Question About Hilbert's Hotel

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thinkandmull said:
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.
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You’re going to have to elaborate, please. Where is the leap? How does that relationship not mean that every natural number has exactly one corresponding odd number (and even number) and vice versa? And how does THAT not mean that the three sets are the same size?
Further, all the odd numbers will forever,** going up to **infinity, be a PART of the whole (odd plus even)
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Yep. The set of all odd numbers and the set of all even numbers are proper subsets of the set of all natural numbers, both being contained within the latter. Nevertheless, and quite counter-intuitively, all three sets are the same size.

I’m not saying it’s not weird as heck, just that it’s true.

Usagi
 
rossum said:
Which is irrelevant. Cantor believed that 2 + 2 = 4; do you think that his belief must automatically be wrong because of where he died?

rossum
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He’s been accused of trying to put things together that don’t go together and thus ruining his mind. Bertrand Russell said that after he wrote his Principles of Mathematics he could think as clearly on most other subjects. That suggests to me he too may have been thinking of things that were quite right.

Imaginary numbers, to give another example, may have use functionally, but that there is not square root of a negative number is basic math. To say otherwise is to be a mathematical relativist and then at that point why are we even having this conversation?
 
Usagi said:
You’re going to have to elaborate, please. Where is the leap? How does that relationship not mean that every natural number has exactly one corresponding odd number (and even number) and vice versa? And how does THAT not mean that the three sets are the same size?

Yep. The set of all odd numbers and the set of all even numbers are proper subsets of the set of all natural numbers, both being contained within the latter. Nevertheless, and quite counter-intuitively, all three sets are the same size.

I’m not saying it’s not weird as heck, just that it’s true.

Usagi
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As I point out on page 9 of this thread, the 2n-1 function doesn’t lead to a one to one correspondence.

The function makes you think of the correspondence as:
1 1
2 3
3 5
4 7
5 9

when in reality it still is:

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

See my point? 2n-1 is just a slight of hand
 
thinkandmull said:
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)
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I’m not sure how you or others will receive this analogy, but you might think of the set of odds and their one to one correspondence with the naturals – as taking place in a kind of “accelerated way”. 1 3 5 7 9 … generates greater numbers faster because of the presiding function y = y+2; initialize y at -1 and use y as an accumulator to generate next y in sequence. For the naturals y= -1; but y = y+1. With iteration of the function call, the co varying X → + infinity; parallel sets are generated and hence the 1 to 1 correspondence.
If you want to see the same kind of concept, look into the Fibonacci numbers.
 
Michael19682 said:
I’m not sure how you or others will receive this analogy, but you might think of the set of odds and their one to one correspondence with the naturals – as taking place in a kind of “accelerated way”. 1 3 5 7 9 … generates greater numbers faster because of the presiding function y = y+2; initialize y at -1 and use y as an accumulator to generate next y in sequence. For the naturals y= -1; but y = y+1. With iteration of the function call, the co varying X → + infinity; parallel sets are generated and hence the 1 to 1 correspondence.
If you want to see the same kind of concept, look into the Fibonacci numbers.
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Both sets would accelerate at the same time, one still being a part of the other. That’s what I meant at the start of this thread when I said the whole natural numbers have more “push forward” towards infinity than the odd numbers
 
George Gamow saws this proves that all points on a plane are equal to all the points on any segment. (He says you can similarly prove that all points in a cube are equal to all points on any segment):

"Suppose that the position of a certain point on the line is given by some number, say 0.75120386… We can make from this number two different numbers selecting even and odd decimal signs and putting them together. We get this: 0.7108… AND 0.5236…

Measure the distances given by these numbers in the horizontal and vertical direction in our square, and call the point so obtained the ‘pair-point’ to our original point on the line. In reverse, if we have a point in the square the position of which is described by, let us say, the numbers: 0.4835… and 0.9907, we obtain the position of the corresponding ‘pair-point’ on the line by merging these two numbers: 0.49893057 "

These seems like very arbitrary math to me. All the points on a cube can be stretch out infinitely towards the horizon. That cannot have a one to one correspondence with a simple segment. The lines between them cannot be drawn
 
thinkandmull said:
These seems like very arbitrary math to me.
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Look at the words “seems” and “to me” in there. I am not Christian so the three-in-one concept of the Trinity seems like very arbitrary theology **to me/].

Gamow was correct. You are not.

rossum**
 
thinkandmull said:
Both sets would accelerate at the same time, one still being a part of the other. That’s what I meant at the start of this thread when I said the whole natural numbers have more “push forward” towards infinity than the odd numbers
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I do see what you meant; but in set theory they do not really accelerate the same, because in countingit is the covariant, the loop iterator, that determines the two sequences, as it is involved in the f(x) = x + 1; or f(x) = 2x + 1.
For every increment of one unit X, the naturals increase by one set element and so do the odd numbers – the clue to understanding the theory is not to worry about the actual values of the elements per se, but to see them without judging that, and to see only their bare elemental sense, i.e. the number 1,000,000 is no more an element qua element than is the number 89. It is only when you compare values that you get into trouble.
 
thinkandmull said:
As I point out on page 9 of this thread, the 2n-1 function doesn’t lead to a one to one correspondence.

The function makes you think of the correspondence as:
1 1
2 3
3 5
4 7
5 9

when in reality it still is:

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

See my point? 2n-1 is just a slight of hand
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I do not see your point. Your lists show which elements of the two sets are the same, but that is not what we want to know. All we care about is the “count” of elements in each set. The fact that some of the elements are the same is actually a distraction.

