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 September 10th, 2017, 09:14 AM #1 Member   Joined: May 2013 Posts: 36 Thanks: 1 are the real numbers truly more infinite than the rationals In Hilbert's hotel, he has a infinite number of guests, the whole numbers staying in an infinite number of rooms. An infinitely long bus comes along with infinite passengers, the rationals. To fit them in, he makes each guest in his hotel move to the room 2*room number. then he has an infinite number of buses containing an infinite number of people in each bus arrive. the polynomial numbers. (Bus 1 could be square root of all the rational numbers, bus 2 the cube root, bus 3, the fourth root, and so on.) To fit them in, he puts all the people in bus 1 in 2^bus seat number, all the people in bus 2 in 3^bus seat number and so on with all the primes. now the only thing left would be the non-polynomial numbers, or otherwise known as the transcendental numbers. Those also come in on an infinite number of buses with an infinite number of people. (pi, e, and so on.) to fit those in we put them in composite powers. bus 1 (2*3)^bus seat number, bus 2 (2*5)^bus seat number, bus 3 (2*7)^bus seat number, bus 4 (3*5)^bus seat number and so on. now all the numbers in the real set have been accounted for, haven't they? Last edited by skipjack; September 10th, 2017 at 09:34 AM.
 September 10th, 2017, 09:30 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2662 Math Focus: Mainly analysis and algebra You are assuming that each new set is countable before you try to accommodate it in the hotel. The transcendental numbers are not countable.
 September 10th, 2017, 10:50 AM #3 Senior Member   Joined: Jun 2015 From: England Posts: 915 Thanks: 271 The study of this subject starts with Cantor's Continuum Hypothesis. (Which might be considered a strange name considering what it actually says) Continuum Hypothesis -- from Wolfram MathWorld
September 10th, 2017, 11:08 AM   #4
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 Originally Posted by studiot The study of this subject starts with Cantor's Continuum Hypothesis. (Which might be considered a strange name considering what it actually says) Continuum Hypothesis -- from Wolfram MathWorld
Aren't the real numbers uncountable regardless of the truth value of CH?

How would CH have any relevance to the OP's question?

Last edited by Maschke; September 10th, 2017 at 11:17 AM.

 September 10th, 2017, 12:00 PM #5 Senior Member   Joined: Jun 2015 From: England Posts: 915 Thanks: 271 I said it's where this started. I didn't say its the be all and end all of the subject. And I'm happy for Phillip to decide for himself whether it is of interest or not. Last edited by studiot; September 10th, 2017 at 12:04 PM.
September 10th, 2017, 01:10 PM   #6
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 Originally Posted by studiot I said it's where this started.
But that's false, historically and factually. Historically, Cantor's first proof of the uncountability of the reals was published in 1874; and he proposed CH in 1878. And factually, the reals are uncountable regardless of the status of CH or any assumption you could make about its truth value.

https://en.wikipedia.org/wiki/Georg_...theory_article

https://en.wikipedia.org/wiki/Continuum_hypothesis

Quote:
 Originally Posted by studiot I didn't say its the be all and end all of the subject.
I'm just wondering why you brought it up at all, since it's totally unrelated to the OP's question of whether the reals are indeed uncountable.

Quote:
 Originally Posted by studiot And I'm happy for Phillip to decide for himself whether it is of interest or not.
Why not point him to the Hilbert Nullstellensatz and let him decide if it's helpful? That has as much relevance to his question as CH.

If you misspoke yourself I don't mean to poke you. If you're just a little confused, I'd be glad to explain it. But why double down on an error?

The reals are uncountable. Cantor gave three proofs. CH is the statement that there's no uncountable cardinal strictly between the naturals and the reals. Whether we take it as true or false (which we are free to do since it's independent of the other axioms of set theory) the reals are still uncountable.

Last edited by Maschke; September 10th, 2017 at 01:22 PM.

September 10th, 2017, 02:09 PM   #7
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 The reals are uncountable. Cantor gave three proofs. CH is the statement that there's no uncountable cardinal strictly between the naturals and the reals. Whether we take it as true or false (which we are free to do since it's independent of the other axioms of set theory) the reals are still uncountable.
Of course they are, and I have not said otherwise.

Equally the issue of whether there is an uncountable cardinal between the naturals and the reals only makes sense if the reals are uncountable in some way to make their cardinality greater than that of the naturals, just as Phillip asked.

On the same day that I offered Phillip this prompt towards his investigation, I suggested the phrase 'Thermal Resistance in relation to electrical theory' to someone else on another site.
Within a few hours I had a reply saying that was just the right phrase for the questioner to research and he had thus been able to complete the solution of his problem.

In my experience questioners don't often ask their questions in the nice tight phraseology that textbook and exam questions are (hopefully) written and rarely say exactly what they mean.

September 10th, 2017, 02:29 PM   #8
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 Originally Posted by studiot On the same day that I offered Phillip this prompt towards his investigation, I suggested the phrase 'Thermal Resistance in relation to electrical theory' to someone else on another site. Within a few hours I had a reply saying that was just the right phrase for the questioner to research and he had thus been able to complete the solution of his problem.
Exactly right. You have expertise and insight into applied math that enables you to get to the heart of a question in that area. My point exactly.

Last edited by Maschke; September 10th, 2017 at 02:31 PM.

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