[kenberg]

I am pretty sure this infinite hotel is a completely dead end.

mathematical: Add 1 to the integers 1,2,3... and get 2,3,4... opening up room 1

Physical Reality: I have never seen a hotel built this way

So?


I seriously doubt, but I cannot be certain, that Hilbert maintained that you could conclude on the basis of thought that a universe with infinitely many stars was impossible. Whatever the physical arguments for ot against, conceptually it seems perfectly possible that the universe could be built that way. I would not expect Hilbert to claim that proper thinking, mathematical or otherwise, would show such a notion to be nonsense.

The problem with the hotel is that you get into issues of how you could build a hotel with infinitely many rooms, wouldn't you run out of door handles and so on. All more or less irrelevant to any significant conclusion.

The possible infinitude of stars seems far more to the point. Physics may provide evidence, although probably it is impossible to settler conclusively, but it seems fine to go with the idea that either finitely many or infinitely many is perfectly sensible to contemplate.



I don't know what Hilbert's point was but I really doubt you can make much of anything out of thinking about the physical impossibility of a hotel with infinitely many rooms. I would guess he was just illustrating that mathematics needs to be used with some care in application to the real world. I certainly would agree with that.

[PassedOut]

After work I thought about this thread a bit more and thought I should clarify one of the things I said:

PassedOut, on Apr 6 2009, 01:56 PM, said:

To be honest, I don't really understand "actual infinite" and don't see any reason to spend time working on it, given all the worthwhile things I've left to learn (and given the relative shortness of the portion of my life that remains).

Using these definitions:

luke warm, on Apr 5 2009, 02:43 PM, said:

in any case, the terms are usually defined:

an actual infinite is a set of things that cannot be added to
a potential infinite is a set of things that can be added to

I don't understand "actual infinite" here because the whole idea of infinite, to me, entails limitless expansion.

Suppose you define a set of all the non-overlapping 1-cm circles that fit within a 1-m circle. Would that set constitute an "actual infinite?"

If so, what does that have to do with the word "infinite?" If not, why does the set not meet the definition?

I suspect that you've had similar thoughts about this, Jimmy, and have resolved them in some way to your satisfaction. I'd appreciate it if you'd help me out here.

[helene_t]

PassedOut, on Apr 7 2009, 04:55 AM, said:

I don't understand "actual infinite" here because the whole idea of infinite, to me, entails limitless expansion.

To me the distinction feels very natural but maybe it's just me or maybe it's difficult to explain.

Craig's definition would be something like this, in terms of an example:

Suppose this thread is going to end after some number of posts. The number of posts this thread will actually reach will therefore be finite. However, since there is no constraint on the number of posts a priori, the number of posts the thread could potentially reach is unlimited, AKA potentially infinite.

That example was with a stochastic variable which has an infinite domain but finite realizations. Another example could be time:

Suppose (for the sake of the argument) that time is unbounded in the forward direction. Suppose this thread goes on forever and accumulates some ten new posts per day. Then the number of posts this thread will reach at some point is infinite, but since the year "+infinity a.d." will never appear in calendars, the number of posts it will contain at any actual time point is finite.

A third example could involve a model of the universe in which time is discrete but space is dense. The set of points in space a particle could potentially traverse is finite, but the set of points it actually traverses is finite.

I am sure you are familiar with these concepts, but you just call them something else.

Applied to Hilbert's Hotel, think of a hotel to which a newly build section is added every year, and the construction goes on forever. That would be "potentially" infinite.

It occurs to me that the concept of "potential infinite" does not relate to the real world, but maybe Craig has an idea of the real world which is very different from mine

[PassedOut]

helene_t, on Apr 7 2009, 06:32 AM, said:

Suppose this thread is going to end after some number of posts. The number of posts this thread will actually reach will therefore be finite. However, since there is no constraint on the number of posts a priori, the number of posts the thread could potentially reach is unlimited, AKA potentially infinite.

The concepts are familiar but this terminology seems deceptive to me.

Using these definitions:

luke warm, on Apr 5 2009, 02:43 PM, said:

in any case, the terms are usually defined:

an actual infinite is a set of things that cannot be added to
a potential infinite is a set of things that can be added to

when do we say that the set of posts "cannot be added to?" When BBF closes or archives the thread?

If a set always has a finite limit, even though that limit is left unspecified (but always less than, say, one trillion posts), in what sense is it infinite?

PassedOut, on Apr 7 2009, 04:55 AM, said:

Suppose you define a set of all the non-overlapping 1-cm circles that fit within a 1-m circle.

Isn't this a clear example of a set that "cannot be added to?"

