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

[kiwiMark]

Your logic is flawed because 9... represents an infinite number of 9's, not a number of 9's tending to infinity.

holy fuck and the penny drops

maybe I just needed to sleep on this

[surviva316]

Or another way of putting it is you're saying that

infinitelyxinfinity > Axinfinity > infinity (where A>1)

And I'm saying they're all equal.

What happens when you divide each of those terms by infinity?

If you say that it becomes infinity > A > 1, then you have proven that the greater than relationship you presented in fact correct, in which case not all infinites are equal.

If you say that you cannot assume that, then you are saying that infinity/infinity is not necessarily 1, but all equivalent values DO necessarily equal 1 when divided by each other, so then you are saying that not all infinites are equal.

Conclusion: Not all infinites are equal, regardless of your answer to the question.

Anyway, if I'm not mistaken, infinityXinfinity > Ainfinity > infinity is a false assumption. I think that 2(inf) can be "less than" just infinity (I realize I'm playing really fast and loose with the terminology from a mathematical standpoint). Infinity's too general of a term to know for sure. To use the Christian analogy, that would mean that two human Christian souls > God Himself.

I'll jump in from a more conceptual, non-mathematically, possibly flawed perspective and see if I can help. Let's imagine that we referred to all very big numbers (like anything greater 12,736) as simply "more than I can be fucked to count." So how many grains of sand are on the beach? More than I can be fucked to count, I'll tell you that much. How many USD is the US in debt? Again, more than I can be fucked to count. Are there more grains of sand on the beach than there are dollars that the US is in debt? Obviously the correct answer here is "I don't know."

Obviously if you subtract some number that is more than I can be fucked to count from the US debt, then it is more likely that this will leave more room in the debt or actually make for a surplus than it will make the debt exactly zero, because "more than I can be fucked to count" is just an entire massive class of numbers, most of which are not equivalent.

I'm sure it's not a perfect mathematical analogy, but it might be conceptually helpful to the idea of infinity. It's just anything that isn't finite; that doesn't mean that they're all some number that's just the greatest number ever wrought (in fact, it most definitely does NOT ever mean that). Some non-finite amount of something can (and most likely are) be different from another random non-finite amount of something else.

[rong]

You can't divide something by infinity. It is not a number. 2infinity is just infinity. No number is greater than infinity. It just means continues forever. Either it does or it doesn't. There's no in-between, no greater than, no comparing two of them, there is nothing to compare. I get the sets concept and what you are demonstrating. But the mistake made is in the definition of the sets. It's as if you are assuming they are finite when you compare them.

[rong]

Lol surviva at your analogy for understanding infinity that assumes its finite for simplicity.

[rong]

Also I do believe the real and natural number sets have cardinality, its just complicated to demonstrate.

Eg. You give me 0.1, I give you 1, you give me 0.2, u give you 2. Now you yet to get clever and give me 0.15, I give you 3. You give me 0.15789, I give you 4. You can go on to infinity, interestingly enough, so can I.

[Savy]

What happens when you divide each of those terms by infinity?

Remember when I said that infinity isn't a number it was a concept and you didn't really seem to take my post seriously.

Well, that question is similar to asking what happens when we divide 8 by potato.

Also I do believe the real and natural number sets have cardinality, its just complicated to demonstrate.

Eg. You give me 0.1, I give you 1, you give me 0.2, u give you 2. Now you yet to get clever and give me 0.15, I give you 3. You give me 0.15789, I give you 4. You can go on to infinity, interestingly enough, so can I.

That's wrong, and someone has already posted the logic behind why it is wrong.

The reason in easier logic (but by no means anywhere near a proof, as I am just looking at positive real numbers) is that the natural numbers are countable.

1, 2, 3, 4, 5, 6, 7.....

Whereas to count the real numbers no matter what number you pick we get more, so much so that they aren't countable.

0.1, 0.2, ohh wait we can also have 0.15 and loads of others numbers, let's try 0.01.

