Question from a Texan Textbook

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Question from a Texan Textbook

#1  Postby The Plc » Sep 09, 2015 10:44 am

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Re: Question from a Texan Textbook

#2  Postby Thomas Eshuis » Sep 09, 2015 11:17 am

The mathz it hurts.
Translation please? :shifty:
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Re: Question from a Texan Textbook

#3  Postby Nicko » Sep 09, 2015 11:57 am

It's a very weird - but obvious - corollary of the fact that all integers are rational but not all rational numbers are integers. The more usual way of looking at it would be to say that integers ("whole numbers") are a subset of rational numbers (any number that can be made by dividing two integers).
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Re: Question from a Texan Textbook

#4  Postby Newmark » Sep 09, 2015 12:10 pm

But it's factually wrong. There is a one-to-one correspondence between the integers and the rationals, as they are both countable sets of the same cardinality (aleph-0)*. Infinities may be a bit counter-intuitive, as a set may contain as many elements as a proper subset of said set, but anyone who can't be bothered to look this up has no business writing a math book.

For anyone who's interested, wiki on countable sets is a good place to start...

[EDIT] *Or rather, it's the other way around, but the cardinality of the sets should given you a hint even if you don't understand the proof.
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Re: Question from a Texan Textbook

#5  Postby Evolving » Sep 09, 2015 12:18 pm

However, it’s not correct. The set of rational numbers is countably infinite, which is the same thing as what the question is asking, and you can count them, for example, diagonally:

- you arrange the rational numbers in each interval of 1 around its middle point, like this: ½, 1/3, 2/3, ¼, ¾…, leaving out the ones like 2/4 where numerator and denominator have a common factor; the same for the next interval up (1½, 1 1/3, 1 2/3, 1¼…) , the next interval down (- ½ etc), and so on alternating between positive and negative intervals
- and you count like this: the first rational number from the central interval between 0 and 1; the first number from the next interval up, then from the next interval down; then back to the central interval for the second number there, followed by the second number in the next interval up (from 1 to 2) and down (from -1 to 0), followed by the first number in the intervals outside them (from 2 to 3 and from -2 to -1); then back to the central interval for the third number and so on, ad infinitum.

Obviously you’re going to get to any rational number you care to name eventually, and for any integer there’s going to be a specific rational number assigned to it.

EDIT: Ninja'ed by newmark.
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Re: Question from a Texan Textbook

#6  Postby scott1328 » Sep 09, 2015 12:23 pm

Fundamentalists have a problem with advanced mathematics.

http://boingboing.net/2012/08/07/what-d ... ntali.html

And we all know that fundamentalists dominate the text book selection in Texas.
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Re: Question from a Texan Textbook

#7  Postby Evolving » Sep 09, 2015 12:32 pm

Actually I missed out a bit, because you'd never get round to counting the integers themselves with my algorithm. But never fear: just insert the integers into the algorithm such that, whenever you count the first fraction in an interval (27½, for instance), you first count the integer at the lower bound of that interval (27).
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Re: Question from a Texan Textbook

#8  Postby Newmark » Sep 09, 2015 12:36 pm

scott1328 wrote:Fundamentalists have a problem with advanced mathematics.

http://boingboing.net/2012/08/07/what-d ... ntali.html

And we all know that fundamentalists dominate the text book selection in Texas.

:this:

Indeed. It seems very much like the kind of error that Craig does when he tries to argue that "actual infinities" cannot exist (which he must do to support his version of the KCL). Hausdorff's bible blogg has a better write-up on that subject than I can manage :)
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Re: Question from a Texan Textbook

#9  Postby Greyman » Sep 09, 2015 1:37 pm

scott1328 wrote:Fundamentalists have a problem with advanced mathematics.

http://boingboing.net/2012/08/07/what-d ... ntali.html
:what: ... o.kay. :doh: I hadn't quite realised that the problem they had with it was that they thought modern math theories were evil. :crazy:
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Re: Question from a Texan Textbook

#10  Postby Shrunk » Sep 09, 2015 2:13 pm

Newmark wrote:
scott1328 wrote:Fundamentalists have a problem with advanced mathematics.

http://boingboing.net/2012/08/07/what-d ... ntali.html

And we all know that fundamentalists dominate the text book selection in Texas.

:this:

Indeed. It seems very much like the kind of error that Craig does when he tries to argue that "actual infinities" cannot exist (which he must do to support his version of the KCL). Hausdorff's bible blogg has a better write-up on that subject than I can manage :)


That guy seems to really know his shit. He does a great job of breaking down the bafflegab that apologists rely upon to give their arguments a veneer of intellectual respectability. e.g. This response in the comment section:

Perhaps the difficulty here is that infinity can mean different things, and it is being used in different ways here. When we talk about infinite regress we mean one thing, when you say "God is infinite" you mean something completely different.

