Can an actual infinite exist?

[QUOTE="inoperativeRS"][QUOTE="domatron23"]

What rule of logic states that an actual infinite must conform with rules of addition and subtraction etc.

danwallacefan

An actual infinite can not conform to any of those rules due to the nature of itself, mathematically speaking. Subtraction, addition, multiplication and division are all more or less meaningless in relation to an infinite number.

Yeah, just wanted to get in a mathematical perspective somewhere. In relation to your argument I'd actually agree with you to a certain extent, in that I think actual infinities are theoretical concepts which are necessary for describing the world but also can't actually exist due to the obvious limitations of our universe. Unless our universe is closed both time and space are potential infinities since both will continue to grow forever though.

And it seems to me that the rejection of an actual infinite carries with it just as many things that are unintuitive as the acceptance of such a thing. For example, if there does not exist any actual spatial infinite, then there must be a barrier that demarcates the end of space - but then one could ask the obvious question: what the heck is beyond that barrier? Even if the answer is "nothing", empty space is still space, which means that there could not even be empty space in that direction.

It's really little different for time, too - if there is an "absolute zero" in terms of time, then, well, what happened before that point?

GabuEx

AFAIK spacetime solves both those problems? As the universe 'expands' it does not expand into previously existing space but rather creates new space. Since it expands at the speed of light - that is how fast the photons will travel after all - there is no way to get 'outside' of the universe, and hence the notion of something outside of the universe is meaningless. There is no such thing as 'space' outside of it.

As for time, it is tied to space and thus it simply did not exist before Big Bang or whatever created space occurred, and hence the question 'What happened before the Big Bang?' is just as meaningless as 'What is north of the North Pole?' since they contradict the definition of terms used (example borrowed from Hawking).

I know you probably know more about this stuff than me Gabu so please do inform me about any errors in my logic. :P

[QUOTE="ChiliDragon"]The Dungeon Dimensions. Be very frightened.GabuEx

Is that a reference to something that I'm not getting? :P

danwallacefan I'm still not comprehending your response to my criticism of the Hilbert's Hotel problem. How is Hilbert's Hotel contradictory and why is it not a broad logical possibility? And what does it mean for something to be a broad logical possibility as opposed to a strict logical possibility?domatron23

Now "strict logical possibility" merely means that nothing within the definition is contradictory. But "broad logical possibility" means that the thing in question contains no contradictions. 

For instance, without the modal perfection proof, we can only say that God is strictly logically possible because the given definition of God is not contradictory. But that doesn't prove that a God would not be contradictory. It is said to be strictly, but not certainly broadly logically possible. 

Either those should be nines or my maths teachers have made a big mistake somewhere down the line :PFunky_Llama

I fervently deny your assertion.

domatron23

Oops, yes, I did of course mean 0.999... :P

AFAIK spacetime solves both those problems? As the universe 'expands' it does not expand into previously existing space but rather creates new space. Since it expands at the speed of light - that is how fast the photons will travel after all - there is no way to get 'outside' of the universe, and hence the notion of something outside of the universe is meaningless. There is no such thing as 'space' outside of it.

As for time, it is tied to space and thus it simply did not exist before Big Bang or whatever created space occurred, and hence the question 'What happened before the Big Bang?' is just as meaningless as 'What is north of the North Pole?' since they contradict the definition of terms used (example borrowed from Hawking).

I know you probably know more about this stuff than me Gabu so please do inform me about any errors in my logic. :P

inoperativeRS

Eh, neither of those explanations really does it for me. In terms of space, you could ask the question - if you were some theoretical photon at the very edge of the universe, what would you see? I can't think of any answer that would make sense.

Similarly to the notion of the beginning of time - everything that is within cause and effect is within time, such that one can say "X caused Y", implictly saying "X preceded Y". If there was a beginning of time, then whatever caused time to come into existed must have occurred before the beginning of time - but how would anything happen without a period of time within which for it to happen?

Terry Pratchett's Discworld books. :) Basically the Disc is covered by a sky dome and outside that is the universe. The universe is finite and ends, and there is a huge barrier where it does end. Behind that barrier is the Dungeon Dimensions, inhabited by creatures who have absolutely no understanding of reality, but crave it, want it, and desperately try to achieve it. They exist of random body parts assembled at random (the prettier ones look like very ugly versions of demons), and deep deep hatred for all things alive. It's assumed that they are jealous of things that are "real". They try to join reality whenever they can, but because of what they are and how many they are, if they ever succeed it'll be a bit like the ocean joining the passengers on the Titanic, but much worse. And yes, they are Pratchett's version/parody of all Lovecraftian creatures. :)ChiliDragon

Ahh, Terry Pratchett, should've known. :P I love his stuff.

Hilbert's Hotel merely uses infinite set theory and applies it to the real world, and thereby shows that self-contradictions arise once we posit the existence of an actual infinite.danwallacefan

You keep saying that it's contradictory, but what exactly is contradictory about it? The only problem I see that has been raised is the same sort of problem that one will run into if one thinks of most any infinite in finite terms - just like I said as many have done in denying that 0.999... equals 1.

[QUOTE="domatron23"]danwallacefan I'm still not comprehending your response to my criticism of the Hilbert's Hotel problem. How is Hilbert's Hotel contradictory and why is it not a broad logical possibility? And what does it mean for something to be a broad logical possibility as opposed to a strict logical possibility?danwallacefan

But you haven't actually established this yet. This is the very point that we're arguing over and you can't just use the premise that actual infinites are self-contradictory to conclude that they do not exist without begging the question.

Now "strict logical possibility" merely means that nothing within the definition is contradictory. But "broad logical possibility" means that the thing in question contains no contradictions.danwallacefan

Okay those definitions are fine. You still haven't established that the Hilbert's hotel problem produces a contradiction though and so saying that it does or invoking the law of non-contradiction wont work until you demonstrate how it is so.

For instance, without the modal perfection proof, we can only say that God is strictly logically possible because the given definition of God is not contradictory. But that doesn't prove that a God would not be contradictory. It is said to be strictly, but not certainly broadly logically possible. 

danwallacefan

Bah, you and the modal ontological argument. That's yet another thing that you haven't demonstrated to be sound at all.

 

[QUOTE="inoperativeRS"]

AFAIK spacetime solves both those problems? As the universe 'expands' it does not expand into previously existing space but rather creates new space. Since it expands at the speed of light - that is how fast the photons will travel after all - there is no way to get 'outside' of the universe, and hence the notion of something outside of the universe is meaningless. There is no such thing as 'space' outside of it.

As for time, it is tied to space and thus it simply did not exist before Big Bang or whatever created space occurred, and hence the question 'What happened before the Big Bang?' is just as meaningless as 'What is north of the North Pole?' since they contradict the definition of terms used (example borrowed from Hawking).

I know you probably know more about this stuff than me Gabu so please do inform me about any errors in my logic. :P

GabuEx

Eh, neither of those explanations really does it for me. In terms of space, you could ask the question - if you were some theoretical photon at the very edge of the universe, what would you see? I can't think of any answer that would make sense.

Similarly to the notion of the beginning of time - everything that is within cause and effect is within time, such that one can say "X caused Y", implictly saying "X preceded Y". If there was a beginning of time, then whatever caused time to come into existed must have occurred before the beginning of time - but how would anything happen without a period of time within which for it to happen?

But you haven't actually established this yet. domatron23

How so? What happens when you take half of the guests out? You take an infinite amount of guests out, but there is still an infinite amount of guests, which means that there no fewer people in the hotel. Why am I able to fill the rooms of the hotel even though its at half-capacity? These are self-contradictions, and disprove broad logical possibility. Your only response is that these are simply how infinite sets behave, they dont behave like finite sets. But, of course, this is irrelavent because all things which exist in reality must obey the laws of logic. 

