TM,So a thing cannot be a part of a whole and the whole at the same time.
What do you say to what I showed. Given the variable of time being both intrinsic to the needs of the shift and extrinsic to the hotel, Cardinality is moot where the guests arrive in an infinite series whose final term cannot be ascertained.There is a lot of Dunning-Kruger in this thread.
If thinkandmull is having trouble with the concept that the cardinality (the “size”) of the set of all odd numbers is the same as that of the natural numbers, what will he make of the proof that the set of all rational numbers (i.e. all numbers that can be expressed as the fraction of two natural numbers: 1/2, 3/7, 235673/256777654 are all rational numbers) is countably infinite and therefore the cardinality of that set is the same as the set of natural numbers, in spite of the fact that there are infinitely many rational numbers in the interval between any two rational numbers (the rational numbers are said to be dense rather than discrete). The same is true for the set of algebraic numbers (i.e. all numbers which can be expressed as the root of a finite, non-zero polynomial in one varable with rational coefficients). Indeed the cardinality of the set of rational numbers is the same as the cardinality of the set of any natural number raised to the power of all natural numbers (e.g. {13459^1, 13459^2, 13459^3 …} = {13459, 181144681, 243802621579 …} and this true no matter how big the base is.
Whereas the cardinality of the set of real numbers can be proven to be greater than these and that proof is ingenious and satisfying.
I am always amused and bemused in equal measure by people who casually consider some mathematical or scientific concept and who, failing to understand it, blame their misunderstanding on some elementary mistake that they believe has been made by the best minds who for decades have considered, accepted and enlarged on the concept, rather than blaming their misunderstanding on their own ignorance of the subject.
I think you misunderstand the concept of Hilbert’s Hotel which is *precisely *about the cardinality of certain infinite sets (to be exact, countably infinite sets). When discussing Hilbert’s Hotel, cardinality cannot be moot as this is the very point that the concept makes. Furthermore, time is irrelevant to the idea - the rearrangement of the guests in the hotel can be taken as instantaneous. If an infinite number of guests arrive at the hotel, they can all be accomodated.What do you say to what I showed. Given the variable of time being both intrinsic to the needs of the shift and extrinsic to the hotel, Cardinality is moot where the guests arrive in an infinite series whose final term cannot be ascertained.
1…1
3…3
4…4
5…5
3…3
6…6
8…8
7…7
random…random
With each guests arrival you could lengthen the shift to that many terms, but since you cannot know a repeat pattern nor know the last term, you can never accomplish the shift
I’m sorry - I don’t understand the point you are trying to make.What does the variable ‘random’ represent, a whole number? Yes. What number?
I have no idea why you think people need to answer the point you have made which seems to be based on misunderstandings both of the concept of Hilbert’s hotel and the point of the OP. No-one is suggesting that Hilbert’s hotel provides a proof that the Universe had no beginning. On the contrary, we were discussing the fact that William Lane Craig points to what he thinks is the absurdity of an actual Hilbert’s hotel as proof that there cannot be an infite time or an infinite series of events in the Universe. Others in the thread argued that Craig’s position fails because Hilbert’s hotel in not absurd on the face of it, as it does not defy reason or logic. The rest of the thread has then been occupied by people trying to explain to one contributor why the cardinality of any countably infinite set is the same - aleph null.I dare say that anyone who wants to post additional items must first answer what I’ve shown. (It renders the entire Hotel moot insofar as its capacity to prove no beginning is concerned.)
My point is that an infinite number of guests cannot be “taken instantaneously” when there is no pattern available for the proprietor to use for room shifts or as you call it “rearrangement”.I think you misunderstand the concept of Hilbert’s Hotel which is *precisely *about the cardinality of certain infinite sets (to be exact, countably infinite sets). When discussing Hilbert’s Hotel, cardinality cannot be moot as this is the very point that the concept makes. Furthermore, time is irrelevant to the idea - the rearrangement of the guests in the hotel can be taken as instantaneous. If an infinite number of guests arrive at the hotel, they can all be accomodated.
