Gracious of you given the mess I made of the middle bit.
The terminology here has lost me. Let me pick through it and see if I can make sense of what you mean.

I agree with this - B (the subset) is not identical to the original set A. I don't know what a totality is. What I am taking the term to mean is that it must be understood as a unit, not as some union of disjoint components. This is very likely not what you mean though so perhaps you should spell out what a totality is.
It means that the two non-identical objects A and B can be placed in one-to-one correspondence - I'd use cardinality rather than numerosity.
I don't know what much of this means, I'm afraid. By identity do you mean correspondence? What does 'when |B|' mean? What is the 'natural correlation'? What do you mean by 'establish the set'?
It is very different to that - there are no new rooms built. The set {original guests}U{new arrival} has the same cardinality as {rooms} - nothing about the rooms changes, all that changes is how we allocate guests to rooms (they all move up one).

No room is built, rather the guest originally in room 1 goes to room 2, the guest originally in room 2 goes to room 3. What's the problem or peculiarity? Where is the guest who was originally in room 1010234? He's in 1010235. Where did the guest who is now in room 20202 come from? He was originally in 20201. Who is now in room 1? Nobody - the new guy can go in there.

EDIT: Maybe I misunderstood your objection here. You could create a hypothetical where the hotel expanded (by co-opting rooms in the hotel across the street or something). However this would be understood as taking a union of two sets, not of 'adding to a totality' or somesuch - these other terms are not yet defined.
You've kind of lost me again there. I don't really like using terms like 'totality' and 'completed sets' without defining them. I'm sure they can be made sensible, but until that is done we are likely to rely on commonsense intuitions derived from finite sets and this may easily lead us to impute qualities to the set of rooms or either set of guests which are not properties of infinite sets.

Remember that 'adding' is not strictly something we do with sets. We can form a union - doing so results in a new set of guests (who can also be accomodated in the unchanged hotel).

EDIT: Ok further re-reading. Remember that for two infinite sets to have the same cardinality (be the same size) there must be a one-to-one correspondence between them. The fact you can make other correspondences (like moving everyone to the room double their current number, leaving all odd-numbered rooms blank or somesuch) doesn't change the cardinality of the hotel or the set of guests. When you say "| rooms | > | guests |" you are making an error - you are thinking of the new set being "infinity + one" or something which is not defined. You do not 'add' sets, nor do you even add to them. You take unions of sets. The cardinality never changes because all throughout the process any of the important sets at any moment can be placed in one-to-one correspondence with the natural numbers. They all have a cardinality of aleph-0.

Okay, so obviously bunny's post is more useful, but let me see if I can point out what seems to be an error in how you are thinking of one-to-one correspondence. Here's the definition of an actual infinite set given by bunny:

Now, you seem to agree with us that in our putative examples of infinite sets there exists a one-to-one correspondence with some proper subset (or with the natural numbers). However, you claim that a contradiction arises because we can also place these sets in a two-to-one correspondence with some proper subset. So in your example we could have 2 elements in set A for every 1 element in set B.

I suspect that you view this as a contradiction because you are assuming that for any two given well-ordered sets, if you use each element only once, there is only a single mapping relationship that can hold between them. But yet with infinite sets this doesn't seem to be the case and so you think that there is something incoherent in the idea of an infinite set.

Okay, the first thing to say in response to this idea is that this assumption is not included in the definition of an infinite set. Notice that it says only of such sets that they can be placed in a one-to-one correspondence with the natural numbers.

Second, I think this assumption makes some intuitive sense with finite sets. After all, at some point you run out of elements to place into correspondence, so it seems like there is a single kind of mapping available (to the mathematicians, excuse my loose talk here). However, the whole point of saying something is infinite is just that you never actually do run out of elements--there's always more. So it is this very breakdown in the usual pattern of this kind of mapping (really we are just talking about cardinality here) that we are trying to capture when defining infinite sets. And this is what the definition does--it highlights the distinguishing behavior of infinite from finite sets.

Ha! You did a much better job of this bit than me - I said exactly the opposite (wrong) thing using much more complicated language. It was very irritating.

This is probably where I'm drawing a wrong conclusion. I'm thinking:

If the hotel is an actual infinite and the hotel is full, then the hotel is an actual infinite of full rooms.

Am I missing something here?

What bunny said.