This is the very point that we're arguing over and you can't just use the premise that actual infinites are self-contradictory to conclude that they do not exist without begging the question.domatron23

How is it question begging to assume that the 3 laws of logic apply to actual infinites? Is the negation not equally question begging? But, of course, you're hung up on this point and it only proves strict logical possibility. 

Okay those definitions are fine. You still haven't established that the Hilbert's hotel problem produces a contradiction though and so saying that it does or invoking the law of non-contradiction wont work until you demonstrate how it is so. domatron23

Simple: I can fill up the entire hotel even if it only has 50% occupancy just by shifting people around. That's self-contradictory. 

Bah, you and the modal ontological argument. That's yet another thing that you haven't demonstrated to be sound at all.

domatron23

Well, AFAIK for space to exist (according to physics) there needs to exist distances between objects, so if there are no objects there is no space. Now if someone somehow hypothetically managed to get 'outside' of the universe he would literally create space since he would be an object and thus distances between objects could be measured. The photon at the edge would probably see nothing (literally) but would be the edge of the universe at that point, there is no way to get past it. It does sound like an answer that just avoids the question since it seems like there is some kind of 'space' outside of the universe and physics simply doesn't define as such. I find it quite weird in a sense but thinking about spacetime as the surface area of a balloon helps somewhat - it isn't really expanding as much as it's stretching out, it certainly isn't expanding into something outside of it (the balloon is here helpful only if you think of the universe as a two dimensional area).

inoperativeRS

Well, if the extent of space is defined by the objects occupying it, then it seems to me that that would basically say that there is an infinite amount of theoretically available space - meaning that an infinite really does exist.

space you can't have time (which is quite logical really) so 'the time before space was created' is a paradox. I think the general concensus among scientists is that we simply don't know enough about the Big Bang to say with certainty whether or not it was the real beginning of time, and what could possible have come before it (any such time would have had to be accompanied by space though).

inoperativeRS

Right, I understand that for there to be time you need something to move through time, but that still doesn't answer the question of how something could have caused time to exist if it itself does not exist within time.

How so? What happens when you take half of the guests out? You take an infinite amount of guests out, but there is still an infinite amount of guests, which means that there no fewer people in the hotel. Why am I able to fill the rooms of the hotel even though its at half-capacity? These are self-contradictions, and disprove broad logical possibility. Your only response is that these are simply how infinite sets behave, they dont behave like finite sets. But, of course, this is irrelavent because all things which exist in reality must obey the laws of logic.danwallacefan

Again, you're thinking of an infinte in finite terms. If you have an infinite number of rooms and an infinite number of occupants, then yes, you can take the hotel from half to full capacity without adding anyone new on account of the fact that you have an infinite amount of both. This doesn't work if you have a finite number of each because eventually you'll run out of rooms or people, but you will never run out of either with an infinite number of them. The math on this subject is very rigorous, and there is no contradiction at all in what you're saying - it is unintuitive, perhaps, but certainly not contradictory.

[QUOTE="inoperativeRS"]

Well, AFAIK for space to exist (according to physics) there needs to exist distances between objects, so if there are no objects there is no space. Now if someone somehow hypothetically managed to get 'outside' of the universe he would literally create space since he would be an object and thus distances between objects could be measured. The photon at the edge would probably see nothing (literally) but would be the edge of the universe at that point, there is no way to get past it. It does sound like an answer that just avoids the question since it seems like there is some kind of 'space' outside of the universe and physics simply doesn't define as such. I find it quite weird in a sense but thinking about spacetime as the surface area of a balloon helps somewhat - it isn't really expanding as much as it's stretching out, it certainly isn't expanding into something outside of it (the balloon is here helpful only if you think of the universe as a two dimensional area).

GabuEx

Well, if the extent of space is defined by the objects occupying it, then it seems to me that that would basically say that there is an infinite amount of theoretically available space - meaning that an infinite really does exist.

Well it sounds like the two of you are not doing physics, but metaphysics. The claim "space exists if there is distance between two objects" doesn't sound scientific at all, but strictly metaphysical. Perhaps if we reduced it to our measurements of spacetime, it would be more interesting and scientific. Moving further, this would not prove the existence of an actual infinite because there would first have to be an infinite amount of objects, or two objects with an infinite amount of space between them. But there isn't really any way of proving this. 

space you can't have time (which is quite logical really) so 'the time before space was created' is a paradox. I think the general concensus among scientists is that we simply don't know enough about the Big Bang to say with certainty whether or not it was the real beginning of time, and what could possible have come before it (any such time would have had to be accompanied by space though).

inoperativeRS

I would have to disagree with this spacetime realism you're giving us on a few grounds

1: As far as science is concerned, time only deals with our measurements of change. What time is and how it works really can't be answered by science

2: This sort of spacetime realism you're giving me, of course, was disproven about 40 years ago with experiments using quantum mechanics. These experiments found that there is in fact a causal relation between two waves/particles of light moving in the opposite direction away from eachother. On einstinian relativity, This is impossible for the simple reason that change cannot happen faster than the speed of light. 

All this suggests that the quantum interpretation, not the Einstinian interpretation, of gravity is correct.

So at best we can say that there was no physicality, no change in physical objects, before space existed. I will have to contest your notion that no time, no change whatsoever, could happen before space existed. 

Again, you're thinking of an infinte in finite terms. If you have an infinite number of rooms and an infinite number of occupants, then yes, you can take the hotel from half to full capacity without adding anyone new on account of the fact that you have an infinite amount of both. This doesn't work if you have a finite number of each because eventually you'll run out of rooms or people, but you will never run out of either with an infinite number of them. The math on this subject is very rigorous, and there is no contradiction at all in what you're saying - it is unintuitive, perhaps, but certainly not contradictory.

GabuEx

This is a rather illuminating admission you have just made GabuEx. This statement "you can take the hotel from half to full capacity without adding anyone new on account of the fact that you have an infinite amount of both." really all boils down to this statement "if you have an infinite set, you can add to it without adding to it". This is flatly self-contradictory, and thus disproves broad logical possibility.

danwallacefan

Ah, but that's the thing: you haven't increased the number of occupants at all when you take it from 50% capacity to 100% capacity. You don't need to add a single thing in order to achieve that goal. It is mathematically provable that the sets {2, 4, 6, 8, ...} and {1, 2, 3, 4, ...} are the same size. Thus, if you take each number to be a room in the hotel and if you take the number's presence in the set to indicate that the corresponding room is filled, then the fact that those two sets are the same size indicates that you can have the hotel at both 50% capacity and 100% capacity without changing the number of occupants in the hotel.

Again, this all comes back to what I said before: you are attempting to force logic surrounding finite sets onto infinite sets, and it just doesn't work. Of course you cannot take a hotel from 50% capacity to 100% capacity without adding any new occupants - if you have a finite number of rooms and occupants. However, the same logic does not carry over one bit when dealing with an infinite number of rooms and occupants.

Weird? Sure. Unintuitive? You bet. Contradictory? Nope.

then the fact that those two sets are the same size indicates that you can have the hotel at both 50% capacity and 100% capacity without changing the number of occupants in the hotel.GabuEx

This, of course, is flatly contradictory. 

Again, this all comes back to what I said before: you are attempting to force logic surrounding finite sets onto infinite sets, and it just doesn't work. Of course you cannot take a hotel from 50% capacity to 100% capacity without adding any new occupants - if you have a finite number of rooms and occupants. However, the same logic does not carry over one bit when dealing with an infinite number of rooms and occupants.

Weird? Sure. Unintuitive? You bet. Contradictory? Nope.

GabuEx

Logic, rationality, and argumentation collapse if we seriously think that there is some aspect of reality which is not subject to the basic 3 rules of logic. 