I’m sorry - I don’t understand the point you are trying to make.e
.
It doesn’t matter who supports your position. If you can’t disprove my position, than in this circle my position has the upper hand. 2n-1 was expressing the proportion between the two sets, but it doesn’t mean they are equal infinitiesTM,
I am truly amazed that a person that had a high SAT score for math cannot understand the concept of one to one correspondence and what it means.
Would you please answer one question for me, a simple math question: What does (2n-1) signify?.
Thanks
Yppop
Rational numbers, if they are all true numbers, are greater in number than natural numbers. I haven’t denied thatThere is a lot of Dunning-Kruger in this thread.
If thinkandmull is having trouble with the concept that the cardinality (the “size”) of the set of all odd numbers is the same as that of the natural numbers, what will he make of the proof that the set of all rational numbers (i.e. all numbers that can be expressed as the fraction of two natural numbers: 1/2, 3/7, 235673/256777654 are all rational numbers) is countably infinite and therefore the cardinality of that set is the same as the set of natural numbers, in spite of the fact that there are infinitely many rational numbers in the interval between any two rational numbers (the rational numbers are said to be dense rather than discrete). The same is true for the set of algebraic numbers (i.e. all numbers which can be expressed as the root of a finite, non-zero polynomial in one varable with rational coefficients). Indeed the cardinality of the set of rational numbers is the same as the cardinality of the set of any natural number raised to the power of all natural numbers (e.g. {13459^1, 13459^2, 13459^3 …} = {13459, 181144681, 243802621579 …} and this true no matter how big the base is.
Whereas the cardinality of the set of real numbers can be proven to be greater than these and that proof is ingenious and satisfying.
I am always amused and bemused in equal measure by people who casually consider some mathematical or scientific concept and who, failing to understand it, blame their misunderstanding on some elementary mistake that they believe has been made by the best minds who for decades have considered, accepted and enlarged on the concept, rather than blaming their misunderstanding on their own ignorance of the subject.
No. Wrong again. There are as many rational numbers as there are positive integers. Another one of Cantor’s elegant proofs. See here: An easy proof that rational numbers are countable.Rational numbers, if they are all true numbers, are greater in number than natural numbers. I haven’t denied that
This is quite irrelevant to the mathematics of transfinite numbers. In other words, your point is mathematically irrelevant.My point is that an infinite number of guests cannot be “taken instantaneously” when there is no pattern available for the proprietor to use for room shifts or as you call it “rearrangement”.
Nope - no idea what a “little volley of response” means in this context.If you like I’ll walk you through my position. But understand, it requires a little volley of response.
Nope - no idea what a “presiding function” is. All analytic functions are differentiable and all complex differentiable functions are analytic. Are you sure that you have the means to enter into this discussion on the mathematics of Cantor’s transfinites?as an opener practice shot, consider the piece wise graphs of calculus. there is no single presiding function that describes them,
Opening question:
He moves the existing guests into rooms (n+3) (so the guest in room 1 goes to room 4, the guest in room 2 goes to room 5. and so on) leaving rooms 1, 2 and 3 empty for the new guests. Simples.What rearrangement shall the proprietor make in the case of Hilbert’s Hotel where guests arrive in “sets of three” that must all be accommodated with contiguous rooms because they are families of three whom want suites, as the term goes?
How do you know that this is so?
- Eternal time may not be irrational. 2) eternal causation is
Yes. Simple indeed. Since you answered the question I thought most salient I won’t for now dwell on the others.He moves the existing guests into rooms (n+3) (so the guest in room 1 goes to room 4, the guest in room 2 goes to room 5. and so on) leaving rooms 1, 2 and 3 empty for the new guests. Simples.
