Errors From Cantor to Hilbert

I will make a brief comment on the paradox called Hilbert's Hotel, since it has gotten a fair amount of attention recently. Hilbert imagines a hotel with infinite rooms. On one night, the hotel is found to be full. Hilbert then asks, "what if someone else comes to reception, begging a room?" What do we do? This paradox has been used to justify various types of transfinite math. Entire forests have been felled commenting on the mathematical implications of this paradox. However, it can be dismissed with one comment. If the infinite hotel is full, then no one can come begging a room. They are all already in the hotel. It is a contradiction in terms to imagine that you can add one to infinity in the first place. Where did the one come from? How is it possible for a quantity to be off an infinite number line? The simple and direct answer is that it is not possible. You cannot add one to infinity, because an infinite set is a complete set. An infinite set is complete in the fullest sense, meaning that there exists nothing outside the set. If you have an infinite set of people, then all people are in that set. You cannot postulate another person. Hilbert's Hotel is not a paradox, it is a very bad logical mistake, from the first paragraph. It is based on the same terrible mistake that underlies all transfinite math. The mistake is believing that the word "transfinite" can mean something. What it means in practice is really "transinfinite." Mathematicians believe that something can exist beyond infinity.
      If you accept the addition of 1 to infinity, then it means that you don't understand infinity to begin with. All the math that takes place in the transinfinite is quite simply false. Notice that I do not say it is physically baseless, or mystical, or avant garde, or any other half-way adjective. It is false. It is wrong. It is a horrible, terrible mistake, one that is very difficult to understand. It is further proof that Modern math and physics have followed the same path as Modern art and music and architecture. It can only be explained as a cultural pathology, one where self-proclaimed intellectuals exhibit the most transparent symptoms of rational negligence. They are outlandishly irrational, and do not care that they are. They are proud to be irrational. They believe—due to a misreading of Nietzsche perhaps—that irrationality is a cohort of creativity. Or it is a stand-in, a substitute. A paradox therefore becomes a distinction. A badge of courage. A brave acceptance of Nature's refusal to make sense (as Feynman might have put it.) If we somehow survive this cultural pathology, the future will look upon our time in horror and wonderment. How did we ever reach such fantastic levels of intellectual fakery and denial, especially in a century steeped in the warnings of Freud to beware of just this illness?

and then a skeleton popped out

>>1
Your complaint seems slightly carriage before horse-ish.  I find it more disturbing that an infinite number of rooms can ever be considered "filled."

The problem is taking an object of the mind and pretending that it makes sense in a physical sense.

How is it possible for a quantity to be off an infinite number line? The simple and direct answer is that it is not possible. You cannot add one to infinity, because an infinite set is a complete set.

I don't understand at all what you're getting at, or what you mean by "complete".  Are you suggesting that we can't take the integers as an infinite set, because the set of naturals is "complete" and cannot be extended?
Can you take a power set of an infinite set?  That's transfinite math.  What about measure theory, or even simple integral calculus?  These disciplines rely on taking "measurements" of infinite sets, and have been proven effective when applied to the real world.  Is it just a coincidence that such manipulations of the infinite happen to apply to the real world?
You might want to look into finitism, a philosophy which I'm not too familiar with myself at the moment.
http://en.wikipedia.org/wiki/Finitism

>>5
When he says "complete" I think he means all things that could be put into the set are already included in the set.  His argument is if an infinite set of rooms are filled with an infinite number of "guests" - identified as all things that can approach the reception desk and request a room - then there can not be a mysterious late-coming guest, even if it previously was not identified.  This guest was already included in the set.

I believe it is specious but my argument is that an infinite set can always accommodate another infinite set, regardless of how encompassing/large one of the two is perceived.  The hotel, hence, should always be in a state of "becoming occupied" and the guests in a state of "checking in."  I don't deny the practicality of transfinite math just that it suffers from really bad metaphors/analogies.

my argument is that an infinite set can always accommodate another infinite set, regardless of how encompassing/large one of the two is perceived

I'm not sure how to interpret "accommodate".  It may seem like nit-picking but math is about precision.

No set has an injection into it from its own powerset.  For a more concrete example, there is no injection from the reals into the naturals.

If you want to critique something, you should first understand fully the thing you're critiquing and the arguments for it, in this case, set theory (cardinal arithmetic in particular).  Infinite sets meet the criterion of being sets in ZFC (and other axiom systems I'm sure), but they are merely abstract mathematical objects, like a group or a category.  How they can be correctly applied is what's really up for debate (as you said yourself, there are many bad analogies).