[QUOTE="GabuEx"]

[QUOTE="inoperativeRS"]

Well, AFAIK for space to exist (according to physics) there needs to exist distances between objects, so if there are no objects there is no space. Now if someone somehow hypothetically managed to get 'outside' of the universe he would literally create space since he would be an object and thus distances between objects could be measured. The photon at the edge would probably see nothing (literally) but would be the edge of the universe at that point, there is no way to get past it. It does sound like an answer that just avoids the question since it seems like there is some kind of 'space' outside of the universe and physics simply doesn't define as such. I find it quite weird in a sense but thinking about spacetime as the surface area of a balloon helps somewhat - it isn't really expanding as much as it's stretching out, it certainly isn't expanding into something outside of it (the balloon is here helpful only if you think of the universe as a two dimensional area).

danwallacefan

Well, if the extent of space is defined by the objects occupying it, then it seems to me that that would basically say that there is an infinite amount of theoretically available space - meaning that an infinite really does exist.

Well it sounds like the two of you are not doing physics, but metaphysics. The claim "space exists if there is distance between two objects" doesn't sound scientific at all, but strictly metaphysical. Perhaps if we reduced it to our measurements of spacetime, it would be more interesting and scientific. Moving further, this would not prove the existence of an actual infinite because there would first have to be an infinite amount of objects, or two objects with an infinite amount of space between them. But there isn't really any way of proving this. 

space you can't have time (which is quite logical really) so 'the time before space was created' is a paradox. I think the general concensus among scientists is that we simply don't know enough about the Big Bang to say with certainty whether or not it was the real beginning of time, and what could possible have come before it (any such time would have had to be accompanied by space though).

inoperativeRS

I would have to disagree with this spacetime realism you're giving us on a few grounds

1: As far as science is concerned, time only deals with our measurements of change. What time is and how it works really can't be answered by science

2: This sort of spacetime realism you're giving me, of course, was disproven about 40 years ago with experiments using quantum mechanics. These experiments found that there is in fact a causal relation between two waves/particles of light moving in the opposite direction away from eachother. On einstinian relativity, This is impossible for the simple reason that change cannot happen faster than the speed of light. 

All this suggests that the quantum interpretation, not the Einstinian interpretation, of gravity is correct.

So at best we can say that there was no physicality, no change in physical objects, before space existed. I will have to contest your notion that no time, no change whatsoever, could happen before space existed. 

The question of quantum and relativistic gravity is a different beast altogether. What my space argument boils down to is that there really is no space (as we think of it) per se outside of our (finite) universe and the objects and measurable distance comment was more of an attempt to give some kind of explanation for it, however inadequate it might be. The fact that we need measurable distances to be able to know that space exists is a scientific definition though AFAIK. The question of whether or not we need to be able to measure space for it to exist is metaphysical I guess.

The comments about time and quantum mechanics is just a way to say that 'change' can happen outside of our three traditional dimensions I think? Then I'd have to agree but the stuff about quantum and relativistic gravity is fairly irrelevant except for the fact that quantum interpretations tend to be correct for small systems where relativity fails.  Relativity and spacetime can't be trusted in the extreme conditions that existed around the time of the big bang so my theory does fail in that sense. For change to happen there needs to be something or somewhere for it to happen on/in though IMO, which would indicate the existance of - if not traditional, three dimensional space - at least of some kind of... place (?) if time was to exist. Not terribly scientific again. :P

This, of course, is flatly contradictory.danwallacefan

Only if you're asserting such a thing of finite sets. It is mathematically provable that any two countably infinite sets are necessarily the same size. Which is exactly as one would expect.

Suppose you have a set X, and suppose you are adding a single element to that set. The percentage change in the size of X by this addition is given by 1 / |X|. It is thus trivial to see that, as |X| goes to infinity, the percentage change in the size of X goes to 0 - thus, if a set is infinite, then it makes perfect sense that adding a new item would not change its size - if it did change its size, then its size would be finite.

The question that must be asked is "why can't a finite hotel increase in percent capacity without adding new people?" The answer is simple: you run out of people.

Suppose you have two sets: P = {1, 2} representing the people, and R = {1, 2, 3, 4} representing the rooms. To assign people to rooms is equivalent to defining a function f : P -> R such that f is an injection. By definition of injection, this means that every element of R has exactly one element of P mapped to it (in other words, that every room has at most one occupant, and that every occupant is in only one room). Once you have defined f(1) and f(2), then you are done - it is impossible to fill more than two rooms without breaking the requirement that f must be an injection. Percent capacity between P and R is thus fixed at 50%.

This is, however, not the case if P and R are both countably infinite. If that is the case, then you can never run out of either rooms or people. As a result, things become drastically different. Percent capacity is defined for finite sets P and R simply by dividing |P| by |R|, but for countably infinite sets P and R, such a division no longer makes sense - percent capacity is instead simply defined by the definition of the function f. Since you will never run out of rooms into which people can be placed, you can leave as many rooms empty as you like and still satisfactorily define an injection that places every single person in a room.

As a result, the concept of "capacity" when dealing with infinite sets has, really, a very different and much less useful meaning than when dealing with finite sets. With finite sets, it denotes the maximum number of people that can be accomodated by the number of rooms, but with infinite sets, the concept of "maximum" is really quite meaningless (provided that the levels of infinity in both sets are the same).

As I've already said, you are forcing onto infinite sets logical reasoning that includes in its premises the assumption that everything under consideration is finite and with a meaningful maximum capacity defined by the size of the containing set. That reasoning cannot be used to produce any meaningful results at all about infinite sets for the fundamental reason that its core assumptions are false in such a situation.

Again, I repeat: there is no contradiction here whatsoever. Infinite sets are totally different than finite sets. All of the above is completely mathematically provable and rigorous.

Its not as if the 3 axioms of logic only apply to finite sets. If you had even a basic rudimentary understanding of them, you would realize that they apply to ALL REALITY. This is why infinite sets are not broadly logically possible.

Logic, rationality, and argumentation collapse if we seriously think that there is some aspect of reality which is not subject to the basic 3 rules of logic.danwallacefan

Enumerate, if you will, the three axioms of logic, and show precisely how they have been violated.

Enumerate, if you will, the three axioms of logic, and show precisely how they have been violated.

GabuEx

You said that infinite sets are exempt from this. 

This disproves broad logical possibility. 

they all boil down to the law of non-contradiction, namely A=/= not A. This principle governs reality.

You said that infinite sets are exempt from this.

This disproves broad logical possibility.

danwallacefan

Well, first, the law of non-contradiction says that A and not A is necessarily false, not that A and not A are not equal.

I have to question whether or not you've read any of my posts, however. I have never once said that the law of non-contradiction does not apply to infinite sets. What I said is that there is no contradiction in infinite sets.

The idea that one ought not to be able to add a new occupant to Hilbert's hotel really comes from the intuitive thought that, if every room in the hotel is full, then there is no available room for a new occupant. This is clearly the case with finite sets - if you have sets P = {1, 2, 3} and R = {1, 2, 3}, with function f : P -> R defined as f(x) = x, then if one adds the element 0 to P and then re-defines f as f(x) = x + 1 (analogous to what is done in Hilbert's thought experiment), then it is clear that f(3) = 4 does not properly map to any element in R.

However, this is clearly not the case if we instead define P = {1, 2, 3, ...} and R = {1, 2, 3, ...}, again with f: P -> R defined as f(x) = x. If, now, one adds the element 0 to P and re-defines f as f(x) = x + 1, then the above is no longer true - there exists no x in P such that f(x) is not in R. Thus, the intuitive reasoning that one ought not to be able to add an element to P falls apart when both P and R are countably infinte.

Now, it can be said that if the operation of moving of occupant x to room x + 1 takes a nonzero length of time, then it is the case that this process will continue for an infinite length of time - but that is certainly not a contradiction either.