Here the relevant bit is Craig arguing claiming that he thinks that the actual infinite is metaphysically impossible, but not logically impossible. First, I'll note that as I understand duffe's argument, it actually implies that the actual infinite is logically impossible--this is why I said earlier that I thought he was going beyond Craig.

Second, even if this is a real distinction (which I'm a bit skeptical of), it doesn't seem to apply in this case. The examples of metaphysical impossibility given by Craig are of statements which are either nonsensical or proven false by basic arithmetic. Neither of these apply to the infinite set, which is clearly defined and consistent with basic mathematics.

However, as I've said before, I really do think it could be the case that an actual infinite is impossible, I just think that the burden of proof is on the person making this claim, and that the counter-intuitive results of the set-theoretic definition presented by Craig doesn't meet this burden.

I am not sure why you included this quotation. This addresses a specific argument used by Quine and Putnam for thinking that some mathematical objects exist. Thus, it would be relevant to bunny's mathematical Platonism, but not to the question of whether a physical infinite exists.

Yeah, that may be my main problem. With…

A: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, … n …
B: 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, … n …

… I should be correlating 6 in A and 9 in B with 2?

No this is correct at the start. The physical act of moving all the guests "up one" is not a set operation though, it is a reassignation of the correlation between two infinite sets (guests and rooms) - as discussed previously it is one-to-one but it is not onto, this is not part of the requirement for two sets to have the same cardinality.

All of the rooms are full. Then one of the rooms isn't full.

This may not make sense to you, but that's because you're used to there being a certain number of rooms. If there are n full rooms, and one of those rooms is emptied, then there are now n-1 full rooms. Thus, someone must be displaced in order to empty a room.

That doesn't work because there is no n. There is no number of elements in an infinite set. You can't use the concept of number of elements because the set is unbounded, and any number of elements represents a bound. There is no answer to the question "how many rooms are there in the hotel?"

Thus, any reasoning that relies on the concept of "how many," as well as any reasoning that relies on the concept of a "last room," or on what happens when that "last room" is reached, does not applies. The reasoning that makes this type of thing impossible in a finite hotel falls under these categories. When we remove the restriction on the categories, we also remove the impossibility of such processes.

I'm worried that you're taking the wrong point from my post. There is no significance to the meaning of the numbers in either set A or B (or with the set of natural numbers). For instance, set A could be : {1, 1, 1, 1, 1,...} and set B: {2, 2, 2, 2, 2,...}. All we are doing if we put these into one-to-one correspondence is taking the first 1 from set A and drawing a line to the first 2 in set B, then drawing a line from second 1 in set A to the second 2 in set B, then drawing a line from the third 1 and so on. It doesn't matter what "1" or "2" stand for, all that matters is that they refer to discrete elements of the respective sets.

If we take the set of all natural numbers, we have a set with an infinitely large number of elements. By looking at the behavior of this set, we can identify the properties of the infinite. One of the properties of this set we've discovered is that the cardinality of some subsets is the same as the cardinality of the set as a whole. We discover this by doing the one-to-one correspondence bit. But this isn't really meant to tell us what the cardinality of the infinite set is--rather it tells us that whatever it is, it is the same for both sets, because for each element in set A we can draw a line to an element in set B.

Since this property is unique to infinite sets, it is used as a definition of the infinite set.

I'm working on it:

X: the Hilbert Hotel is an actual infinite - counting numbers (1,2,3,4…).
Y: an actual infinite of rooms - odd-counting numbers (1,3,5,7…).
Z: an actual infinite of guests - even-counting numbers (2,4,6,8…).

X has no vacancy (is full) iff Y and Z.
X has vacancy iff Y and ¬Z.

LOL!

It wouldn't even surprise me if someone who isn't even as old as a teenager knows the answer given the current widespread accessibility of good resources.

No worries. Thanks to the patience of Bunny and yourself, (and both of your obvious commitment to the pure dialectic in controversial/polemic issues), I think I'm getting it in my own 'mathematical ebonic' sort of way. So, I'll retract my assertion that set theory leads to a logical contradiction. I was wrong. What I deemed a logical contraction within set theory, is a conflict between identity theory and set theory when applied to an actual infinite.

(fwiw: my metaphysics isn't tied to this issue one way or the other, so I'm just kind of exploring it without any sort of bias. I don't consider Craig to be a philosophical dolt, like some do, so if he's hanging his hat so to speak on this particular point, personally, I consider that merit for some exploration or consideration. But at the end of the day, from a theist perspective, if I have to side with 'expert opinion' I'd probably stick with the philosophical rigor of Plantinga and Aquinas who both affirm that we can't conclude an actual infinite is impossible. So, if an actual infinite is possible I'm just trying to figure out why its possible, as in a necessarily not refuter, from a logical standpoint.)

