Question About Hilbert's Hotel

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yppop said:
I decided to listen to Craig:

His conclusion: Since an infinite number of events can’t exist, the number of events must be finite, therefore reality must have had a beginning.

His argument: An infinite number of events can’t exist because, the existence of an infinity is absurd; it is absurd because if an infinity of rooms are full how can you add another infinity of guests to it?

However, his argument is based on a really bad assumption, namely, that all the rooms in the infinite hotel are full. This assumption is bad because:

It is the nature of the infinite that it can’t be filled. This a concept that gives so much trouble for people like ThinkandMull who haven’t thought through this concept, but once grasped helps to clarify the ideas that seem paradoxical, when they are not if one accepts the rigorously defined set theory. I myself, join with the vast majority of the world’s mathematicians, physicists, and other scientifically trained persons, and accept the logic of set theory.

And I also believe that the universe is finite and had a beginning, but not because of Craig’s specious argument.
Yppop
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I agree with *everything *said in this post.
 
Furthermore Craig’s cosmological Kalam argument has been heavily criticised by several philosophers including Quentin Smith, Graham Oppy and Adolf Grunbaum. But the best analysis of Craig’s support for the second premise, in which he claims that an actual infinity is an absurdity, is that of Wes Morriston which is available on-line here: stripe.colorado.edu/~morristo/craig-on-the-actual-infinite.pdf
 
If “it is the nature of the infinite that it can’t be filled” absolutely, than how can one infinity be greater than another? If infinity means “absolutely everything” than there is no one infinity greater than another. A math book I have says there are more “shapes of curves” than points on a line. However, apply Cantor here. The book has pictures of some squiggly curves. Line up 5 of them with five points on a line, and then think how they go to infinity, and thus they are equal! In denying that odd numbers are a **part ** of natural numbers, Cantor has erased the concept of “larger infinity” all together
 
If there are specific areas of math i need to research in order to overturn my simply logical reasoning on this, do tell. But my opponents on here have been pretending that they have a simple answer or answers for me on this, when in reality my arguments have been successful. I always try hard to be honest intellectually and see what the other side has to say. Few people are willing to give the other side as fair a hearing as they give there own beliefs.
 
On Craig, the eternity of time is irrelevant to his case, although his specific argument is that no actual infinity exists (points on a line?) The question is how there could be an infinity of causes. Like a pendulum swinging back in forth from eternity, never having had a first position. How is that possible?
 
thinkandmull said:
If there are specific areas of math i need to research in order to overturn my simply logical reasoning on this, do tell.
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There are. We have.
But my opponents on here have been pretending that they have a simple answer
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Cantor’s maths is not simple. It does answer your questions, but you have to learn more mathematics first precisely because it is not simple. Start with basic set theory.

rossum
 
The book I have is One Two Three… Infinity!

Its not really a math book per se, but a science book, although its first chapter is on math.

It tries to prove that all the points in a solid are equal to all the points on a segment. When I get a chance I will quote his “proof”. I am not sure if the proof works or not. If it does, then a segment has as many points as an infinite line, because a solid can be stretched into an infinite line. Weird
 
thinkandmull said:
So you admit there is no simple answer to my objections?
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There are some things in mathematics that are not simple. For example, there is no simple proof of the Riemann Hypothesis.

Merely because you want something to be simple does not mean that it is simple. Cantor’s maths is not simple. You will need to do some work to understand it. If you do not want to do the work then I suggest that you keep quiet about a subject where your knowledge is insufficient.

rossum
 
You actually tried to provide the proof on here, but it didn’t work. If it takes lot more premises than have been provided on this thread, then that’s fine
 
thinkandmull said:
If “it is the nature of the infinite that it can’t be filled” absolutely, than how can one infinity be greater than another? If infinity means “absolutely everything” than there is no one infinity greater than another. A math book I have says there are more “shapes of curves” than points on a line. However, apply Cantor here. The book has pictures of some squiggly curves. Line up 5 of them with five points on a line, and then think how they go to infinity, and thus they are equal! In denying that odd numbers are a **part ** of natural numbers, Cantor has erased the concept of “larger infinity” all together
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TM,
I think it is admirable that you have George Gamow’s book “One, Two Three, Infinity”. I scanned the first chapter in my copy and unfortunately, don’t believe that his explanations are going to be simpler than what has been posted here in this thread by your adversaries.

Fortunately, his conclusions are the same as those presented here by a number of respondents. I hope you have a change of heart, because the magnificence of the infinity concept, especially the relationship between aleph (0) and aleph (1), between rational numbers and real numbers, between discrete space and continuous space, allows one to develop a model that depicts: the reality of hylomorphism; the presence of both material and spiritual substance; the omnipresence of God; the only plausible answer for what existed before the big bang and still exists beyond the universal border; the relationship between mind and matter; and a plausible explanation of life and the soul. I contend that a person cannot begin to understand those dichotomies without first understanding Cantor’s conclusions

Gamow was an outstanding scientist and writer. You could do well by him. Might want to try, " The Mystery of Aleph" ,by Amir Aczel. Unfortunately it isn’t on Kindle.

Yppop
 
I am going to talk to some people about George Gamow’s attempt to prove all points on a cube correspond to all points on any segment.