The idea that one ought not to be able to change the percent capacity from 50% to 100% without changing the number of occupants is similar, and it comes from the similarly intuitive thought that, since the number of occupants and rooms is fixed and unchanging, then the percent capacity must also be fixed and unchanging, since the percent capacity is simply the number of occupants divided by the number of rooms. This, again, is clearly the case with finite sets - if you have sets P = {1, 2, 3} and R = {1, 2, 3, 4, 5, 6}, with a given injection f : P -> R, then it is clearly impossible to have more than three ordered pairs in f - you can only have as many ordered pairs in an function as there are elements in its domain. Whether f = {(1, 1), (2, 2), (3, 3)} or f = {(1, 6), (2, 5), (3, 4)} is immaterial - in every case you will have three and only three empty rooms, because you have a fixed number of occupants to allocate.

Again, however, this is not the case if we have P = {1, 2, 3, ...} and R = {1, 2, 3, ...}. If we define f : P -> R as f(x) = 2x, then clearly, every odd room has no occupant, as for all y in {1, 3, 5, ...}, there is no x such that f(x) = y. If, however, we instead define f as f(x) = x, then clearly every room has an occupant, as there is no y in R such that f(x) =/= y.

As above, if the operation of moving the occupant in room x to room x / 2 takes a nonzero length of time, then it is the case that this process will continue for an infinite length of time - but, again, that is not a contradiction.

To repeat, this is not to say that the law of non-contradiction does not apply here; it is to say that there is no contradiction to be found. Any attempts to find contradictions here are fundamentally attempts to apply to infinite sets logic that is applicable only to finite sets. It is the exact same sort of reasoning that leads one to say that 0.999... cannot be equal to 1 on account of the fact that there is a rightmost 9 "at infinity", when that is clearly not the case.

All you said, GabuEx, is that somehow, the apparent contradictions are not contradictions. You never gave any argument for this premise other than "well that's just how infinite sets behave, they dont behave like finite sets".

That's not quite an argument, certainly not an argument for broad logical possibility. Perhaps for, once again, strict logical possibility, but not broad logical possibility. Hilbert's hotel is "counterintuitive" not just because we are applying finite rules to infinite sets, but because our intuitions are rooted in logic. 

also, one last point you made, can I see some defense of your premise that .999...=1? You stated this is "clearly the case". 

All you said, GabuEx, is that somehow, the apparent contradictions are not contradictions. You never gave any argument for this premise other than "well that's just how infinite sets behave, they dont behave like finite sets".

That's not quite an argument, certainly not an argument for broad logical possibility. Perhaps for, once again, strict logical possibility, but not broad logical possibility. Hilbert's hotel is "counterintuitive" not just because we are applying finite rules to infinite sets, but because our intuitions are rooted in logic.

danwallacefan

Well, let's recap:

- I have shown how you can accurately model the situation of assigning people to rooms through the use of two sets and a function between them.

- I have shown how the intuitive statements are true for finite sets about what is and is not the case about a hotel and its occupants.

- I have shown how they are not true for countably infinite sets.

So: if a hotel and its occupants can be accurately modeled using mathematical objects, if this math correctly shows that what you say is true in the case of a finite number of occupants and rooms, and if this math then also shows that what you say is false in the case of a countably infinite number of occupants and rooms... then what, exactly, do I still need to show?

Would you like me to build you a hotel with an infinite number of rooms and then gather up an infinite number of people and then show you that this is indeed the truth?

also, one last point you made, can I see some defense of your premise that .999...=1? You stated this is "clearly the case".

danwallacefan

1/3 = 0.333...

3 * (1/3) = 3 * 0.333...

1 = 0.999...

1/3 = 0.333...

3 * (1/3) = 3 * 0.333...

1 = 0.999...

GabuEx

1/3 = 0.333...

3 * (1/3) = 3 * 0.333...

1 = 0.999...

GabuEx

That makes sense. Though wouldn't 0.333... be asymptotic to 1/3 but not actually equal to it? Same thing with 0.999... wouldn't it just approach a point that is infintesimally close and yet still not equal to 1.

Not sure on this one just putting it out there.

That makes sense. Though wouldn't 0.333... be asymptotic to 1/3 but not actually equal to it? Same thing with 0.999... wouldn't it just approach a point that is infintesimally close and yet still not equal to 1.

Not sure on this one just putting it out there.

domatron23

That would be the case if you were building it step by step by adding on one 3 or one 9 at a time, but the notation "0.333..." indicates that every single digit to the right of the decimal place is already in place.  Like I said before, people intuitively feel that 0.999... cannot possibly be equal to 1 because, as their logic goes, there is a rightmost 9 "at infinity"... but there is no such thing.

...Anyway, this was not exactly the driving force behind what I was trying to say. :P  Although I suppose that understanding the reason why 0.999... = 1 requires the same understanding as what is needed to see that there is no contradiction in Hilbert's hotel.

[QUOTE="domatron23"]

But you haven't actually established this yet. danwallacefan

How so? What happens when you take half of the guests out? You take an infinite amount of guests out, but there is still an infinite amount of guests, which means that there no fewer people in the hotel. Why am I able to fill the rooms of the hotel even though its at half-capacity? These are self-contradictions, and disprove broad logical possibility. Your only response is that these are simply how infinite sets behave, they dont behave like finite sets. But, of course, this is irrelavent because all things which exist in reality must obey the laws of logic.

I agree that all things which exist in reality must obey the laws of logic I just don't agree that Hilbert's hotel demonstrates any such disobedience in regards to actual infinites. In this case your use of terms like "half" and "fewer" are problematic when applied to an actual infinite.

You expect that by removing half of an infinite number of guests that you will then have fewer than you did before. This uses the implicit premise that halving the members of a set will reduce the number of members in that set. Now of course this premise is intuitive because it applies to finite sets that we are used to working with but like I established back on page 1 arithmetic functions do not alter the value of infinity and so when you halve the number of guests in Hilbert's hotel you do not reduce their number. In this way it is not contradictory to remove half of the guests and yet still have the same number left in the hotel. Thus you have not established a contradiction, you have not exposed a violation of the law of non-contradiction and you have not disproved the broad logical possibility of actual infinites.

The same thing goes when you try to say that it is contradictory for the hotel to be at half capacity and yet still be full. You're using an implicit premise that half the capacity of a hotel with infinite rooms is less than the full capacity of a hotel with infinite rooms which you know by now is incorrect.

[QUOTE="domatron23"]This is the very point that we're arguing over and you can't just use the premise that actual infinites are self-contradictory to conclude that they do not exist without begging the question.danwallacefan

How is it question begging to assume that the 3 laws of logic apply to actual infinites? Is the negation not equally question begging? But, of course, you're hung up on this point and it only proves strict logical possibility.

Okay I probably could have worded that a bit better. I didn't mean that you were begging the question by suggesting that the logical laws apply to actual infinites I just meant that you were begging the question by suggesting that actual infinites are contradictory because they are contradictory.

In hindsight though you weren't really making an argument to that effect you were just explaining your reasoning. My bad.

[QUOTE="domatron23"]

Okay those definitions are fine. You still haven't established that the Hilbert's hotel problem produces a contradiction though and so saying that it does or invoking the law of non-contradiction wont work until you demonstrate how it is so. danwallacefan

Simple: I can fill up the entire hotel even if it only has 50% occupancy just by shifting people around. That's self-contradictory.

It would be if 50% of an infinite number of guests was less than 100% of an infinite amount of guests but as I explained above it isn't. In the case of infinity 50% and 100% and 25% etc, although they are different terms, all express the same value. That is to say that although 50% of infinity has a different sense to 100% of infinity it has the same reference.*

And that means that filling up Hilbert's Hotel with only 50% occupancy is not self contradictory because "full" and "50% full" are terms of equivalent meaning as far as infinity is concerned.

[QUOTE="domatron23"]

Bah, you and the modal ontological argument. That's yet another thing that you haven't demonstrated to be sound at all.

danwallacefan

I didn't prevent you from continuing to argue over Maydole's version of the ontological argument (which was the argument that brought up the problem of existence as a predicate) you abandoned it for the Liebnizian argument for divine possibility. By all means go back to the CU and resurrect the ontological argument thread if you want to demonstrate why your argument is sound and how my criticisms horribly misunderstood the issue.

