if x is an even natural number and if n(x) is the number of natural numbers in the interval [1,x] and e(x) is the number of even natural numbers in the interval [1,x], then:
lim___e(x)/n(x) = 1/2
x->inf
which means that as x becomes greater and greater, the ratio of even numbers to natural numbers in the [1,x] interval tends to 1/2 (actually in this case, it remains 1/2 indefinitely, because e(x)=0.5x and n(x)=x so e(x)/n(x)=0.5 no matter how much x is).
But this isn't the same as saying that the infinity of even numbers divided by the infinity of natural numbers equals 1/2. You have to distinguish between something that tends to infinity and something that is infinite.
There is no practical difference between the set of natural numbers and the set of all the grains of sand on an infinite beach. There is similarly no difference between the set of all even natural numbers and the set of grains of sand. So why should there be a difference between the set of natural numbers and the set of even natural numbers?
BUT THAT DOESN'T MAKE SENSE MATH IS STUPID!!!
The point is that you would not even question any of this if you had never learned as a child to count the elements in a set to find out how big the set is.
If instead they had taught you from the get go that one set is as big as another if and only if there is a bijective function between the two sets, then it would be 100% natural for you to say that the set of even natural numbers is as big as the set of natural numbers, that the set of real numbers is bigger than the set of natural numbers, that the set of real numbers between 0 and 1 is as big as the set of all real numbers and that the set of points on ray A is as big as the set of points in the entire universe.
Unlearn the preconceptions and transcend yourself!
Hey, totally grunching the thread here. I read daviddem's first reply that explained cardinality and it made perfect sense. I've done an MA in Econ, which isn't nearly as math-y (we get looked down on by math types) as some disciplines, but however. I hadn't really needed to understand cardinality but have been made familiar with these concepts before. Anyway.
One thing that has always bugged me:
In the second video (the one with the clearer explanation and proof of the infiniteness of the set (0,1) being larger than that of the set of all natural numbers), the proof explicitly excludes the number 9 in the construction of 'x'.
This is justified as a way to avoid ambiguities like 0.499999... = 0.5
I had personally always thought that 0.499999... != 0.5
because we can always choose a sufficiently small number 0.000....01 such that 0.5 minus this number equals 0.49999...
Am I wrong? This has always pissed me off.
Hey surviva, if it helps, Galileo had the same problem you have. Only instead of using the example of the set of natural numbers and the set of even natural numbers, he used the set of natural numbers and the set of the squares of natural numbers {0,1,4,9,16,25,36,...}. This problem is called "Galileo's paradox", and the solution to it is just as I said: dump the old notion of "size" and replace it with cardinality. Or rather, extend, generalize the notion of size to that of cardinality.
See here:
http://en.wikipedia.org/wiki/Two_New_Sciences#Infinity
and here:
http://en.wikipedia.org/wiki/Galileo%27s_paradox
From the first article, Galileo's conclusion:
And following this, the remark of the author of the article:Quote:
We can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal," greater," and "less," are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number.
Quote:
This conclusion, that ascribing sizes to infinite sets should be ruled impossible, owing to the contradictory results obtained from these two ostensibly natural ways of attempting to do so, is certainly a consistent resolution to the problem but less powerful than that used nowadays. In contemporary mathematics, the problem is resolved instead by only considering Galileo's first definition of what it means for sets to have equal sizes; that is, the ability to put them in to one-to-one correspondence. This turns out to yield a way of comparing the sizes of infinite sets that is free from contradictory results.
This is already discussed at length in some posts in this same thread. I am not doing this again, but essentially, yes, you are wrong.Quote:
Originally Posted by Penneywize
0.4999... equals 0.5 because 9... represents an infinite number of 9's, not "some finite number of 9's that we can make as large as we want" or "a number of 9's that tends to infinity". Being infinite and tending to infinity are two different things.
I'm trying to count my sperm but am struggling a bit, any help?
If there's any hope left for the world it'll be 0.Quote:
Originally Posted by kiwiMark
lolQuote:
Originally Posted by ImSavy
It's OK: if it's somewhere he can count it (i.e his wrist), there should not be much risk of reproduction.
How would he have gotten sperm on his wrist, and how would he get it to stay there?
You focus on Galileo's paradox.Quote:
Originally Posted by surviva316
Also, Galileo's literally one of my top 5 favorite people in history...certainly my favorite sellout ever, so I will take that as solace.