I recommend reading a set theory text and analyzing the arguments laid out there if you really want to understand what's going on.  A particularly good place to start is Derek Goldrei's text:
http://www.amazon.com/Classic-Theory-Independent-Chapman-Mathematics/dp/0412606100/

>>1
I call bullshit. Infinity plus one is just infinity.

Also isn't infinity people different than an infinite set of people?

Not an expert on formal math here, just intrigued.

>>8
Infinity plus one is just infinity.
This is for the most part correct.  To be more specific, there are different kinds of infinity.  An infinite cardinal plus one is equal to the same infinite cardinal, as you said.  An infinite ordinal, on the other hand, will be equal to a different infinite ordinal upon adding one.  The result is still infinite.

Also isn't infinity people different than an infinite set of people?
I would argue no.  Naïvely, a set is just a collection of things.  Is there a difference between an infinite number of people and a collection of an infinite number of people?  The former (to me) sounds like just taking the elements of the latter and grouping them together, which is how the set was defined in the first place.

>>9
Ok... but this still seems presumptuous:
An infinite set is complete in the fullest sense, meaning that there exists nothing outside the set.
Err.. the fact that because there are infinity people, that every person in existence must be included just doesn't sit well with me. Suppose the hotel were full with an infinite number of men. Couldn't a woman come asking for a room?

>>10
I completely agree.  There actually is no such set; i.e. no set of all sets.  The collection of all sets is what is known as a proper class (too big to be a set).  Otherwise, that set would be a member of itself, and that would lead to all sorts of Russelian paradoxes.  (If you're not familiar with Russel's paradox, just think:  does the set of all sets that do not contain themselves contain itself?)

You're joking right?

A hotel with infinite rooms?

Surely you see the paradox already, right?
No box can have infinite space, which is why the FINITE space within the box of a hotel will become full eventually.

The only way a hotel can have infinite rooms is for the hotel to be always making another room which would invalidate the prior assertion that the rooms would become full.

Someone didn't fully understand the depth of what "infinite" means. Seriously, a hotel "a box" can never be infinite. Which is why I say, if the space of our universe is as it is asserted to be then our universe is a box which would then mean that the space within our universe is finite. However, I would include that whatever space our universe is contained within may be infinite.

:/

maybe it's bigger on the inside

No box can have infinite space, which is why the FINITE space within the box of a hotel will become full eventually.
No anything can really have infinite anything. At least, not anything physical.

A fractal can have infinite one dimensional space though. And a three dimentional fractal can have infinite two dimensional space, surface area that is.

Anyway, my point is it's easy enough to postulate infinite space and even a model from which it arises so I don't see how that in and of itself is a paradox.

>>14
After rereading that maybe I'm onto something...

Replace "hotel" with a Koch snowflake and "guests" with line segments drawn by an infinitely fine pencil.

...Although that seems to lead to more problems. Food for thought anyway. It might make things easier to visualize.

>>15
VISUALIZE MY ANUS!

That's easy, suppose your infinite hotel has room's numbered 1, 2, 3, and so on, with each room currently occupied by its number.  Well, this is an infinite hotel that is full, but 0 doesn't have a room.  Infinity does not mean everything, just more than finitely many.

And this doesn't give a paradox because infinity should not be treated as a real number.

>>17
Why doesn't zero (and negative numbers and decimal numbers and imaginary numbers, etc.) have its own room?  That seems like an artificial constraint to your exercise.

I don't see the problem.  The postulation is a hotel of an infinite number of rooms; that means a hotel that never runs out of unoccupied rooms.  As soon as the hotel becomes "full" the vacancy of the hotel is known and the number of rooms becomes finite and countable.  Infinity can always be made into an artificially finite number - just pick a really big number far beyond your expectations - but you can never make a finite number into an infinite one - add 1, and 1 again, and 1 again, ad infinitum.  So, a hypothetical hotel with an infinite number of rooms can never be "full."  Even an infinite number of people can be housed in a hotel of an infinite number of rooms: for every person, there will always be a vacant room and as soon as that rule no longer applies you have proven that the hotel never possessed an infinite number of rooms in the first place.

A hotel is not in fact a mathematically rigorously defined object. As luck has it though, Hilbert's Hotel isn't a proof but an illustrative Gedankenexperiment.  You might say it appeals to a non-intuitive intuition.  Trying to infer mathematical truths from it is about as effective as trying to expand on modern quantum physics by picking nits on Schrödinger's cat-in-a-box.

>>16
Well, I'm sure it's been widened out significantly. I doubt it has infinite empty space though.

Gedanken, eh.