Infinite infinities

[uncool]

There is no "room infinity." That is the entire point of the "Hilbert Hotel" paradox. No last room, nothing. None at all.

=Uncool-

[Fred56]

OK, so can that "mathematical reality" be woven somehow into it, or is there always some point where logic hits the wall, so to speak?

The problem with the story is that he appears to be the only guest (except for the register). But since he forgets which room, does that deal with the conundrum? Could the clerk say something about extensions to the hotel being underway, so his room has to be changed?

 

Why doesn't his room number (infinity plus one) deal with your objection? How can a hotel with an infinite number of rooms not have a "room infinity"?

 

Perhaps there is no answer.. the idea doesn't "work": there are an infinite number of possible objections to such a proposition...

[JesterArts]

"It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is zero, and that any people you may meet from time to time are merely the products of a deranged imagination."

 

-- Hitchhiker's Guide to the Galaxy.

 

Sorry, couldn't resist.

[Fred56]

There's a logical fallacy: if there are an infinite number of planets, by definition there are an infinite number of both occupied and unoccupied ones. Saying there are a finite number (of infinite planets of any kind) is incorrect.

[Mr Skeptic]

Someone just recently educated me on this matter [of large finite numbers in an infinite set]. The idea is that the supremum of a set is not necessarily in the set. Consider the set of all negative reals. Then the supremum is zero, but zero is not in the set. You can get as close as you like to zero, but no matter how close you get, there will always be a closer number between zero and whatever negative number you chose. Does that make sense for you?

[Fred56]

What happens if the set has its sign changed arbitrarily (that means for no good reason btw)?

[Mr Skeptic]

There's a logical fallacy: if there are an infinite number of planets, by definition there are an infinite number of both occupied and unoccupied ones. Saying there are a finite number (of infinite planets of any kind) is incorrect.

 

Saying that there's a logical fallacy in the Hitchhiker's Guide to the Galaxy is just plain ignorant.

 

Also, you are wrong in this case. Given an infinite number of planets, you would have an infinite number of inhabited planets only if the probability that a planet be inhabited is not infinitesimally small for most of the universe. Consider all the integrals from n to infinity that have a finite value and you'll see what I mean. Or the infinite sum 1/2 + 1/4 + 1/8 + ... = 1.

 

What happens if the set has its sign changed arbitrarily (that means for no good reason btw)?

 

What do you mean?

 

In my example, all the negative numbers would become positive. Then zero would be the greatest lower bound rather than the least upper bound but would still not be in the set [now of all positive reals]. Then you would be unable to find a lowest number in the set of positive reals.

[Fred56]

Also, you are wrong in this case. Given an infinite number of planets, you would have an infinite number of inhabited planets only if the probability that a planet be inhabited is not infinitesimally small for most of the universe.

I see. What if, instead of occupation by some lifeform, you used planet size, or atmosphere, or colour (red ones and blue ones)? Or would there be that same problem with the probability of any planet, say being blue, or having an atmosphere, or whatever?

If there are an indenumerable number of planets, how can we know anything about "how many" of them are a particular size, or colour, or anything?

all the negative numbers would become positive. Then zero would be the greatest lower bound rather than the least upper bound but would still not be in the set [now of all positive reals]. Then you would be unable to find a lowest number in the set of positive reals.

This presumably is only applicable to the "quantity" zero?

Because zero has no "quantity", how can it, in fact, "belong" to anything (especially a set of values, or "quantities") despite what Godel has to say?

[JesterArts]

There's a logical fallacy: if there are an infinite number of planets, by definition there are an infinite number of both occupied and unoccupied ones. Saying there are a finite number (of infinite planets of any kind) is incorrect.

 

Thanks. I should probobly disregard most of Douglas Adam's ideas then!

 

Saying that there's a logical fallacy in the Hitchhiker's Guide to the Galaxy is just plain ignorant.

 

Also, you are wrong in this case. Given an infinite number of planets, you would have an infinite number of inhabited planets only if the probability that a planet be inhabited is not infinitesimally small for most of the universe. Consider all the integrals from n to infinity that have a finite value and you'll see what I mean. Or the infinite sum 1/2 + 1/4 + 1/8 + ... = 1.

 

 

Thats a relief, thanks. Hitchhiker's Guide is gospel to me again.

[uncool]

There's a logical fallacy: if there are an infinite number of planets, by definition there are an infinite number of both occupied and unoccupied ones. Saying there are a finite number (of infinite planets of any kind) is incorrect.

 

Do you mean that there are an infinite number of one of them, or rather, that the number of planets which are either occupied or unoccupied is infinite?

Because one of them could be finite...

=Uncool-

[Fred56]

Sorry, Jim, but this is not logical.

The proposition is that there are an infinite number of somethings.

By definition, there is no way to count these somethings, so if they look different (or are 'observed' to not be identical) then there is necessarily an infinite number of each different kind.

If you started to count them and found they were all different (as far as you could tell not a single one is identical), this would not allow the conclusion that there are a finite number of any of the varieties (found up to then).

Because the number of them is infinite, you need to count all of them to determine that there is only a single representative of each kind...

[ghstofmaxwll]

How the hell do you guys get to write notations in your posts? Is it with latex?