[helene_t]

all sets can be added to of course. Add an orange to the set of Lynn's circles and you get the set
{Lynncircle1, Lynncircle2, ...., LynncircleN, anOrange}

Anyway, that's not in the definition. The definition is not written in mathematical jargon so it is confusing but here it is:

http://www.woodford.redbridge.sch.uk/rs/ye...lberthotel.html

I hope my examples make more sense

[PassedOut]

Thanks.

[Gerben42]

Quote

I am pretty sure this infinite hotel is a completely dead end

Economically yes, but technically no

[helene_t]

A review of Craig's argument is here:
journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=108688

The mistakes in Craig's reasoning are quite elementary. Looks as if he doesn't grasp the concept of infinity at all.

[luke warm]

TimG, on Apr 6 2009, 03:50 PM, said:

luke warm, on Apr 6 2009, 03:09 PM, said:

TimG, on Apr 6 2009, 02:18 PM, said:

luke warm, on Apr 6 2009, 01:28 PM, said:

TimG, on Apr 6 2009, 11:40 AM, said:

What temporal events must be added?

temporal in the sense of historical happenings

I still don't get it. Are trying to add the civil war into the time line?

yes

Why isn't it already there?

it is, but the fact that we're here and now from there shows that there was a beginning

jdonn, on Apr 6 2009, 03:47 PM, said:

Of course you can remove an element from an infinite set. If you can add one to it, you can simply put it back exactly how it was before. That is removing an element, subtracting, whatever you want to call it.

intuition?

helene_t, on Apr 6 2009, 03:36 PM, said:

aleph0 - 1 = aleph0

See? It works!

Seriously, transfinite arithmetics is about operations like 2^aleph0 > aleph0. Adding (or subtracting) ordinary integers from/to transifinites is not so interesting.

But surely you can remove a single element from an infinite set. The set of all integers which are not 7 may be constructed by removing 7 from the set of all integers. Nothing wrong with that.

everything i've read on the subject points to the fact that inverse operations using cantonian transfine ordinals is disallowed (conventionally) because of inconsistentcies encountered... if this isn't true, it isn't true

kenberg, on Apr 6 2009, 08:55 PM, said:

I seriously doubt, but I cannot be certain, that Hilbert maintained that you could conclude on the basis of thought that a universe with infinitely many stars was impossible. Whatever the physical arguments for ot against, conceptually it seems perfectly possible that the universe could be built that way. I would not expect Hilbert to claim that proper thinking, mathematical or otherwise, would show such a notion to be nonsense.

maybe so ken, all we know is what he said, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remains for the infinite to play is solely that of an idea." it seems to me that he's saying that an actual infinite is not to be found in reality, only in the mind... do you understand him differently?

PassedOut, on Apr 6 2009, 10:55 PM, said:

I don't understand "actual infinite" here because the whole idea of infinite, to me, entails limitless expansion.

Suppose you define a set of all the non-overlapping 1-cm circles that fit within a 1-m circle. Would that set constitute an "actual infinite?"

If so, what does that have to do with the word "infinite?" If not, why does the set not meet the definition?

I suspect that you've had similar thoughts about this, Jimmy, and have resolved them in some way to your satisfaction. I'd appreciate it if you'd help me out here.

not to my way of understanding the terms... what you have described is not a thing found in reality, such as rocks or stars or historical events

helene_t, on Apr 7 2009, 06:32 AM, said:

It occurs to me that the concept of "potential infinite" does not relate to the real world, but maybe Craig has an idea of the real world which is very different from mine

no helene, he's saying exactly what you say here

[jdonn]

luke warm, on Apr 7 2009, 12:18 PM, said:

jdonn, on Apr 6 2009, 03:47 PM, said:

Of course you can remove an element from an infinite set. If you can add one to it, you can simply put it back exactly how it was before. That is removing an element, subtracting, whatever you want to call it.

intuition?

I think there is an inverse correlation between how right you are and how clever you think you're being at any given moment. In any case, you're the one using an analogy that doesn't exist in the real world, so don't blame me for doing the best I can with it.

[PassedOut]

luke warm, on Apr 7 2009, 12:18 PM, said:

maybe so ken, all we know is what he said, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remains for the infinite to play is solely that of an idea." it seems to me that he's saying that an actual infinite is not to be found in reality, only in the mind... do you understand him differently?