But the families arrive always in three. Shouldn’t the guest in room 1 go to room 4, and the guest in room 2 go to room 8? etc to accommodate the entire infinite line of families?He moves the existing guests into rooms (n+3) (so the guest in room 1 goes to room 4, the guest in room 2 goes to room 5. and so on) leaving rooms 1, 2 and 3 empty for the new guests. Simples.
No not two. There are an infinite number of ways to accomodate families of an arbitrary number of x members. Families of x members can be accomodated by moving the existing guest in any room n to room n+x, in room n+1 to room n+1+x etc leaving x rooms free, starting at room n, for the family of x members. Note that both n and x can be any positive natural number. This can be done any number of times for an arbitrary number of families and can be continued indefinitely for an unending sequence of families of any number of members. The hotel will always be able to accomodate them, even though it is always full after each arrival.Yes. Simple indeed. Since you answered the question I thought most salient I won’t for now dwell on the others.
What if the guests arrive in and unending pattern of families of three, two and six members (3, 2, and 6) and still require rearrangement to keep suites? Simple again, right?
Hint: I see two way to accomplish the shift. It would help if you gave both.
ok. that point is good. edited .No not two. There are an infinite number of ways to accomodate families of an arbitrary number of x members. Families of x members can be accomodated by moving the existing guest in any room n to room n+x, in room n+1 to room n+1+x etc leaving x rooms free, starting at room n, for the family of x members. Note that both n and x can be any positive natural number. This can be done any number of times for an arbitrary number of families and can be continued indefinitely for an unending sequence of families of any number of members. The hotel will always be able to accomodate them, even though it is always full after each arrival.
Please start with the first guest in the hotel. It makes the conversation simpler and we can proceed. We are not challenging each other, but testing a hypothesis about the Hotel’s inability to accommodate and a particular type of infinite sequence. If you jump ahead, we can’t make progress.No not two. There are an infinite number of ways to accomodate families of an arbitrary number of x members. Families of x members can be accomodated by moving*** the existing guest in any room n*** to room n+x, in room n+1 to room n+1+x etc leaving x rooms free, starting at room n, for the family of x members. Note that both n and x can be any positive natural number. This can be done any number of times for an arbitrary number of families and can be continued indefinitely for an unending sequence of families of any number of members. The hotel will always be able to accomodate them, even though it is always full after each arrival.
what if the sequence is unending and generated randomly in time? how shall the proprietor know what shift to make for the last item in the series if it can only be determined by the random number generator?No not two. There are an infinite number of ways to accomodate families of an arbitrary number of x members. Families of x members can be accomodated by moving the existing guest in any room n to room n+x, in room n+1 to room n+1+x etc leaving x rooms free, starting at room n, for the family of x members. Note that both n and x can be any positive natural number. This can be done any number of times for an arbitrary number of families and can be continued indefinitely for an unending sequence of families of any number of members. The hotel will always be able to accomodate them, even though it is always full after each arrival.
Only if an infinite set of families of three arrive together. In which case guests in room n are moved to room 4n leaving an infinite set of three contiguous rooms, {4n-3, 4n-2, 4n-1} for any n to accomodate the infinite set of families of three.But the families arrive always in three. Shouldn’t the guest in room 1 go to room 4, and the guest in room 2 go to room 8? etc to accommodate the entire infinite line of families?
I haven’t jumped ahead. Hilbert’s hotel starts with an infinite number of rooms containing an infinite number of guests. The hotel is full. That is the starting point, We proceed from there.Please start with the first guest in the hotel. It makes the conversation simpler and we can proceed. We are not challenging each other, but testing a hypothesis about the Hotel’s inability to accommodate and a particular type of infinite sequence. If you jump ahead, we can’t make progress.
When x new guests arrive he makes x rooms free according to the schemes we have already discussed.what if the sequence is unending and generated randomly in time? how shall the proprietor know what shift to make for the last item in the series if it can only be determined by the random number generator?
What variable ‘X’ would he use?