Full hotel is possible in the following cases: Y and Z, ~Y and ~Z
Vacancy is possible in the following cases: Y and Z, Y and ~Z, ~Y and ~Z

Since Z -> Y (assuming all guests are accommodated), ~Y and Z is an impossible condition.

Basically what the Hilbert Hotel says is that we can have a one-to-one correlation here:

So here, every "guest" has a "room." But what if we add one extra guest?

Even though we added an extra guest (3), there is still a one-to-one correlation. By adding one extra guest we don't change the one-to-one correlation - there is still exactly one guest for each room.

Now, imagine that instead of 3 being an extra guest, 3 is a vacancy.

One-to-one correlation, still. Now every guest still has a room and every room has exactly one guest - except room #2, which has a vacancy.

I'm thinking before we can add a guest, necessarily:
Which implies that room 2 correlates with a null.

aside: it's interesting where you initially started off. Got me to thinking in a quantum logical sense: 1/2 vacancy and 1/2 no vacancy. Or, the Quantum Herbert Hotel may have vacancy AND may not have vacancy.

Remember the definition. There are all kinds of correspondences, the fact you can find ones you didn't expect. If there is a one-to-one correspondence then the sets are of the same size.

I don't think this is actually a one-to-one correlation, right (although we could put these two sets into a one-to-one correlation)? 2 isn't correlating with anything here...

I think the way that duffe should be thinking about one-to-one correlation is that it is just a method for comparing the difference in cardinality between two different sets. If it is possible to put two sets in a one-to-one correlation, then the two sets have the same cardinality, if it is not possible, then they don't. Thus, the fact that we could find different correlative relationships between these two sets wouldn't be relevant--that would just be another way of looking at these sets.

So the point about the Hilbert Hotel is that if you add more guests to the full Hotel, and then compare the number of rooms to the new number of guests--it is still the same. We can prove this by putting the amount of guests we have now in a one-to-one correlation with the amount of rooms. If this wasn't the case, then it wouldn't be the Hilbert Hotel, just a really big hotel.

As I understand it, the basic claim being made by Craig is that while this definition of the infinite works in math, it is metaphysically impossible because there is something absurd about there being a physical object or series with the properties of infinite sets.

The problem with this claim is that you can't show this by appealing to either math or logic (since, as admitted by Craig, Hilbert's Hotel is not logically or mathematically contradictory). So you have to rely on your intuitions about what is physically possible to support this claim. However, as noted by bunny, our intuitions about what is physically possible derive from our experience of finite sets, and since we know (by definition) that actual infinite sets would not have the same properties of finite sets, then these intuitions just don't seem very useful in answering the question of whether an actual infinite is metaphysically possible.

Suppose we have an actual infinite set of humans with an actual infinite set of hands as a proper subset. Now because both sets are actual infinites, they necessarily have the same cardinality. Obviously, we'll run into trouble maintaining a 1-1 correspondence if hand 1 corresponds to human 1, hand 2 to human 2, hand 3 to human 3, etc… The only way I'm able to maintain a 1-1 correspondence in reaching a 'completed totality' is accepting hands-without-humans or positing humans-without-hands.

That depends on how we manage the physicality of the actual Hilbert Hotel. We could either remove the occupant from the room and then put the guest in, or we could put the guest in (having 2 people in the room) and then remove the original occupant. But the process can make it complex, since a physical process works through time - when we start switching rooms we create a kind of "wave" of switching occupants that will perpetuate forever.

Well, I'm trying to tie in the idea of an actual hotel with the representations of these series and their "sizes." I don't know how we'd mathematically represent a "vacancy."

Yeah, he seems very focused on "how" the guests get "moved." But maybe I'm just confusing the issue.

Again, there is no trouble placing the infinite set of human hands in a one-to-one correspondence with the infinite set of humans. First, you draw a line from the first human to the first hand, then from the second human to the second hand, then from the third human to the third hand, and so on.

I've not been exactly sure what you mean by your use of "totality" (or "completed totality") in this thread, but it seems to me that you are thinking of the infinite set of humans (or hands) as a totality in the sense that there is some number how large it is. There isn't--that's why it is an infinite set. So there is no need to posit humans without hands or hands without humans. The one-to-one relationship works because even though there are two hands per human, you never run out of humans with which to place them in the one-to-one correspondence.

from wiki:

Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set.