As for this thread lets keep it on the topic of actual infinites.

[QUOTE="domatron23"]

That makes sense. Though wouldn't 0.333... be asymptotic to 1/3 but not actually equal to it? Same thing with 0.999... wouldn't it just approach a point that is infintesimally close and yet still not equal to 1.

Not sure on this one just putting it out there.

GabuEx

That would be the case if you were building it step by step by adding on one 3 or one 9 at a time, but the notation "0.333..." indicates that every single digit to the right of the decimal place is already in place.  Like I said before, people intuitively feel that 0.999... cannot possibly be equal to 1 because, as their logic goes, there is a rightmost 9 "at infinity"... but there is no such thing.

...Anyway, this was not exactly the driving force behind what I was trying to say. :P  Although I suppose that understanding the reason why 0.999... = 1 requires the same understanding as what is needed to see that there is no contradiction in Hilbert's hotel.

I get what you're saying and I'm not falling into the trap of assuming a right-most 9 but it still seems like something is amiss. Nevertheless I think I've just had a brain wave which relates to the discussion we're having.

My mistake is to say that 0.999... reaches a point infinitesimally close but not equal to 1. I assumed that an infinitesimal difference between two values constituted an actual difference to the effect that its presence made the two values inequal. That's clearly not the case though because an infinitesimal amount, exactly like an infinite amount, does not behave in accordance with normal functions of arithmetic.

For example let x represent an infinitesimally small value and y represent a finite number. x can be no smaller than 2x in the same way that infinity can be no smaller than infinity multiplied by 2. That means that an equation like x+y would yeild a result equal to 2x+y and to 3x+y etc etc all of which means that any infinitesimal amount does not alter the value of a finite number. Therefore although there is an infinitesimal difference between 0.999... and 1 they both represent equal values.

Like you said the same understanding that applies here also applies to Hilbert's hotel. Both cases involve looking past differences that distinguish an inequality in terms of finites but not in terms of infinites.

[QUOTE="GabuEx"]

1/3 = 0.333...

3 * (1/3) = 3 * 0.333...

1 = 0.999...

Funky_Llama

Idk it doesnt feel "ok". :?

[QUOTE="Funky_Llama"][QUOTE="GabuEx"]

1/3 = 0.333...

3 * (1/3) = 3 * 0.333...

1 = 0.999...

Teenaged

Idk it doesnt feel "ok". :?

Watch this then.

I know what you mean about it feeling weird but it makes sense when you think it over.

[QUOTE="Funky_Llama"][QUOTE="GabuEx"]

1/3 = 0.333...

3 * (1/3) = 3 * 0.333...

1 = 0.999...

Teenaged

Idk it doesnt feel "ok". :?

let x=0.999...

10x=9.999...

9x=10x-x=9.999...-0.999...=9

x=9x/9=9/9=1

therefore 0.999...=1

Watch this then.

I know what you mean about it feeling weird but it makes sense when you think it over.

domatron23

Yeah, it is rather counterintuitive. What about...

let x=0.999...

10x=9.999...

9x=10x-x=9.999...-0.999...=9

x=9x/9=9/9=1

therefore 0.999...=1

Funky_Llama

I know that it is mathematically correct as shown in both Funky's example and the videos explanation (and I know that thats why I hate math :P), but isnt it a paradox?

I know that it is mathematically correct as shown in both Funky's example and the videos explanation (and I know that thats why I hate math :P), but isnt it a paradox?

Teenaged

I know that it is mathematically correct as shown in both Funky's example and the videos explanation (and I know that thats why I hate math :P), but isnt it a paradox?

Teenaged

Well, all you really have to realize is that any real small value you can come up with always will be larger than the difference between 0.999... and 1. Since the difference is smaller than any real number it follows that they essentially have to be the same number. Or something like that. :P

isnt it a paradox? Teenaged

No. It's just two ways of writing the same number. Same as how 1/3 = 0.333...

[QUOTE="Teenaged"][QUOTE="Funky_Llama"][QUOTE="GabuEx"]

1/3 = 0.333...

3 * (1/3) = 3 * 0.333...

1 = 0.999...

domatron23

Idk it doesnt feel "ok". :?

Watch this then.

I know what you mean about it feeling weird but it makes sense when you think it over.

but his second proof sound airtight

I'm still confused as to how this "same reasoning" is responsible for me having doubts about the broad logical possibility of actual infinites. 

I'm still confused as to how this "same reasoning" is responsible for me having doubts about the broad logical possibility of actual infinites. 

danwallacefan

Because if you can understand that there is no rightmost 9 at infinity in 0.999... that prevents it from being equal to 1, then you can also understand that there is no rightmost person at infinity that has no room to occupy when you add another person to Hilbert's hotel.  The logic in both cases is identical.

If you want to show what is wrong with my assertion that there is no contradiction in Hilbert's hotel, then you can show one of two things: how my mathematical model does not accurately depict the situation in Hilbert's hotel, or how the math used to reach the conclusion is faulty.  Either one of the two will do.

I'm still confused as to how this "same reasoning" is responsible for me having doubts about the broad logical possibility of actual infinites. 

danwallacefan

You think that 50% occupancy of Hilbert's hotel =/= 100% occupancy for the same reasons that you think 0.999... =/= 1. Basically both problems seem contradictory because you fail to recognize why they express equal terms.

You think that 50% occupancy of Hilbert's hotel =/= 100% occupancy

domatron23

Well, to be clear, the situation of 50% occupancy is not exactly the same as 100% occupancy, the difference of course being the rooms which are and are not occupied.  However, the key distinction is that, when you have countably infinite rooms and people, then the question of percent occupancy is really just a question of configuration, not a question of the absolute ratio of people to rooms.

[QUOTE="domatron23"]

You think that 50% occupancy of Hilbert's hotel =/= 100% occupancy

GabuEx

Well, to be clear, the situation of 50% occupancy is not exactly the same as 100% occupancy, the difference of course being the rooms which are and are not occupied.  However, the key distinction is that, when you have countably infinite rooms and people, then the question of percent occupancy is really just a question of configuration, not a question of the absolute ratio of people to rooms.

also, why say that there's no person at room infinity in Hilbert's hotel since the switching goes on forever? Why say that it is room switch followed by room switch (etc.)? Why not say that all the room switches take place simultaneously?

can you unpack that statement "just a question of configuration"?

danwallacefan

If you put people in every second room, then you're at 50% capacity; if you put people in every room, then you're at 100% capacity.  So it's really just a function of how you allocate people to rooms.

also, why say that there's no person at room infinity in Hilbert's hotel since the switching goes on forever? Why say that it is room switch followed by room switch (etc.)? Why not say that all the room switches take place simultaneously?

danwallacefan

There is no "room infinity".  That's exactly what I'm talking about - the same logic that leads people to feel that there's a rightmost 9 "at infinity" in 0.999... is what leads people to feel that there must be a "room infinity" in Hilbert's hotel whose occupant cannot find a new room.  But this is not the case - the rooms go on forever and ever.

The reason why it's put forth as a room switch followed by a room switch is just for the ease of imagining what's going on.  If they all took place simultaneously, which they could, then absolutely nothing changes in terms of the resulting configuration - you still have all the same people in all the same rooms at the end.

If you put people in every second room, then you're at 50% capacity; if you put people in every room, then you're at 100% capacity.  So it's really just a function of how you allocate people to rooms.GabuEx

Eh, so is there a difference between 50% and 100% occupancy? 

There is no "room infinity".  That's exactly what I'm talking about - the same logic that leads people to feel that there's a rightmost 9 "at infinity" in 0.999... is what leads people to feel that there must be a "room infinity" in Hilbert's hotel whose occupant cannot find a new room.  But this is not the case - the rooms go on forever and ever.

The reason why it's put forth as a room switch followed by a room switch is just for the ease of imagining what's going on.  If they all took place simultaneously, which they could, then absolutely nothing changes in terms of the resulting configuration - you still have all the same people in all the same rooms at the end.

