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The solution to induction
Posted: 29 June 2011 05:10 AM   [ Ignore ]   [ # 46 ]
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dougsmith - 29 June 2011 03:57 AM
Write4U - 28 June 2011 08:29 PM

I was thinking of Hilbert’s Hotel with infinite number of rooms and an infinite number of guests, yet he is always able to accommodate one more guest, merely by moving each guest into the next room. Thus the hotel can always accommodate an infinite +1 number of guests

Thus my musing if that would not translate into an alternating infinite number of even numbered and odd numbered rooms, by accommodating 1 more arriving guest.

“Infinite +1” is provably equal to infinity. That is, it’s of provably the same size. “Infinite +1” is not a meaningful number. Indeed, that was precisely the point of Hilbert’s proof: an infinitely large hotel can accommodate any finite number of additional guests (or even a countably infinite number of guests) without expanding the hotel. That is to say, a countable infinity plus a finite number equals a countable infinity. And a countable infinity plus another countable infinity equals a countable infinity.

For more on this see the Wiki on Hilbert’s ‘paradox’ (which isn’t a real paradox in the logical sense).

Ok, I understand that, but what about the odd/even problem? Can a countable infinity be odd or even and under what circumstance?

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Posted: 29 June 2011 06:00 AM   [ Ignore ]   [ # 47 ]
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W4U, infinity is not a number, it is a concept, therefore you cannot perform mathematical operations on infinity.

From my computer’s online dictionary, in mathematics infinity is:

a number greater than any assignable quantity or countable number

Trying to use infinity in calculations is as meaningless as dividing by zero. Indeed, when scientists get infinities as answers they know they have a problem with their theories.

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Posted: 29 June 2011 06:21 AM   [ Ignore ]   [ # 48 ]
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Write4U - 29 June 2011 05:10 AM

Ok, I understand that, but what about the odd/even problem? Can a countable infinity be odd or even and under what circumstance?

To understand that you really have to understand that notions of “odd” or “even” are meaningless when talking about infinities.

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Posted: 29 June 2011 06:27 AM   [ Ignore ]   [ # 49 ]
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DarronS - 29 June 2011 06:00 AM

W4U, infinity is not a number, it is a concept, therefore you cannot perform mathematical operations on infinity.

It’s an interesting question, Darron, but I don’t think I’d quite put it that way. You can perform mathematical operations on infinity, but the answers you get are strange.

So a countable infinity plus any countable number equals a countable infinity. A countable infinity plus a countable infinity equals a countable infinity. These are mathematical operations, and provable, but counterintuitive.

A countable infinity times any countable number equals a countable infinity. (A countable infinity plus a countable infinity is just a countable infinity times two).

Not sure about a countable infinity times a countable infinity. That might get you to the first range of uncountable infinities.

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Posted: 29 June 2011 06:30 AM   [ Ignore ]   [ # 50 ]
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The problem with your logic, Doug, is that infinity is, by definition, beyond anything countable, therefore there is no such thing as a countable infinity.

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Posted: 29 June 2011 06:49 AM   [ Ignore ]   [ # 51 ]
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DarronS - 29 June 2011 06:30 AM

The problem with your logic, Doug, is that infinity is, by definition, beyond anything countable, therefore there is no such thing as a countable infinity.

question

I don’t know where you’re getting that from. It’s not from the mathematics. See, e.g., Cantor’s Theorem.

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Posted: 29 June 2011 07:13 AM   [ Ignore ]   [ # 52 ]
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Cantor’s Theorem allows that the counting will never end when counting an infinity. Mathematicians and philosophers get themselves all tizzied up trying to play with infinity, but no one has ever been satisfied with such exercises. The problem is, as I mentioned earlier, infinity is not a number, it is a concept. Treating infinity as a number leads to all sort of paradoxes, such as the power set of a countably infinite set being uncountably infinite. The problem is in the term “countably infinite,” for if a set is infinite it will take an infinite time to count the members of the set, therefore the set is not countable.

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Posted: 29 June 2011 07:21 AM   [ Ignore ]   [ # 53 ]
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DarronS - 29 June 2011 06:00 AM

W4U, infinity is not a number, it is a concept, therefore you cannot perform mathematical operations on infinity.

From my computer’s online dictionary, in mathematics infinity is:

a number greater than any assignable quantity or countable number

Trying to use infinity in calculations is as meaningless as dividing by zero. Indeed, when scientists get infinities as answers they know they have a problem with their theories.

And while your dictionary is right about the assignment of a quantity, many infinite things are countable. You can count the natural numbers - just start counting. You’ll never stop, but you will be counting. Cantor (if I remember correctly) was particularly clever with counting infinite things.

I would agree in the case of arithmetic operations, but “calculation” usually means more than that. Certainly, as I know you know, infinity is used commonly in calculations such as integration, infinite limits (of the intermediate form (infinity/infinity), and infinite series. For example, the limit (as x approaches c) of f(x) = infinity. And the infinite series is defined as, Sum of a sub k (from k=0 to k=infinity) = l (that’s a lower case el).