[Fred56]

How the hell do you guys get to write notations in your posts? Is it with latex?

after reading the tutorial on Latex, it's fairly straightforward...

[fripro]

The Universe is not infinite; however it (the Universe) does exist in infinity FRIPRO

[frost]

I liked Mr Skeptics way to prove that infinity-infinity is not zero.

I would personally define infinite as not a "static number" like for example 0 or 1. 0 is 0 and stays that way. But when we think about infinite it's kinda growing all the times or something like that. And when we think about infinite as a number it never can't be that number cause it's alway bigger. It's like infinite>n in all cases exept of when n=infinite but you guys said that infinite is not a number so we can't take this as an example for infinite=n.

And I think that this could be used also in the case of infinite-infinite is not 0 cause we have a result but can't define it so could it be that there are several numbers or better said infinite numbers of results becouse there are infinite different ways of infinite-infinite (if infinite is something like e "moving" or "unstatic number" like I said previously) so could it be infinite-infinite=infinite?

This is my imagination playing here. :S Not much of mathematic backup so help me!

[ydoaPs]

Sorry, Jim, but this is not logical.

The proposition is that there are an infinite number of somethings.

By definition, there is no way to count these somethings, so if they look different (or are 'observed' to not be identical) then there is necessarily an infinite number of each different kind.

If you started to count them and found they were all different (as far as you could tell not a single one is identical), this would not allow the conclusion that there are a finite number of any of the varieties (found up to then).

Because the number of them is infinite, you need to count all of them to determine that there is only a single representative of each kind...

How many reals are between 1 and 2?

[Fred56]

How many reals are between 1 and 2?

My understanding is there are an infinite set, which is bounded, and excludes (is closed to), any value equal or less than 1, and equal or more than, 2 (these are the bounds).

Being unbounded in cardinality, it is indenumerable, or uncountable. But there is no member of the natural numbers in it. Its cardinality is infinite (2nd aleph, or something).

This is an example of a set of discrete elements (they are all different), which is a known -there´s no need to count them.

[ydoaPs]

My understanding is there are an infinite set, which is bounded, and excludes (is closed to), any value equal or less than 1, and equal or more than, 2 (these are the bounds).

Being unbounded in cardinality, it is indenumerable, or uncountable. But there is no member of the natural numbers in it. Its cardinality is infinite (2nd aleph, or something).

This is an example of a set of discrete elements (they are all different), which is a known -there´s no need to count them.

Now pick any two of those reals. How many reals are between them?

[Fred56]

The same, uncountable ´number´ of discrete values is between any two such arbitrary bounds, and has the same cardinality (in fact is identical -except for the limits or bounds -the excluded values)... Is this a quiz?

 

These are all sets which are bounded in value but have unbounded cardinality. There´s something of a symmetry, in that you can exclude any cardinal number (by bounding it between any two, or between any two arbitray real values, but each discrete element has a value which means it also has an order , or selectability, within the set. Since by definition all members of such a set are discrete, counting them to determine this trivial fact is, ..well, trivial, or actually pointless (pun intended).

[ydoaPs]

The same, uncountable ´number´ of discrete values is between any two such arbitrary bounds, and has the same cardinality (in fact is identical -except for the limits or bounds -the excluded values)... Is this a quiz?

 

So, if between any two reals between 1 and 2(of which there are an infinite amount) there are an infinite number of reals. Does this not mean there are infinite infinities between 1 and 2?

[Fred56]

It does mean that yes, there are an infinite number of intervals (bounds) that you can define with an infinite number of real values in them (on any interval of the real numbers). Any infinite set also has an infinite number of possible subsets. That´s a general rule, I think. Maybe someone who knows could say if that is the case (I´m not too good at number or set theory).

[ydoaPs]

It does mean that yes, there are an infinite number of intervals (bounds) that you can define with an infinite number of real values in them (on any interval of the real numbers). Any infinite set also has an infinite number of possible subsets. That´s a general rule, I think. Maybe someone who knows could say if that is the case (I´m not too good at number or set theory).

 

So, if you subtract the infinite subset from the infinite set of infinite subsets, then it would still be infinite.....but your op says:

infinity minus infinity is zero

 

IIRC, those two infinities are even the same size.

[Fred56]

Right. But later on I change that particular sweeping statement to "it can be zero", which is true if the subtraction/division is done before any limit is taken (I've done this, and you have too, with a series of infinite terms that cancel so the series converges...).

those two infinities are even the same size

Yea, but what does size mean? they have the same cardinality, but different bounds. If you define the two to have the same bounds, the difference will be zero, won't it?

[MrMongoose]

I think the clearest example is that if infinity is a number and you claim that

 

infinity+infinity=infinity

 

then taking infinity from each side of the equation gives

 

infinity= infinity-infinity

 

Then you claim that infinity-infinity=0

 

so infinity=0

 

Indeed, 0+0=0 and 0-0=0

 

So infinity is equal to zero...

[Fred56]

Actually it depends how you 'define' a particular infinity (there are lots of them -an infinite variety). Math lets you subtract infinities if they are 'undefined'. Two undefined infinities can be equal (or just assumed to be). Likewise with bounds for an infinite set, as above.

I think your definition might run into difficulties with the idea of limits.