Hmm. Didn't you already agree that time can be infinite (using the standard definition of infinite)?

luke warm, on Apr 5 2009, 06:59 PM, said:

PassedOut, on Apr 5 2009, 05:51 PM, said:

In plain English, what I am saying is that time can continue forever, moment after moment (no matter how finely or coarsely one defines "moment") without violating logic. Indeed, this scenario is quite likely, given the evidence we have to date. This is true whether or not you accept the big bang as the starting point for time.

Agreed?

i agree with this *if* there was a beginning, but not otherwise... i don't care what this "beginning" might be

[TimG]

luke warm, on Apr 7 2009, 12:18 PM, said:

TimG, on Apr 6 2009, 03:50 PM, said:

luke warm, on Apr 6 2009, 03:09 PM, said:

TimG, on Apr 6 2009, 02:18 PM, said:

luke warm, on Apr 6 2009, 01:28 PM, said:

TimG, on Apr 6 2009, 11:40 AM, said:

What temporal events must be added?

temporal in the sense of historical happenings

I still don't get it. Are trying to add the civil war into the time line?

yes

Why isn't it already there?

it is, but the fact that we're here and now from there shows that there was a beginning

I guess I'm still missing the point then. If it's there already, why does it need to be added?

As far as there being a beginning because we are here, let's consider that time is a number line and we are at 5. The number line can be infinite in one or both directions. That is, it could have a starting point at zero and continue indefinitely into the positive, or it could have no starting point and continue indefinitely into both positive and negative. If there is a beginning, we can measure our distance from the beginning no matter where we are along the number line, but that doesn't mean that there aren't still an infinite number of points not between us and the origin.

Perhaps the problem is that I am looking at time as all moments, past, present and future. While you are looking at all moment past and present (a finite set) and want to add the future moments. But, I don't see any reason those future moments can't be included in the set of moments we call time.

[helene_t]

luke warm, on Apr 7 2009, 06:18 PM, said:

everything i've read on the subject points to the fact that inverse operations using cantonian transfine ordinals is disallowed (conventionally) because of inconsistentcies encountered... if this isn't true, it isn't true

You may have read somewhere that aleph0-aleph0 is undefined.

This is because, if set A had cardinality Aleph0 and its subset B has cardinality aleph0 as well, you cannot (from that information alone) predict the cardinality of the difference set A\B. If A is the set of integers and B is the set of integers which are not 3 or 4, then card(A\B) = card ({3,4}) = 2, but for other subsets B, card(A\B) might be zero, alephh0, or any positive integer.

This does not mean that we cannot remove B from A, i.e. constructing A\B. We can, but the point is we must know what B exactly is in order to compute card(A\B). Therefore, aleph0-aleph0 is undefined.

If A has larger cardinality than B then card(A-B) = card(A). So in general, transfinite subtraction is ok, with the exception that you cannot subtract a transfinite number from itself.

kenberg said:

The problem with the hotel is that you get into issues of how you could build a hotel with infinitely many rooms, wouldn't you run out of door handles and so on.

Besides, since the size of the hotel would be infinite and communication is limited by the speed of light, it would take an infinity of time to communicate the room changes to all guests. So the new guest would have to wait an infinity of time before his room would be vacant.

[Winstonm]

Quote

maybe so ken, all we know is what he said, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remains for the infinite to play is solely that of an idea." it seems to me that he's saying that an actual infinite is not to be found in reality, only in the mind... do you understand him differently?

If the role for the infinite to play is solely that of an idea, then God, being infinite, cannot be a reality but can only be a creation of the mind - which I think gets back to the heart of the argument.

[Winstonm]

And another question for all you geniuses: the universe is supposed to be expanding. If the universe is defined as all things, including space and time, then into what is the universe expanding? What is beyond the edge of the universe?

OK, now that I've had one more hit of this stuff, I see that the answer is simple - the universe is exanding into a void due to pressure differences between the something and the nothing.

You got any more of this *****?

[luke warm]

PassedOut, on Apr 7 2009, 01:42 PM, said:

luke warm, on Apr 7 2009, 12:18 PM, said:

maybe so ken, all we know is what he said, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remains for the infinite to play is solely that of an idea." it seems to me that he's saying that an actual infinite is not to be found in reality, only in the mind... do you understand him differently?

Hmm. Didn't you already agree that time can be infinite (using the standard definition of infinite)?

luke warm, on Apr 5 2009, 06:59 PM, said:

PassedOut, on Apr 5 2009, 05:51 PM, said:

In plain English, what I am saying is that time can continue forever, moment after moment (no matter how finely or coarsely one defines "moment") without violating logic. Indeed, this scenario is quite likely, given the evidence we have to date. This is true whether or not you accept the big bang as the starting point for time.