0.01, 0.02, ohh wait we can also have 0.015 and loads of other numbers, let's try 0.001

0.001, 0.002, ohh wait ...

As the real numbers are uncountable, that makes them a bigger set. Therefore there is no amount of natural numbers we can use to match up them all, as we could always find a real number in between ANY two real numbers, whereas we can not find a natural number in between ANY two real numbers because natural numbers next to each other, i.e. 1&2, 8&9, 214124&214125 don't have another natural number between them.

[rong]

Actually, thinking further about the cardinality issue, you're right. I can't map one into the other, but this doesn't prove one is larger than the other. It proves that set theory (or w/e) can't deal with sets with infinite cardinality.

[rong]

Savy, you're wrong, one set isn't bigger than the other, both are infinite. Now, if you decide to suddenly stop the infinite count of natural numbers then yes at the point you stop counting there are more real numbers, but the point you stop counting means you stopped that set being infinite. As long as both sets are infinite then neither is larger than the other as neither has a size, you need to stop them being infinite in order to give them a size and do the count.

[Savy]

Actually, thinking further about the cardinality issue, you're right. I can't map one into the other, but this doesn't prove one is larger than the other. It proves that set theory (or w/e) can't deal with sets with infinite cardinality.

The thing is though, it can. You can show the relationship in size between the two iirc.

A way of looking at it is that imagine if I gave person A one apple and person B two apples.

A = 1, B = 2

I repeat this.

A = 2, B = 4

I repeat this an infinite amount of times.

Both of them will have an infinite amount of apples, but person B will always have more apples than person A even though they both have an infinite amount.

[rong]

As the real numbers are uncountable, that makes them a bigger set. Therefore there is no amount of natural numbers we can use to match up them all, as we could always find a real number in between ANY two real numbers, whereas we can not find a natural number in between ANY two real numbers because natural numbers next to each other, i.e. 1&2, 8&9, 214124&214125 don't have another natural number between them.

The bold part, no amount of natural numbers is correct, but an infinite amount, well now we can.

[Pascal]

but infinite isnt a number it's just a concept?

[rong]

The thing is though, it can. You can show the relationship in size between the two iirc.

A way of looking at it is that imagine if I gave person A one apple and person B two apples.

A = 1, B = 2

I repeat this.

A = 2, B = 4

I repeat this an infinite amount of times.

Both of them will have an infinite amount of apples, but person B will always have more apples than person A even though they both have an infinite amount.

No! For any finite figure this is true, but not for
Infinity. It never ends! If you stop at any point to count then what you say is true at that point for that finite amount but if each are infinite then neither is bigger.

[Savy]

but infinite isnt a number it's just a concept?

That's why my example is partially rubbish, but it was just to try and simplify the concept so it'd make more sense.

[rong]

And Savy, your apples example would product two sets that have cardinality and are therefore considered the same size anyway.

[Savy]

And Savy, your apples example would product two sets that have cardinality and are therefore considered the same size anyway.

Read my above post.

People have posted the actual proof for at least comparing the size of natural numbers and positive real numbers in this thread. If you don't believe it, there will be numerous sites on the internet which go into it in great depth. It's definitely right and what you're saying is definitely wrong.

[rong]

I disagree. Sometimes the general consensus is wrong. I get what they're saying, but I think its a limitation of the theory that it can't deal with infinite sets.

[daviddem]

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

If you take something out of this, it should be that the infinite sets of the same family as the natural numbers are "countable" or "listable". You can decide some criteria to order the elements one way or another, and start writing a list of them. This list is never ending, but as you go you will never miss an element:
0
1
2
3
etc
I didn't miss any natural number so far, and I won't for as long as I continue.

You can't do that with infinite sets of the same family as the real numbers. These sets are "uncountable" or "unlistable".

However it is mathematically true that if you define some arbitrary symbol N0 to designate the cardinality of the natural numbers, it is then possible to prove that the cardinality of the real numbers is equal to 2^N0, and this is always bigger than N0.