Look at the finite sets {1,2,3,4,5} and {1,3,5,7,9}. They have some elements in common and some that are not, but they have the same number of elements because we can line them up one to one. And lining them up to count them doesn’t look like this (your method)–

1 1
2
3 3
4
5 5
7
9

But like this (my method)–

1 1
2 3
3 5
4 7
5 9

The process is a little different with infinite sets, because you can’t literally count them all the way to the end, but you can accomplish the same thing by showing that they necessarily line up one to one with the set of natural numbers. That is what the n to 2n and n to 2n-1 relations show. That is far from a red herring.

Usagi
 
Pink_Elephants said:
To any mathematically inclined person, I’ve recently come across Hilbert’s Hotel in a talk given by Christian philosopher William Lane Craig:
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Dear OP et al~
I wouldn’t waste my time on philosopher William Lane Craig who is a member of the Discovery Institute for Intelligent Design. The guy isn’t bright. Nuff said!
 
Usagi said:
I do not see your point. Your lists show which elements of the two sets are the same, but that is not what we want to know. All we care about is the “count” of elements in each set. The fact that some of the elements are the same is actually a distraction.

Look at the finite sets {1,2,3,4,5} and {1,3,5,7,9}. They have some elements in common and some that are not, but they have the same number of elements because we can line them up one to one. And lining them up to count them doesn’t look like this (your method)–

1 1
2
3 3
4
5 5
7
9

But like this (my method)–

1 1
2 3
3 5
4 7
5 9

The process is a little different with infinite sets, because you can’t literally count them all the way to the end, but you can accomplish the same thing by showing that they necessarily line up one to one with the set of natural numbers. That is what the n to 2n and n to 2n-1 relations show. That is far from a red herring.

Usagi
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You are forcing

1 1
2
3 3
4
5 5
7
9

into

1 1
2 3
3 5
4 7
5 9

completely arbitrarily. I am not stupid
 
rossum said:
Look at the words “seems” and “to me” in there. I am not Christian so the three-in-one concept of the Trinity seems like very arbitrary theology **to me/].

Gamow was correct. You are not.

rossum**

His argument is faulty because the points of line equal the points on one of the sides. He adds the other side, then reasons from there to every point in the square. Its a classic example of mathematically having your conclusion in your premise. But if you are unwilling to reason with me, that so be it
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The 2n-1 relation is a random crunching tool you are using to force the odd numbers back to line up with the odd plus even numbers. It doesn’t work
 
thinkandmull said:
You are forcing
1 1
2
3 3
4
5 5
7
9

into

1 1
2 3
3 5
4 7
5 9

completely arbitrarily. I am not stupid
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You may not be stupid but you are the most blatantly stubborn person I’ve run into since I was in the third grade when I tried to convince Wacky Smith that the moon was not made of green cheese. I have to confess that I must be stupid for trying to get you to see what is so obvious about the idea of "one to one correspondence " to so many other persons.
His [Gamow] argument is faulty because the points of line equal the points on one of the sides. He adds the other side, then reasons from there to every point in the square. Its a classic example of mathematically having your conclusion in your premise. But if you are unwilling to reason with me, that so be it
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And Gamow does several flips in his grave over this interpretation of his explanation of Cantor’s very elegant proof of the equality of points in n-dimensional space and the points on the real number line. Throw the book away, you’re wasting your time with it just like I’m wasting my time with this response.
Yppop
 
Instead of giving up, you could have showed us why you believe that the 2n-1 relation is anything but an arbitrary attempt to put numbers where they don’t belong
 
Gamow’s slight of hand is when he says “We can make from this number ( 0.75120386…) two different numbers selecting even and odd decimal signs and putting them together. We get this: 0.7108… AND 0.5236…” The way he selects the later two from the first is irrelevant to the proof, but makes it appear as if he is proving something by it. 0.75120386… on the line lines up with 0.75120386… any one of the sides, but not with all the points in the squares.

As I said before, take a ball (mentally), break it in half, then break one of the halves in two, and continue this process until you can’t go any further. Line these objects up from largest to smallest. Put your finger to the smaller side and ask “what is at the end”? This shows it MUST go on forever. So Cantor is saying that an infinite line, going out there actually forever, has a one to one correspondence to any simple segment. I disagree. Good for me 🙂
 
thinkandmull said:
Instead of giving up, you could have showed us why you believe that the 2n-1 relation is anything but an arbitrary attempt to put numbers where they don’t belong
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Are you serious, man? After all these attempts, you still think the 2n-1 thing is completely arbitrary?

All right, since rossum has quit the field, I will try one more time.

2n-1 (or just 2n for the even numbers) isn’t just something we pulled out of our nether regions. Nor is there a trick we are trying to pull on you.

If you take the natural numbers in order, multiplying each in turn by two and then subtracting one gives you the odd numbers, also in order. As long as no number is skipped or repeated in the first list, no number will be skipped or repeated in the second list. I am a mathematical layperson like you, and cannot construct proofs to show why multiplication and subtraction work the way they do, though I’m sure there exist people who can. But you’re a fellow who values common sense; think on what “odd number” means, and how you’ve always known multiplication and subtraction to work, and you’ll see that this pretty much has to be the case.

Therefore, because there is one unique odd number that can be paired with each natural number, the two sets have to be equal in size, though obviously not identical in membership. If they weren’t equal in size, there would be extra natural numbers that do not generate a unique odd number when 2n-1 is applied, or extra odd numbers that cannot be generated in that way. That’s why I keep asking you to show a case where that happens. In actual fact, even the even numbers that concern you so still pair with odd numbers when 2n-1 is applied.

Usagi
 
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