Question About Hilbert's Hotel

[rossum]

How is that process different from saying 1 2 3 4 5 is greater than 1 3 5?

You are very obviously misusing terminology here. The “greater” operator is not defined between sets, so you should not use it in this context. All it does is to show your lack or relevant knowledge.

You should say: “{1 2 3 4 5} has a greater number of elements than {1 3 5}”. Note also the use of braces, “{ … }” to delimit a set.

Sure, real numbers are an infinity all of their own, exponentially so.

Oh dear. You do not know the correct meaning of “exponentially” either. Please, for your own good, learn more mathematics before continuing this discussion. Your own feet will thank you for it; they need the time for all those bullet holes to heal.

The issue is the seeming lack of a demonstration that all odd numbers have a one to one correspondence to all the natural numbers

False. The one-to-one correspondence has been demonstrated a number of times by myself and others. Here it is again:

Natural    Odd
-------    ---
   1        1
   2        3
   3        5
   4        7
   5        9
       ...
   n       2n-1
       ...

Is there any natural number missing from the first column? Is there any odd number missing from the second column? Does any natural number not have a corresponding odd number? Does any odd number not have a corresponding natural number?

If you answer “no” to any of those questions, then provide an example.

rossum

 

[thinkandmull]

You can red herring your way around with my choice of words but this is just nonsense nothing. I guess most mathematicians don’t really know what infinity is. They can’t see the forest for the trees, literally.

 

[rossum]

You can red herring your way around with my choice of words but this is just nonsense nothing.

Mathematics is very precise in its use of words. In a mathematical statement the correct use of words is not a red herring.

I asked you some questions about the one-to-one correspondence I showed you. You have not answered them. Does that mean that you now accept that both the natural numbers and the odd numbers are in a one-to-one correspondence?

I guess most mathematicians don’t really know what infinity is.

You guess wrong.

rossum

 

[thinkandmull]

Just because you can keep finding one to one correspondences doesn’t mean those sets are equal in size. My points have been clear, you know what I mean in my posts. Try to graph those two sets, like geometry. As I showed, it shows your wrong. It should make sense in geometric form likewise, no?

And you throw in 2n-1 as if it adds to your argument. It makes me wonder if you are trying to fool people. 2 times 2 minus 1 is three. So??

 

[thinkandmull]

And Euclid and Pythagoras weren’t atheists? How do you know? Did they even teach what you are defending?

 

[rossum]

Just because you can keep finding one to one correspondences doesn’t mean those sets are equal in size.

Yes it does mean precisely that. Every element of one set has a corresponding member in the other set, with no members of either set left over.

My points have been clear, you know what I mean in my posts.

I think I know what you mean, but your terminology is imprecise, and I may have misunderstood.

And you throw in 2n-1 as if it adds to your argument. It makes me wonder if you are trying to fool people. 2 times 2 minus 1 is three. So??

That is the correspondence between the integer 2 and the odd number 3. For every integer, follow the rule “double it and subtract one” to get the corresponding odd number.

rossum

 

[thinkandmull]

What’s left over is 6 thru 10. Is this not possible at least?

 

[thinkandmull]

All you’ve done is found an irrelevant proportion.
1 1
2 3
3 5
4 7
5 9

really should be:

1 1
2
3 3
4
5 5
6
7 7
8
9 9
10

 

[thinkandmull]

This is just like when people say they have an special algorithm to get people together thru a dating site. It doesn’t mean anything.

 

[rossum]

All you’ve done is found an irrelevant proportion.
1 1
2 3
3 5
4 7
5 9

really should be:

1 1
2
3 3
4
5 5
6
7 7
8
9 9
10

No. You ignored the mapping n → 2n-1. Again you are wrong. If you ignore part of an argument, then it is not surprising that you do not follow it.

rossum

 

[Tomdstone]

You can red herring your way around with my choice of words but this is just nonsense nothing. I guess most mathematicians don’t really know what infinity is. They can’t see the forest for the trees, literally.

The mathematics of 1-1 correspondence is not “nonsense nothing” which would be apparent to someone who understands the concept and who has taken and passed a few college math courses.

 

[Rhubarb]

1 1
2
3 3
4
5 5
6
7 7
8
9 9
10

What you are showing is how the two sets differ in their definition. The definition of the left hand set is just n, while the definition of the right hand set is 2n-1. The “gaps” you see are because those even numbers are by definition excluded from the right hand set. They don’t play a part of it at all. There is no 1-gap-3-gap-5. It’s just 1-3-5-… Think about how one-to-one correspondence works. If you have the same number of things in one hand as the other, are the sets of things the same size? Of course they are. If you have a different number of things in each hand, then the sets are not the same size - there are left-overs.

 

[Usagi]

What’s left over is 6 thru 10. Is this not possible at least?

But they’re not left over. They are only left over if you are only talking about the finite sets of natural and odd numbers up to 10. Everyone acknowledges that THOSE two sets have different numbers of elements. But we aren’t talking about those. We are talking about the complete, infinite sets of natural and odd numbers.