His proof about all segments having equal points seems to assumes that points line up the same whether straight or diagonal.
 
thinkandmull said:
A math book I have says there are more “shapes of curves” than points on a line. However, apply Cantor here. The book has pictures of some squiggly curves. Line up 5 of them with five points on a line, and then think how they go to infinity, and thus they are equal! In denying that odd numbers are a **part ** of natural numbers, Cantor has erased the concept of “larger infinity” all together
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You are misusing Cantor here and demonstrating that you have not understood the “one-to-one correspondence” idea.

We have not simply been lining up a few examples of two infinite sets and then baldly asserting that the matching-up continues to infinity. If that is what you have been understanding us to say, I completely see why you find our argument ridiculous.

The important thing that is missing in your example here, which I tried to emphasize in my last post, is that there must be some consistent relationship between the elements you are pairing up that allows you to say with confidence that such pairs will continue to exist no matter how far you go. For the natural numbers and the even numbers, the relationship of n to 2n provides that. For any member of the first set, there is a unique member of the second set that pairs with it, with no members skipped or repeated. THAT is why you can say that the two sets have the same cardinality. Your example of curves and points provides no such reason to conclude that the sets line up one-to-one. Indeed, I imagine that Gamow in the book you are reading will sooner or later provide a demonstration that the two sets are NOT of equal cardinality, like the set of natural numbers and the set of real numbers.

Usagi
 
yppop said:
I decided to listen to Craig:

His conclusion: Since an infinite number of events can’t exist, the number of events must be finite, therefore reality must have had a beginning.

His argument: An infinite number of events can’t exist because, the existence of an infinity is absurd; it is absurd because if an infinity of rooms are full how can you add another infinity of guests to it?

However, his argument is based on a really bad assumption, namely, that all the rooms in the infinite hotel are full. This assumption is bad because:

It is the nature of the infinite that it can’t be filled. This a concept that gives so much trouble for people like ThinkandMull who haven’t thought through this concept, but once grasped helps to clarify the ideas that seem paradoxical, when they are not if one accepts the rigorously defined set theory. I myself, join with the vast majority of the world’s mathematicians, physicists, and other scientifically trained persons, and accept the logic of set theory.

And I also believe that the universe is finite and had a beginning, but not because of Craig’s specious argument.
Yppop
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The two bold faced quotes taken together draw an incomplete picture of infinity.

An infinite set is full, in and of itself, but not** by** itself – which is why the Hilbert’s Hotel establishes a rule of shifting the guests over one room and thereby “locks and keys-in” another countable infinity, like the propagation in a nerve cell that will go on forever given its own infinite length under the right electrical conditions… It is no accident that Hilbert leaves no room vacant; else there would be a break in infinity, and you would be talking about 2 hotels simultaneously as opposed to an instantaneous assumption of yet another countable infinity.
 
Usagi said:
You are misusing Cantor here and demonstrating that you have not understood the “one-to-one correspondence” idea.

We have not simply been lining up a few examples of two infinite sets and then baldly asserting that the matching-up continues to infinity. If that is what you have been understanding us to say, I completely see why you find our argument ridiculous.

The important thing that is missing in your example here, which I tried to emphasize in my last post, is that there must be some consistent relationship between the elements you are pairing up that allows you to say with confidence that such pairs will continue to exist no matter how far you go. For the natural numbers and the even numbers, the relationship of n to 2n provides that. For any member of the first set, there is a unique member of the second set that pairs with it, with no members skipped or repeated. THAT is why you can say that the two sets have the same cardinality. Your example of curves and points provides no such reason to conclude that the sets line up one-to-one. Indeed, I imagine that Gamow in the book you are reading will sooner or later provide a demonstration that the two sets are NOT of equal cardinality, like the set of natural numbers and the set of real numbers.

Usagi
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The “n to 2n” idea only shows a relationship between two sets, but doesn’t do away with the fact the there are 5 more numbers in 1 2 3 4 5 6 7 8 9 10 than 1 3 5 7 9
 
thinkandmull said:
The “n to 2n” idea only shows a relationship between two sets, but doesn’t do away with the fact the there are 5 more numbers in 1 2 3 4 5 6 7 8 9 10 than 1 3 5 7 9
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Of course there are, if you stop there. Obviously the set of odd numbers does not contain any even numbers, while the set of natural numbers contains both. And that does make it seem, on the surface, that the latter set must be larger,

Nevertheless, for every natural number n, there is an odd number (2n-1) that uniquely pairs with it. No matter what natural number you look at, there is such a corresponding odd number, and vice versa. The odd number that pairs with 10 is 19, which is not in your set that ends with 9 but is in the set of all odd numbers.

Unless you can demonstrate that “multiply by two and subtract one” stops yielding unique odd numbers at some point, there is one and only one odd number for each natural number, which means the two sets are the same size.

Usagi
 
You are connecting the “n to 2n” relationship as if it forces all the odd numbers to line up to all the positive whole numbers. Its a false logical leap.

Further, all the odd numbers will forever,** going up to **infinity, be a PART of the whole (odd plus even)
 
Cantor wrote: “I realise that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.”

What is so troubling that I side with his opponents?

According to Andy Boyd at the University of Houston, French mathematician-theoretical physicist-engineer-philosopher Henri Poincaré called Cantor’s work “a grave mathematical malady,” while Leopold Kronecker famously called Cantor a charlatan, a renegade, and a “corruptor of youth.”

“Leopold Kronecker was a German mathematician who worked on number theory and algebra. He criticized Cantor’s work on set theory” Wikipedia
 
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