GabuEx

:lol:Oh wait, I'm sorry, you can't subtract from, add to, multiply or divide infinite sets. 

Gabu, this is why mathematicians acknowledge that infinites are just concepts, not things that can exist in reality. 

Eh, so is there a difference between 50% and 100% occupancy? 

danwallacefan

Consider the following sets. [1, 2, 3, 4...] [1, 3, 5, 7...]

Each contains an infinite number of members but the second one has no even numbers which means that it also lacks an infinite amount of members. They represent the same amount but not the same configuration.

Same thing with Hilberts hotel. 100% occupancy means that all rooms are filled, 50% occupancy means that there are an infinite number of rooms which are not full but in both cases there is the same amount of guests in the hotel.

Whether or not there is a difference depends on whether you're talking about the guests or the rooms.

Eh, so is there a difference between 50% and 100% occupancy?

danwallacefan

Only with respect to which rooms are occupied and which aren't.

well sure there's no "room infinity", but how is that a problem if all the room switches take place simultaneously? Furthermore, how is there no change when the occupants switch up rooms simultaneously? and what the heck do you mean by "the same people in the same rooms"? You have an extra person in the Hotel!

danwallacefan

I mean the same people in the same rooms as what you would have if each room change happened one at a time rather than all at once.  If you put person 0 into room 1, then person 1 into room 2, et cetera; or if you simultaneously move every person in room n to room n + 1 and then put person 0 into room 1, the end result is the same - the same people are in the same rooms in the end.

:lol:Oh wait, I'm sorry, you can't subtract from, add to, multiply or divide infinite sets.

danwallacefan

Oh, you can.  It just doesn't change the cardinality of the set to do so when the set is infinite.  Which even intuitively makes sense - if you could make a set larger by adding an item to it, then its size must have been finite.

Gabu, this is why mathematicians acknowledge that infinites are just concepts, not things that can exist in reality. 

danwallacefan

Appeal to authority, and an unspecified authority to boot.  You have not presented one single iota of evidence against what I have provided.  I repeat: you can either show why a hotel's room assignments cannot be accurately modeled with two sets and a function between them, or you can show why the math that I employed was faulty.  Either will prove your case that I have not adequately proven the possibility of the existence of an actual infinite.  So, which will it be?

Personally when it comes to the origin of man im fine, then the origin of life i say well here are possibilities but we cant prove them...then into the origin of the universe...well heres my idea. 

People are too stupid to guess the beginning and get it right. like evolution our crave for the knowledge on the origin of the universe, must come over time, and is like the knowledge of "why am i here" well ur here to go and mate my boy and make life be even more persistent.

also the big bang seems like a sciency version of the universe being made by the apparent god.  

[QUOTE="danwallacefan"]

Eh, so is there a difference between 50% and 100% occupancy?

GabuEx

Only with respect to which rooms are occupied and which aren't.

So there's no difference in how many guests there are? 

I mean the same people in the same rooms as what you would have if each room change happened one at a time rather than all at once.  If you put person 0 into room 1, then person 1 into room 2, et cetera; or if you simultaneously move every person in room n to room n + 1 and then put person 0 into room 1, the end result is the same - the same people are in the same rooms in the end.GabuEx

Eh, who said anything about "room 0"? If there is no room zero, then there's another person in the hotel. 

[QUOTE="danwallacefan"]

:lol:Oh wait, I'm sorry, you can't subtract from, add to, multiply or divide infinite sets.

GabuEx

Oh, you can.  It just doesn't change the cardinality of the set to do so when the set is infinite.  Which even intuitively makes sense - if you could make a set larger by adding an item to it, then its size must have been finite.

But this seems to defeat modus ponens (which is contradictory). If you add something to some given set, then that set becomes larger. 

[QUOTE="danwallacefan"]

Gabu, this is why mathematicians acknowledge that infinites are just concepts, not things that can exist in reality. 

gabuex

Appeal to authority, and an unspecified authority to boot.

Learn what appeals to authority are.

You have not presented one single iota of evidence against what I have provided.  I repeat: you can either show why a hotel's room assignments cannot be accurately modeled with two sets and a function between them, or you can show why the math that I employed was faulty.  Either will prove your case that I have not adequately proven the possibility of the existence of an actual infinite.  So, which will it be?

GabuEx

But this is self-contradictory, for it doesn't quite appreciate the definition of "addition"

[QUOTE="danwallacefan"]

Eh, so is there a difference between 50% and 100% occupancy? 

domatron23

Consider the following sets. [1, 2, 3, 4...] [1, 3, 5, 7...]

Each contains an infinite number of members but the second one has no even numbers which means that it also lacks an infinite amount of members. They represent the same amount but not the same configuration.

Same thing with Hilberts hotel. 100% occupancy means that all rooms are filled, 50% occupancy means that there are an infinite number of rooms which are not full but in both cases there is the same amount of guests in the hotel.

Whether or not there is a difference depends on whether you're talking about the guests or the rooms.

So there's no difference in how many guests there are? 

danwallacefan

If you don't add or remove any in the process, that is correct.

Eh, who said anything about "room 0"? If there is no room zero, then there's another person in the hotel.

danwallacefan

...You're not getting what I'm saying.

I'm saying that you have two possible ways of adding a new person (call him person 0) to the hotel:

1. Move person 0 to room 1.  Move person 1 to room 2. Move person 2 to room 3.  Et cetera.

2. Move person n to room n + 1, all at once.

The end result in both cases is exactly the same.

But this seems to defeat modus ponens (which is contradictory). If you add something to some given set, then that set becomes larger.

danwallacefan

Only if the set is not infinite, as I have said again and again.  You are applying to the infinite logic applicable only to the finite and then trying to wring a contradiction out of it.

And "defeat(s) modus ponens"?  I mean, come on.  First the idea of "violating the law of non-contradiction" and now this, when the real question is nothing more than whether or not a contradiction exists.  I can assure you that I'm well-versed enough in theoretical logic that I can tell when someone is bluffing their case through unnecessarily technical language.

Learn what appeals to authority are.

danwallacefan

I think you could stand to do so first.  An appeal to authority looks like this:

1. Person A is (claimed to be) an authority on subject S.

2. Person A makes claim C about subject S.

3. Therefore, C is true.

Your argument looked like this:

1. Mathematicians are (claimed to be) an authority on infinites.

2. Mathematicians are said to have made the claim that infinites are just concepts, not things that can exist in reality. 

3. Therefore, infinites are just concepts, not things that can exist in reality.

That was a complete and utter textbook appeal to authority.  That you would deny it is baffling.

You admitted that when you add to an infinite set, the set doesn't become larger.

But this is self-contradictory, for it doesn't quite appreciate the definition of "addition"

danwallacefan

You can make the claim that this is contradictory until the cows come home, but that doesn't make it true.  There is absolutely nothing in the definition of addition that says that adding a new item to an infinite set makes the set larger. If you add the contents of set A to set B, all this says is that the contents of A will then be present in set B.  This says nothing about the resulting size of the new set B.

Proving this is simple.  First, the finite case: Let B = {1, 2, 3}.  Then, let B' = B U {0} = {0, 1, 2, 3}.  Now, how do we know that B is not the same size as B'?  Well, let's define a function f : B -> B'.  We can just add the elements in ordinal order: first we add (1, 0) to f, then (2, 1), then (3, 2), then... we've run out of elements in B.  Thus we can see that it is impossible to define a bijection from B to B' - a function that maps every element in B to every element in B' once and only once - because we run out of elements in B.  Therefore, we can conclude that the size of B' is indeed larger than the size of B, because there are still elements in B' left over once we have paired off every element of B with an element of B'.