Yes, it’s true that the set of natural numbers contains every element of the set of odd numbers PLUS MORE (all the even numbers). That is trivially true. We all know that. The weird, counter-intuitive (but demonstrably true) thing is that both sets (and also the set of even numbers, considered separately) are the same “size” or cardinality.

The lists and formulae we’ve been presenting aren’t irrelevant nonsense. They are how the proof works. Every natural number n can be used to generate a corresponding even number by the formula 2n and a corresponding odd number by the formula 2n-1. If we look at the sets so generated in the light of the definitions of natural, even, and odd numbers, we notice that when the formulae are applied to each natural number in order (1,2,3,etc.), they produce every even number in order (2,4,6,etc.) and every odd number in order (1,3,5, etc.) respectively. And while we cannot actually perform the calculation for EVERY number because all three sets are infinite, there is no finite number for which simple multiplication and subtraction suddenly stop working, so we can conclude that those patterns extend as far as we care to take them. And if, for every natural number, there exists a unique corresponding odd number and a unique corresponding even number, without skipping any, then the three sets must be the same “size,” however weird that seems.

Yes, the set of odd numbers will never contain the element “6”; that’s true by definition. But that is a red herring when it comes to the problem in question. We are not concerned with whether the sets contain all the SAME elements (as they manifestly do not) but whether they contain the same number of elements. And we can show, by the above method, that they must.

Usagi

 

[Tomdstone]

Yes, the set of odd numbers will never contain the element “6”; that’s true by definition. But that is a red herring when it comes to the problem in question.

You can red herring your way around…

Red herrings or even silver herrings are not relevant to the concept of countable infinity.

 

[Michael19682]

Perhaps I have no business here, but I found this on the net. Surely someone has commented on it earlier and it is definitely relevant:

Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the “reals.” (This set comprises all numbers that can be represented as a decimal, even if that decimal representation is infinite in length. Hence, 27 is a real number, as is π, or 3.14159….)

to be clear on my reference:

scientificamerican.com/article/strange-but-true-infinity-comes-in-different-sizes/

Fact is, it can be explained as the proverbial “hogwash” applying a little philosophy.
Care to know how?
All I ask is that if you know what I would say, say it yourself and get on with it.

 

[Michael19682]

Let is say that if we use the concept of an idea, as in Plato’s Republic, we make the argument, or at least part of Hilbert’s Hotel moot. In our philosophy, “3” is an idea which does not change under any circumstances. It cannot be said that three elephants is purely “3” any more than three giraffes is “3”. In fact, only God knows what “3” is. But for the sake of argument here, suppose we say that the number of reals between the naturals “3” and “4” is infinite; and furthermore, that each of these reals is represented by a single person with his own unique participatory “idea” of the number “3”. To one person belongs 3.1415 and to another 3.1416 etc. Of course, with each imagination or participation, the number of “child” processes is also infinite every time we add a decimal number. We shall satisfy this by assuming infinitely large families extending infinitely toward heaven in their several generations (“random or otherwise”, we should say for bravado). The idea to make clear is that the parent process of all “3”'s is RELATED TO or PERVADES NATURALLY all his children, and each of these children in turn to infinity pervades their child processes. The gut of the matter is, all these children and grandchildren and “quadrillion children” are human approximations of “3”; just as no three elephants are the same three elephants, no three boats, no three cars, etc.
Additionally, the idea of “3”, along with all the other numbers, is an idea of some parent among ideas. If we are Catholic we acknowledge that ideas come from our participation in the life of the spirit.
We use ideas to express ourselves, most importantly in prayer. Unfortunately, we also are given false ideas by the tempter, the Devil.
To say that God alone knows the size of the universe is possibly true. The Devil might know it also.
The Devil is a spirit, and hence has knowledge.
The point is that we cannot prove what God reveals in the “inexpressible groaning of the spirit” Rm 8:26?
If we follow [the Devil’s] logic, be that as it may, we might arrive at the correct size of the universe or answer about these questions of numbers – but how shall we know in that case if the truth itself is an intentional falsehood of the Devil?
What conclusions do you draw?

 

[Michael19682]

Furthermore, to directly challenge the video’s final assertion: if the universe can be consumed by men for eternity, a world without end, day after day, infinitely transformed by infinitely increasing/arriving life, then it cannot have had a beginning because something must remain to sustain the life, which can go on for eternity.
The argument of the video rests in the well developed infinite set theory, and in the failure to ascertain an actual infinity.
Scripture holds that of the day or hour no one knows except the Father. So therefore there is an end. But this end means among other things that earthly consumption through new arrivals must stop, it does not prove ? that the supply is exhausted. As long as the supply remains, we cannot know if there ever was a beginning --** provided it wasn’t all formed simultaneously**, which the account of creation we have veers away.
Lastly, and here I’ve probably tripped and fallen already, but the hotel is used as a model in infinite set theory so as to disprove infinity. It’s value toward the end of modeling is in** the application of a single process rule**, ie, moving everyone one door down, and basing the falsity of infinity on that rules working. If time, which is continuous, is the true real, then there must be no gaps in its flow; which is to say that the arrival of the guests must not interrupt the set of rooms with an empty room or set of rooms. What if the universe is a clustered oscillation such that it is like families arriving in bunches with** no repeating pattern to the bunching since the stream is infinite in length?** By what “sole rule of shifting” could the proprietor of Hilbert’s Hotel prevent a discontinuity in time (ie, leave no vacant room) if to accommodate the families or clusters a contiguity or continuity of rooms must be maintained? Thus, if God randomly generates the universe the proprietor would have no way of knowing how to shift guests to accommodate families in adjacent rooms? This is analogous to saying that human or other expansive life doesn’t expand continuously, but in spurts according to available energy and resources, which is random insofar as we can’t see God’s providence with perfection, and which does not disprove the continuity of time because when you are counting to infinity, how fast you count is not important.