"Things that change are not infinite, for infinites cannot increase into something they do not already have, or else they would not be infinite in the first place."

This is a perfect example, you seem to be saying that infinities cannot grow, but in certain circumstances infinities can grow. Consider a line in a plane, it can be infinitely long but have no width. It is reasonable to say that the line is infinite, but it certainly has room to grow. I could thicken it up and we would have an infinitely long ribbon. It has changed, it has grown, and it was infinite before and it still infinite.

Now, I'm sure that when you are talking about infinite thing that can't change, you do not have this kind of infinite in mind, you are talking about something else entirely. That is what I'm trying to get at.
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Re: Question from a Texan Textbook

#11  Postby Nicko » Sep 09, 2015 2:53 pm

Newmark wrote:But it's factually wrong. There is a one-to-one correspondence between the integers and the rationals, as they are both countable sets of the same cardinality (aleph-0)*. Infinities may be a bit counter-intuitive, as a set may contain as many elements as a proper subset of said set, but anyone who can't be bothered to look this up has no business writing a math book.

For anyone who's interested, wiki on countable sets is a good place to start...

[EDIT] *Or rather, it's the other way around, but the cardinality of the sets should given you a hint even if you don't understand the proof.


:picard:

Fuck.

You're right, of course.

You are never going to "run out" of integers to assign to rationals.

:oops:

Another reminder - though I shouldn't need one - that "obvious" conclusions are usually wrong when dealing with infinite sets.
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Re: Question from a Texan Textbook

#12  Postby scott1328 » Sep 09, 2015 11:31 pm

The set of natural numbers is a proper subset of the integers which is in turn a proper subset of the rationals. The rationals are considered countable because there is a one-to-one mapping of the natural numbers to the rationals and vice versa. And not, merely because the natural numbers never "run out"

On the other hand there is no two-way one-to-one mapping for rational numbers (or natural numbers) to the real numbers, even so, the natural numbers are a proper subset of the rationals which are indeed a proper subset of the real numbers.

Furthermore, the real numbers are a proper subset of the complex numbers, but there exists a one-to-one two way mapping of the real and complex numbers. Hence these two sets are said to the same size, which size, aka cardinality, is said to be C.
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Re: Question from a Texan Textbook

#13  Postby crank » Sep 10, 2015 12:31 am

Real numbers don't exist, at least aren't real as in 'in the real world'. For arbitrary real, you can't name it, or even point to it in any way. It has infinite precision, so it contains infinite information. Ideas of Gregory Chaitin, fascinating guy, tends to being quite animated.



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Re: Question from a Texan Textbook

#14  Postby scott1328 » Sep 10, 2015 12:55 am

crank wrote:Real numbers don't exist, at least aren't real as in 'in the real world'. For arbitrary real, you can't name it, or even point to it in any way. It has infinite precision, so it contains infinite information. Ideas of Gregory Chaitin, fascinating guy, tends to being quite animated.




Nonsense.

Real numbers exist every bit as much as the number one. One is in fact a real number.
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Re: Question from a Texan Textbook

#15  Postby crank » Sep 10, 2015 1:02 am

Please read a little closer.
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Re: Question from a Texan Textbook

#16  Postby scott1328 » Sep 10, 2015 1:07 am

crank wrote:Please read a little closer.

Same to you
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Re: Question from a Texan Textbook

#17  Postby crank » Sep 10, 2015 1:10 am

I'll elaborate. Think of every number you can enumerate, write down, give a formula for, single out in any way, even point to, in the [0 1] number line. Take a magic wand that can select a point on that line, just slice through the line, you choose a point. You have zero probability of ever slicing at one of those numbers.

Saying real numbers don't exist, that is for arbitrary x in real.
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Re: Question from a Texan Textbook

#18  Postby crank » Sep 10, 2015 1:14 am

scott1328 wrote:
crank wrote:Please read a little closer.

Same to you


If this interests you, go hear Chaitin, I'm no mathematician, didn't claim to be one, did't even claim I think he's right, I said it's an idea from an interesting guy. I think I understand most of what he discusses, but it ain't the easiest shit to get your head around, and I am not a mathematician.
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Re: Question from a Texan Textbook

#19  Postby scott1328 » Sep 10, 2015 1:33 am

How many vowels in this sentence?

If you can answer that question then you have demonstrated the existence of a real number.
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Re: Question from a Texan Textbook

#20  Postby crank » Sep 10, 2015 1:54 am

It isn't that no real exists, the class doesn't exist, as in, as I said earlier, some arbitrary x in reals. Not being a mathematician, I doubt seriously I'm saying it properly, but it's understandable. Why bug me, it's Chaitin who says this, listen to him, see if you can understand. It's obvious you're not even trying to understand what I'm saying.
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