This is not the case for an infinite set, however.  Here, let B = {1, 2, 3, 4, ...}.  Then, let B' = B U {0} = {0, 1, 2, 3, 4, ...}.  Now, we can again define a function f : B -> B' in the same way.  We can add (1, 0) to f, then (2, 1), then (3, 2), then (4, 3), then... well, this is going to go on forever.  In general, we can add (x, x - 1) to f such that f(x) = x - 1.  And here you can see that there is no element in B' that does not have an element in B, and it is also clearly the case that every element in B' has only one element in B that is mapped to it.  Thus, f is a bijection, and we can conclude that B and B' are precisely the same size.  They do certainly have different elements in them, but they differ in only finitely many elements, which makes the size differential 0 - completely as expected, really, given that one would not expect a finite to impact an infinite.

Really, your entire argument here is just making the utterly baseless assertion that addition to a set necessarily must increase the size of that set, even if the set is infinite. If you want your argument to hold water, it's time for you to prove that assertion.

If you don't add or remove any in the process, that is correct.

So why did you say that when you add to an infinite set, it doesn't become larger, just below?

[QUOTE="danwallacefan"]

Eh, who said anything about "room 0"? If there is no room zero, then there's another person in the hotel.

GabuEx

...You're not getting what I'm saying.

I'm saying that you have two possible ways of adding a new person (call him person 0) to the hotel:

1. Move person 0 to room 1.  Move person 1 to room 2. Move person 2 to room 3.  Et cetera.

2. Move person n to room n + 1, all at once.

The end result in both cases is exactly the same.

well not exactly. Like you said, if process 1 is used, then the shift never happens because its constantly happening and this will take an infinite amount of time (or rather, will never happen). But if process 2 is used, then the absurdities in Hilbert's Hotel and various other examples will arise. 

Only if the set is not infinite, as I have said again and again.  You are applying to the infinite logic applicable only to the finite and then trying to wring a contradiction out of it.

But this is a simple rule of logic is it not? To add to some set, any set, is to take something external to that set and add to that set, thereby expanding that set. 

And "defeat(s) modus ponens"?  I mean, come on.  First the idea of "violating the law of non-contradiction" and now this, when the real question is nothing more than whether or not a contradiction exists.  I can assure you that I'm well-versed enough in theoretical logic that I can tell when someone is bluffing their case through unnecessarily technical language. GabuEx

I'm "bluffing my case through unnecessarily technical language"? Okay, lemme dumb it down for you. 

If you add X to Y, Y becomes larger. 

You have added X to Y

Therefore, Y isn't larger?

Modus ponens. Learn about it. 

I think you could stand to do so first.  An appeal to authority looks like this:

1. Person A is (claimed to be) an authority on subject S.

2. Person A makes claim C about subject S.

3. Therefore, C is true.

Your argument looked like this:

1. Mathematicians are (claimed to be) an authority on infinites.

2. Mathematicians are said to have made the claim that infinites are just concepts, not things that can exist in reality. 

3. Therefore, infinites are just concepts, not things that can exist in reality.

That was a complete and utter textbook appeal to authority.  That you would deny it is baffling. GabuEx

Here you are simply misunderstanding my point. How you managed to interpret my blanket statement "Mathematicians accept that infinites are just concepts with no bearing on reality" as "Mathematicians accept this, therefore this is correct" is beyond me. 

But I'll play along and assume that I really did say that

So what? Take a remedial debate ****Gabu, an appeal to authority is only fallacious if the authority to which we are appealing is not an actual authority. 

The consensus of mathematicians on a mathematical issue, of course, IS an authority. 

You can make the claim that this is contradictory until the cows come home, but that doesn't make it true.  There is absolutely nothing in the definition of addition that says that adding a new item to an infinite set makes the set larger. GabuEx

yeah, its ANY consistent set.

If you add the contents of set A to set B, all this says is that the contents of A will then be present in set B.  This says nothing about the resulting size of the new set B.

Proving this is simple.  First, the finite case: Let B = {1, 2, 3}.  Then, let B' = B U {0} = {0, 1, 2, 3}.  Now, how do we know that B is not the same size as B'?  Well, let's define a function f : B -> B'.  We can just add the elements in ordinal order: first we add (1, 0) to f, then (2, 1), then (3, 2), then... we've run out of elements in B.  Thus we can see that it is impossible to define a bijection from B to B' - a function that maps every element in B to every element in B' once and only once - because we run out of elements in B.  Therefore, we can conclude that the size of B' is indeed larger than the size of B, because there are still elements in B' left over once we have paired off every element of B with an element of B'.

This is not the case for an infinite set, however.  Here, let B = {1, 2, 3, 4, ...}.  Then, let B' = B U {0} = {0, 1, 2, 3, 4, ...}.  Now, we can again define a function f : B -> B' in the same way.  We can add (1, 0) to f, then (2, 1), then (3, 2), then (4, 3), then... well, this is going to go on forever.  In general, we can add (x, x - 1) to f such that f(x) = x - 1.  And here you can see that there is no element in B' that does not have an element in B, and it is also clearly the case that every element in B' has only one element in B that is mapped to it.  Thus, f is a bijection, and we can conclude that B and B' are precisely the same size.  They do certainly have different elements in them, but they differ in only finitely many elements, which makes the size differential 0 - completely as expected, really, given that one would not expect a finite to impact an infinite.

Really, your entire argument here is just making the utterly baseless assertion that addition to a set necessarily must increase the size of that set, even if the set is infinite. If you want your argument to hold water, it's time for you to prove that assertion.

GabuEx

Now, let's use this analogy more in line with the question of actual infinites, because your analogy is just appealing to two different sets on different axis on a graph while smuggling in the conclusion that Set B really has elements of Set A, but is still not expanded.   

Set 1={2, 4, 6, 8...}

Set 2={1, 3, 5, 7...}

Now, if we added these 2 sets to eachother, we get Set 3, {1, 2, 3, 4...}

Now, is set 3 larger than sets 1 or 2? If not, then how is this not contradictory? If yes, then how so, if they are both, to use your words, "countably infinite"?

I'm "bluffing my case through unnecessarily technical language"? Okay, lemme dumb it down for you. 

If you add X to Y, Y becomes larger. 

You have added X to Y

Therefore, Y isn't larger?

Modus ponens. Learn about it. 

danwallacefan

[QUOTE="domatron23"][QUOTE="danwallacefan"]

Eh, so is there a difference between 50% and 100% occupancy? 

danwallacefan

Consider the following sets. [1, 2, 3, 4...] [1, 3, 5, 7...]

Each contains an infinite number of members but the second one has no even numbers which means that it also lacks an infinite amount of members. They represent the same amount but not the same configuration.

Same thing with Hilberts hotel. 100% occupancy means that all rooms are filled, 50% occupancy means that there are an infinite number of rooms which are not full but in both cases there is the same amount of guests in the hotel.

Whether or not there is a difference depends on whether you're talking about the guests or the rooms.

Talking about adding to and subtracting from infinites is a logically nonsensical use of language. Think of it like Hawking's "north of the north pole" example.

So if you say that you've added to an infinite number then you're just talking jibba jabba. 

EDIT: I'll demonstrate what I mean with the chunk of your post that Funk_Llama asked about.

If you add X to Y, Y becomes larger.

danwallacefan

If Y is an infinite number and if we understand an infinite number to be one than which there is nothing larger then the sentence essentially says "If you add X to a number than which there is nothing larger, it becomes larger".

Which you can plainly see is logical claptrap.

So why did you say that when you add to an infinite set, it doesn't become larger, just below?

danwallacefan

...Because it doesn't.  I have no idea what you're talking about here.

well not exactly. Like you said, if process 1 is used, then the shift never happens because its constantly happening and this will take an infinite amount of time (or rather, will never happen). But if process 2 is used, then the absurdities in Hilbert's Hotel and various other examples will arise. 

danwallacefan

No, no they don't.  In both cases, the exact same people end up in the exact same room.  If you wish to dispute this, then find me a person who ends up roomless in process 2 who does not meet the same fate (eventually) in process 1.