Also, it is often the case that scientists hope to get infinity as an answer - or, at least one of the infinite number of infinities.

Here are the first two chapters of a cool book called A Brief History of Infinity (if you haven’t already read it)... LINK

Edit to add: I see some other posts came up before I posted this - ignore repetition.

[ Edited: 29 June 2011 07:23 AM by traveler ]
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Posted: 29 June 2011 07:21 AM   [ Ignore ]   [ # 54 ]
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DarronS - 29 June 2011 07:13 AM

Cantor’s Theorem allows that the counting will never end when counting an infinity. Mathematicians and philosophers get themselves all tizzied up trying to play with infinity, but no one has ever been satisfied with such exercises. The problem is, as I mentioned earlier, infinity is not a number, it is a concept. Treating infinity as a number leads to all sort of paradoxes, such as the power set of a countably infinite set being uncountably infinite. The problem is in the term “countably infinite,” for if a set is infinite it will take an infinite time to count the members of the set, therefore the set is not countable.

I’m not sure what you mean by “paradoxes” in this context. These are proofs as well established as any in mathematics, and I know plenty of mathematicians (including those who taught me) who are quite satisfied with them. The results are strange and (in a sense) counterintuitive, but that’s like saying that quantum mechanics is “paradoxical” so we should not take its results seriously.

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Posted: 29 June 2011 07:23 AM   [ Ignore ]   [ # 55 ]
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OK, it seems I need to study infinity a bit more before I go off making up my mind about things. Thanks for the online education, I’ll start with the book traveler recommended.

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Posted: 29 June 2011 07:26 AM   [ Ignore ]   [ # 56 ]
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DarronS - 29 June 2011 07:23 AM

OK, it seems I need to study infinity a bit more before I go off making up my mind about things. Thanks for the online education, I’ll start with the book traveler recommended.

And if you understand it after researching, please explain it to me! Feynman admitted that he didn’t understand it, so that makes me smile.  cheese

Just because I used it in calculus, doesn’t mean I ‘get’ it…

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Posted: 29 June 2011 07:30 AM   [ Ignore ]   [ # 57 ]
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From what little I know of Calculus mathematicians and scientists use limits to approach infinity, not reach it. From what I’ve read of Feynman’s works, he was motivated to develop QED to eliminate infinities in quantum theory. Is it deductive reasoning when a scientist decides to develop a new theory simply because he doesn’t like the answers the old one gave? Or is that neither inductive nor deductive?

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Posted: 29 June 2011 08:29 AM   [ Ignore ]   [ # 58 ]
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DarronS - 29 June 2011 07:23 AM

OK, it seems I need to study infinity a bit more before I go off making up my mind about things. Thanks for the online education, I’ll start with the book traveler recommended.

If you can stomach the formalisms, any intro book on Foundations of Mathematics, or of logic in a math department, should have quite a bit in it about infinities. E.g., usually these books will begin with the construction of the natural numbers, which are a countable infinity, and then onto the real number line which is infinitely dense and hence uncountable.

Cantor’s continuum hypothesis is that there is no infinite set that lies between the naturals and the reals.

It can then be proven that there are sets (indeed, an uncountably infinite number of them) that are larger than the set of real numbers. Hence the real number set is only the first (smallest) uncountable set, assuming the continuum hypothesis. That gets into a discussion of the cardinality of infinite sets, where cardinality is standardly formalized by a set’s aleph number. Natural numbers are aleph zero, reals are at aleph one.

[ Edited: 29 June 2011 08:34 AM by dougsmith ]
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Posted: 29 June 2011 10:21 AM   [ Ignore ]   [ # 59 ]
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DarronS - 29 June 2011 07:23 AM

OK, it seems I need to study infinity a bit more before I go off making up my mind about things. Thanks for the online education, I’ll start with the book traveler recommended.

N.B., that book is really just an entertaining history book and the first two chapters are free to look at. I would recommend a math book. Specifically, I would recommend getting whatever calculus text you’re going to be using in school and get a start on it. Calculus concepts are for the most part very simple to understand. What gets most people is a weak foundation in algebra. If you have mastered algebra at the college level, then calculus is more fun than scary.

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Posted: 29 June 2011 12:59 PM   [ Ignore ]   [ # 60 ]
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DarronS - 29 June 2011 06:00 AM

W4U, infinity is not a number, it is a concept, therefore you cannot perform mathematical operations on infinity.

From my computer’s online dictionary, in mathematics infinity is:

a number greater than any assignable quantity or countable number

Is that not what I originally posited?

Infinity is always 1 (at least) larger than the largest (countable) number.

I stipulate that the number 1 may be debatable mathematically, but the idea was “larger than”.

[ Edited: 29 June 2011 01:02 PM by Write4U ]
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