I was thinking 'completed' is used to distinguish between an actual infinite and a potential infinite, in that the latter can be added to (n + 1) and hence it is not completed.
Yeah, I get we won't run out of humans. Maybe the verbiage in the wiki definition with "can form an actual, completed totality," is throwing me off, but with the humans/hands scenario it seems like, "cannot possibly form an actual, completed totality," is more fitting.

I haven't looked over the page, but in my experience - every time I know a lot about a topic, the wikipedia page is not quite right. In maths, definition is everything - without knowing exactly what totality/completed/etcetera means we really can't evaluate if this is true/false/meaningless.
Also, note that their use of 'actual infinite' is almost certainly not consistent with the definition of an actual infinite which I provided earlier - my attempt to encapsulate what I have presumed NotReady and Craig mean by an actual infinite.

Again, in your thinking you must be clear what 'adding' to a set is - the only thing we can do to sets is take unions, complements, intersections, etcetera. Your reference to (n+1) is worrying - if you label the members of an actual infinite (as I've defined it) with their corresponding natural number 1,2,3,... there is no 'biggest' element. As Original Position mentioned a couple of posts ago - there is no number corresponding to the 'size' of an infinite set - it doesn't get bigger when a finite number of extra elements are also included in it (via unions or somesuch).

Hmmm...let me try to be a bit more precise. Here is the definition of a set given by Cantor:

So here's the question, is there a set of all the natural numbers? If you say yes, then there is an actual infinite set (in the Aristotelian sense, not in the sense of physically existing). To say that it is "completed" or "actual" in this case is just to say that it is a set. Aristotle's potential infinity is not a whole and so cannot be understood in set-theoretic terms. Then, if we accept that this is a set, then we can prove that the cardinality of this set is the same as the cardinality of the set of all even natural numbers, i.e. we can prove that Hilbert's Hotel is logically coherent.

The problem with your objections has been that you seem to be claiming that a proper subset of an infinite set cannot have the same cardinality as the set as a whole (or that Hilbert's Hotel is not coherent). But this result is a theorem. So if you want to disagree with it, then you'll have to go back earlier and show which axiom you think is false, or how the proof is invalid.

In other words, in order to make your claim sensible you have to either reject the possibility of infinite sets or reject the methods of set theory more generally.

There is another question also, which is whether an actual infinite can be physically instantiated. Craig's actual view seems to be that while actual infinite sets are mathematically correct they in some way are metaphysically impossible, but this doesn't seem to be yours.

My understanding of an 'actual' infinite is that the members of the set actually exist, hence the set is complete. Whereas a 'potential' infinite is created by counting - there's no actually existing 10 until 9+1, no 11 until 10+1, ad infinitum - hence a potential infinite is not complete. So, with actual infinites I'm not thinking n+1; I'm thinking that one actual infinite set has the same number of members as another actual infinite set: an infinite number.

My point with the actual infinite humans, human/hand scenario is:
I don't find it weird that: | humans | = | pair of hands |;
I do find it weird that: | humans | = | hands |;
And really weird that: | pair of hands| = | hands |.

I get what you said earlier about never running out of humans to correlate hands 1-1 with. But that strikes me as saying that the justification for a 1-1 correlation of two actual infinites is that we'll never run out of correlates, not that the members of each set actually do correlate 1-1.

Well, I'm not sure where I'm at with this right now. The conclusions are weird and counter-intuitive, but thinking about it they're not more weird than what relativity and QM are telling us. I mean, at one point time dilation would have seemed 'absurd,' so just because it seems that way, doesn't mean it can't possibly be mathematically real or physically actual. On the other hand, at the moment I'm not quite sure what I need to accept or conclude to say an actual infinite is mathematically real and metaphysically possible.

Trouble is, I can conceptualize an actual infinite existent as a boundless whole. For example, I can imagine space as a boundless and actually existing whole and I've given quite a bit of thought to God in a like manner. It's when I try to conceptualize that whole as a composition or sum of members as with a set, that's causing me issues. In other words, I'm not sure if the boundless can have proper parts, constituents, members, etc… There's a reason why theologians conclude that God is simple (non-composite) and I'm not sure if the same reasons wouldn't apply to an actual infinite.