But this is a simple rule of logic is it not? To add to some set, any set, is to take something external to that set and add to that set, thereby expanding that set.

danwallacefan

Yes...

...but if the set is infinite, then the change in size is zero.

I'm "bluffing my case through unnecessarily technical language"?

danwallacefan

Considering that the ultimate question is "Is there a contradiction?", and considering that you are attempting to use that language to ignore that question and instead assume an answer to it off the bat, yes.

Okay, lemme dumb it down for you. 

If you add X to Y, Y becomes larger. 

You have added X to Y

Therefore, Y isn't larger?

danwallacefan

The ultimate question we are addressing is precisely that very first premise.  That you are unwilling to even acknowledge that it is in question, and that you instead insist on asserting that I am violating basic logical principles by utterly assuming its truth and proceeding from there, is curious.

The percent increase in Y by adding the contents of X to Y is given by the simple division |X|/|Y|, where |A| is the size of a given set A.  Suppose that the size of X is 1, and that the size of Y is also 1.  In this case, the percent increase is 100% - you have doubled the size of Y.  Now, suppose you add another item to Y.  Now, the percent increase is 50%.  If you keep going, the percent increase steadily decreases - 33%, 25%, 20%, and so on.  Therefore, it is easy to see how, if the size of Y is infinite, the percentage increase from adding an item to it is precisely zero.

Thus, no, modus ponens has not been defeated for the simple fact that the conditional statement is false.  Adding an item to an infinite set does not change the size of the set.  If it did, then the set would not be infinite.

Modus ponens. Learn about it.

danwallacefan

If one were to learn about it, one would presumably discover that "if A, then B" must be true before B can be derived from A.

Here you are simply misunderstanding my point. How you managed to interpret my blanket statement "Mathematicians accept that infinites are just concepts with no bearing on reality" as "Mathematicians accept this, therefore this is correct" is beyond me.

danwallacefan

I will leave any final judgment regarding what you said and what it meant to the observer.

But I'll play along and assume that I really did say that

So what? Take a remedial debate ****Gabu, an appeal to authority is only fallacious if the authority to which we are appealing is not an actual authority. 

The consensus of mathematicians on a mathematical issue, of course, IS an authority.

danwallacefan

Debate cIass?  I was under the impression that we are discussing mathematics and logic.  Or was all that mention of the law of non-contradiction and modus ponens just for show?

This is not a "debate".  One may have a debate over an "ought", or one may have a debate over an "is" that can never be conclusively proven, but we are in neither scenario.  When it comes to mathematics and logic, there is only one credible authority: the rigorously constructed argument.  And you have failed to provide any such argument at all, instead simply making the same unsupported assertion again and again.

If you wish to prove that an infinite cannot exist in reality, then I have already given you two paths that you can take.  You can either show why a hotel's room assignments cannot be accurately modeled by two sets and a function between them, or you can show why the math that I have employed is faulty.  Either will do - but failure to attempt either is a tacit admission both that the model is accurate and that the math is sound, which will leave your argument in rather dire straits.

If you could explain, in layman's terms, How your appeal to a coordinate plain works here, that would be great.

danwallacefan

It's simple: to determine whether or not sets A and B are the same size, you simply match elements from A with elements from B.

If you can show that every element of A can be matched with one and only one element of B, then you have shown that the size of A is less than or equal to the size of B.  This intuitively makes sense, as it would indicate that there are at least as many elements in B as in A.

If you can show that every elements of B can be matched with one and only one element of A, then you have shown that the size of A is greater than or equal to the size of B.   This also intuitively makes sense, as it would indicate that there are at least as many elements in A as in B.

If you can show that both of these can be done simultaneously, then you have shown that the size of A is equal to the size of B, as the only way that a number can be both greater than or equal to and less than or equal to a number is if it is precisely equal to that number.

And this is precisely what I have done: I have shown that a mapping can be made between B and B' such that every element in B is mapped with one and only one element of B', and also such that every element of B' is mapped with one and only one element of B - thus proving that they are the same size.

Now, let's use this analogy more in line with the question of actual infinites, because your analogy is just appealing to two different sets on different axis on a graph while smuggling in the conclusion that Set B really has elements of Set A, but is still not expanded.   

Set 1={2, 4, 6, 8...}

Set 2={1, 3, 5, 7...}

Now, if we added these 2 sets to eachother, we get Set 3, {1, 2, 3, 4...}

Now, is set 3 larger than sets 1 or 2? If not, then how is this not contradictory? If yes, then how so, if they are both, to use your words, "countably infinite"?

danwallacefan

Well, first, I should explain the phrase "countably infinite", since it sounds as though that might be an unfamiliar term.  A set that is countably infinite is one which contains an infinite number of elements, but in which every single element is enumerable.  The set of natural numbers is an example of that - one can simply count off the numbers in it: 1, 2, 3, and so on.  The alternative is an uncountably infinite set.  This is a set like the real numbers, where elements within the set are themselves infinite, and therefore not every element is enumerable.

I use the qualifying phrase "countably infinite" rather than just infinite, because there is a fact about uncountably infinite sets that is at face value curious - namely, that they are strictly larger than countably infinite sets.  I could rigorously prove this, but it's not relevant to this discussion - suffice it to say that it's because there are two orders of infinity at play rather than just one; not only are there an infinite number of elements, but such a set contains elements which are themselves also infinite.  Thus, in order to preserve mathematical rigor, I need to specify precisely what order of infinity I am talking about.

But, let's go back to your example.  In it, we have two sets: A = {2, 4, 6, 8, ...} and B = {1, 3, 5, 7, ...}.  The question is asked: is the union of these sets C = A U B = {1, 2, 3, 4, ...}, larger than either of the two sets alone?

The answer is no.  And the answer to the followup question - how is this not contradictory? - goes back to what I said above.  If a set is already infinite, then adding more elements to it does not change its size, despite the fact that it changes the elements within the set.  And though in the example before only tackled the case of adding a single item to it, it is in fact the case that even adding up to a countably infinite number of items to the set does not change its size.  If it were the case that 2 * |A| > |A|, then |A| would be finite.

I could set up a mapping just like the one I showed earlier in order to prove this, but I'm sure that is an exercise that you yourself could do - it's not hard, really.  You just need to figure out what function would transform {2, 4, 6, 8, ...} into {1, 2, 3, 4...}.

So, in conclusion, is C larger than A or B? No.

But, is this a contradiction?  No.  It's just the nature of infinity.

Talking about adding to and subtracting from infinites is a logically nonsensical use of language. Think of it like Hawking's "north of the north pole" example.

So if you say that you've added to an infinite number then you're just talking jibba jabba. 

domatron23

Well, that's not really quite true.  It is plainly meaningful to add to or subtract from a set with an infinite number of elements.  For example, if I have the set {1, 2, 3, 4...} and I add 0 to it, then the resulting set {0, 1, 2, 3, 4, ...} clearly has a new element in it.  And, similarly, if I remove 1 from that set, then the resulting set {0, 2, 3, 4, ...} clearly has lost an element it previously had.

The crux of the matter, however, is that when you have an infinite number of elements in a set, then the resulting change in size is 0 - precisely as one would expect, really, considering that the amount of elements that were already present was, of course, infinite.

EDIT: I'll demonstrate what I mean with the chunk of your post that Funk_Llama asked about.

[QUOTE="danwallacefan"]

If you add X to Y, Y becomes larger.

domatron23

If Y is an infinite number and if we understand an infinite number to be one than which there is nothing larger then the sentence essentially says "If you add X to a number than which there is nothing larger, it becomes larger".

Which you can plainly see is logical claptrap.

Well, let's be clear here: strictly speaking, there is no such thing as an "infinite number".  If there is a number, there is always a number that will be greater than it.  Infinity is not a number; it is a concept.  That is the reason why the idea of adding something to infinity itself is perhaps less than meaningful - it would be akin to attempting to add 1 to "largeness", and expecting a meaningful result.