 

[thinkandmull]

But they’re not left over. They are only left over if you are only talking about the finite sets of natural and odd numbers up to 10. Everyone acknowledges that THOSE two sets have different numbers of elements. But we aren’t talking about those. We are talking about the complete, infinite sets of natural and odd numbers.

Yes, it’s true that the set of natural numbers contains every element of the set of odd numbers PLUS MORE (all the even numbers). That is trivially true. We all know that. The weird, counter-intuitive (but demonstrably true) thing is that both sets (and also the set of even numbers, considered separately) are the same “size” or cardinality.

The lists and formulae we’ve been presenting aren’t irrelevant nonsense. They are how the proof works. Every natural number n can be used to generate a corresponding even number by the formula 2n and a corresponding odd number by the formula 2n-1. If we look at the sets so generated in the light of the definitions of natural, even, and odd numbers, we notice that when the formulae are applied to each natural number in order (1,2,3,etc.), they produce every even number in order (2,4,6,etc.) and every odd number in order (1,3,5, etc.) respectively. And while we cannot actually perform the calculation for EVERY number because all three sets are infinite, there is no finite number for which simple multiplication and subtraction suddenly stop working, so we can conclude that those patterns extend as far as we care to take them. And if, for every natural number, there exists a unique corresponding odd number and a unique corresponding even number, without skipping any, then the three sets must be the same “size,” however weird that seems.

Yes, the set of odd numbers will never contain the element “6”; that’s true by definition. But that is a red herring when it comes to the problem in question. We are not concerned with whether the sets contain all the SAME elements (as they manifestly do not) but whether they contain the same number of elements. And we can show, by the above method, that they must.

Usagi

“we cannot actually perform the calculation for EVERY number because all three sets are infinite”. That’s where your flaw here. I don’t take math classes because I know the principles, and finding another formula there and another there is just tedious. I’ve read Euclid for example and I don’t have an interest in doing Ptolemy. But if you two have taken the classes you should be able to prove your case, which you haven’t. The fact remains that the two lines of numbers go on forever, but one is a part, not equal to, the other. Your position leads to a denial of the truth of math, especially geometry. You speak of “proofs” in the plural, but you’ve only provided one, and the correspondence it shows is irrelevant to this discussion. Poor logic

 

[thinkandmull]

Furthermore, to directly challenge the video’s final assertion: if the universe can be consumed by men for eternity, a world without end, day after day, infinitely transformed by infinitely increasing/arriving life, then it cannot have had a beginning because something must remain to sustain the life, which can go on for eternity.
The argument of the video rests in the well developed infinite set theory, and in the failure to ascertain an actual infinity.
Scripture holds that of the day or hour no one knows except the Father. So therefore there is an end. But this end means among other things that earthly consumption through new arrivals must stop, it does not prove ? that the supply is exhausted. As long as the supply remains, we cannot know if there ever was a beginning --** provided it wasn’t all formed simultaneously**, which the account of creation we have veers away.
Lastly, and here I’ve probably tripped and fallen already, but the hotel is used as a model in infinite set theory so as to disprove infinity. It’s value toward the end of modeling is in** the application of a single process rule**, ie, moving everyone one door down, and basing the falsity of infinity on that rules working. If time, which is continuous, is the true real, then there must be no gaps in its flow; which is to say that the arrival of the guests must not interrupt the set of rooms with an empty room or set of rooms. What if the universe is a clustered oscillation such that it is like families arriving in bunches with** no repeating pattern to the bunching since the stream is infinite in length?** By what “sole rule of shifting” could the proprietor of Hilbert’s Hotel prevent a discontinuity in time (ie, leave no vacant room) if to accommodate the families or clusters a contiguity or continuity of rooms must be maintained? Thus, if God randomly generates the universe the proprietor would have no way of knowing how to shift guests to accommodate families in adjacent rooms? This is analogous to saying that human or other expansive life doesn’t expand continuously, but in spurts according to available energy and resources, which is random insofar as we can’t see God’s providence with perfection, and which does not disprove the continuity of time because when you are counting to infinity, how fast you count is not important.

"What if the universe is a clustered oscillation such that it is like families arriving in bunches with no repeating pattern to the bunching since the stream is infinite in length? "

Why can’t there be a pattern in an infinity?

If time always was, it would be a denial of the present reality to say you can’t add one to an infinite series

 

[Tomdstone]

I don’t take math classes …

That may be a problem.