Agreed?

i agree with this *if* there was a beginning, but not otherwise... i don't care what this "beginning" might be

here we go again... i don't know if it's semantics or if there's really a lack of communications, but i don't view "time going on forever" to be an actual infinite *if there were a starting point*, i.e. if there was a bb from which the measurement of time began... that means that time (or space/time if you prefer) and thus the universe isn't eternal - although *something* is... it is the fallacy of composition to assert (and i don't say you do this) that because a set (in this case, of future events) might "go on forever" it is actually infinite - infinity goes not only forward

TimG, on Apr 7 2009, 02:11 PM, said:

Perhaps the problem is that I am looking at time as all moments, past, present and future.  While you are looking at all moment past and present (a finite set) and want to add the future moments.  But, I don't see any reason those future moments can't be included in the set of moments we call time.

they can be if there were a beginning... i hate to keep rehashing the same arguments, which is why i tried to introduce hilbert's quotes into the discussion... even if helene is right that someone such as craig has a faulty understanding of an actual infinite in reality, she'd have to say the same for hilbert - and he obviously believed that such an infinite is not possible in reality...

helene_t, on Apr 7 2009, 04:41 PM, said:

luke warm, on Apr 7 2009, 06:18 PM, said:

everything i've read on the subject points to the fact that inverse operations using cantonian transfine ordinals is disallowed (conventionally) because of inconsistentcies encountered... if this isn't true, it isn't true

You may have read somewhere that aleph0-aleph0 is undefined.

But of course you can remove an element from an infinite set. You can even remove an infinite proper subset from another infinite set.

why then are subtraction and division conventionally disallowed? maybe you're right, helene, but others have said what i said above - that the reason division and subtraction is conventionally disallowed is because of inconsistencies... you seem to say there are no such inconsistencies... it's true that some people have gotten around this by redefining the terms (for example, changing 'inconsistent' to 'paradoxical'), but even they don't flat out deny the problems

"We don't subtract a0 from both sides of a0 + 1 = a0 because, if we did, set theory would be inconsistent!"

"But, as is well known, set theories such as ZF, NBG and the like were in various ways ad hoc. Hence, a number of people including da Costa (1974), Brady (1971), (1989), Priest, Routley, and Norman (1989), considered it preferable to retain the full power of the natural abstraction principle (every predicate determines a set), and tolerate a degree of inconsistency in set theory."

[helene_t]

Winstonm, on Apr 8 2009, 12:09 AM, said:

And another question for all you geniuses: the universe is supposed to be expanding.  If the universe is defined as all things, including space and time, then into what is the universe expanding?  What is beyond the edge of the universe?

OK, now that I've had one more hit of this stuff, I see that the answer is simple - the universe is expanding into a void due to pressure differences between the something and the nothing. 

You got any more of this *****?

No, the universe is not expanding into anything. The universe is all there is. There is nothing "outside", not even empty space, void, whatever.

lukewarm said:

We don't subtract a0 from both sides of a0 + 1 = a0 because, if we did, set theory would be inconsistent!

Right, as I said you cannot subtract a transfinite number from itself.

But the axioms of set theory require any set of the form A\B where B is a subset of A, to exist. This is very elementary stuff. It is certainly not something different mathematicians disagree about. If you have read somewhere that set subtraction involving infinite sets is disallowed then you must have misunderstood it.

[TimG]

luke warm, on Apr 7 2009, 06:10 PM, said:

infinity goes not only forward

Does this mean that the set of whole numbers is not infinite (because it is bounded by zero)?

[jdonn]

luke warm, on Apr 7 2009, 06:10 PM, said:

it is the fallacy of composition to assert (and i don't say you do this) that because a set (in this case, of future events) might "go on forever" it is actually infinite - infinity goes not only forward

How does someone even respond to this without saying LOL? Do you know what 'forever' means? Where do you get this stuff? If a set of events that goes on forever is not infinite then how many events are there, 999? Three bajillion? If something with a starting point that goes only forward can not be infinite, please tell me how many positive integers there are.

[jdonn]

Winstonm, on Apr 7 2009, 06:09 PM, said:

And another question for all you geniuses: the universe is supposed to be expanding.  If the universe is defined as all things, including space and time, then into what is the universe expanding?  What is beyond the edge of the universe?

OK, now that I've had one more hit of this stuff, I see that the answer is simple - the universe is exanding into a void due to pressure differences between the something and the nothing. 

You got any more of this *****?

The universe has no boundary. It is not expanding 'into' anything. There is no edge, if you travelled extremely fast you would keep going and potentially even end up back where you started without ever changing directions. This is a good explanation.