So the cardinality of the set of all real numbers is greater than the cardinality of the set of all natural numbers, whether you like it or not.

And in math, another way of saying that the cardinality of a set is greater than the cardinality of another is to say that the first set is more numerous than the second. Two sets that have the same cardinality are equinumerous.

So while both the set of natural numbers and the set of real numbers are infinite, they are not equinumerous.

[Savy]

I disagree. Sometimes the general consensus is wrong. I get what they're saying, but I think its a limitation of the theory that it can't deal with infinite sets.

If you aren't an expert in set theory this is such a ridiculous statement to make it actually annoys me.

[rong]

I'm ok with that.

[seven-deuce]

If you aren't an expert in set theory this is such a ridiculous statement to make it actually annoys me.

Are you an expert in set theory?

[rong]

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

If you take something out of this, it should be that the infinite sets of the same family as the natural numbers are "countable" or "listable". You can decide some criteria to order the elements one way or another, and start writing a list of them. This list is never ending, but as you go you will never miss an element:
0
1
2
3
etc
I didn't miss any natural number so far, and I won't for as long as I continue.

You can't do that with infinite sets of the same family as the real numbers. These sets are "uncountable" or "unlistable".

However it is mathematically true that if you define some arbitrary symbol N0 to designate the cardinality of the natural numbers, it is then possible to prove that the cardinality of the real numbers is equal to 2^N0, and this is always bigger than N0.

So the cardinality of the set of all real numbers is greater than the cardinality of the set of all natural numbers, whether you like it or not.

I agree with the uncountable part. I agree that cardinality of the set of all real numbers is bigger. What I dispute is the interpretation that this means one infinity is.bigger than the other. And further I'm stating that if that's what this branch if mathematics states then it shows it is unable to deal with infinite sets.

[Savy]

Are you an expert in set theory?

No, I'm not the one saying it's wrong though. See the difference?

[daviddem]

One infinity is not "bigger" than the other and one set is not "bigger" than the other in the conventional sense of the term. I guess they maybe use that term in the videos or in articles to try and explain the idea and give a feel to people of what cardinality and numerosity of infinite sets means. It is extremely difficult to explain math and physics concepts to non mathematicians with every day language and without equations and mind boggling definitions, that is why they used this term "bigger" which, strictly speaking, is incorrect.

It is however correct to say that one set is more numerous than the other or that one set has bigger cardinality than the other if you stick to the definition of numerosity and cardinality that mathematicians have adopted for infinite sets, to extend as naturally as possible these notions from finite sets to infinite sets.

[seven-deuce]

No, I'm not the one saying it's wrong though. See the difference?

No need to throw a hissy fit, i was only asking a question. Jeez.

[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.

[a500lbgorilla]

So gimmel, does it make sense now?

[givememyleg]

no

[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]

Remember when I said that infinity isn't a number it was a concept and you didn't really seem to take my post seriously.

Well, that question is similar to asking what happens when we divide 8 by potato.

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.

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.

[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.

[MadMojoMonkey]

^ gangsta

[daviddem]

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?

[surviva316]

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.

[daviddem]

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.

[surviva316]

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?

[surviva316]

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.

[daviddem]

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?

[surviva316]

BUT THAT DOESN'T MAKE SENSE MATH IS STUPID!!!

[daviddem]

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!

[Penneywize]

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.

[daviddem]

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:

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.

And following this, the remark of the author of the article:

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.

[daviddem]

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.

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.

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.

[kiwiMark]

I'm trying to count my sperm but am struggling a bit, any help?

[Savy]

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.

[daviddem]

If there's any hope left for the world it'll be 0.

lol

It's OK: if it's somewhere he can count it (i.e his wrist), there should not be much risk of reproduction.

[surviva316]

How would he have gotten sperm on his wrist, and how would he get it to stay there?

[daviddem]

How would he have gotten sperm on his wrist, and how would he get it to stay there?

You focus on Galileo's paradox.

[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.