Maths question

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 Joined: Sat Oct 18, 2014 12:49 pm
Maths question
When I was 12 or 13, I had a conversation with a physics professor. He asked me whether there were more real numbers between 0 and 1 or more whole numbers greater than zero. I said "the same". He argued that there were more real numbers between 0 and 1 than whole numbers greater than zero. His "proof" was that you divide the set of real numbers between 0 and 1 in two and each half has an infinite number of real numbers. And you can do this infinite times. As to whole numbers greater than zero, take any range and divide by two and each half has only a finite number of whole numbers, therefore there are infinitely more real mumbers between 0 and 1 than whole numbers greater than zero. My reaction was: but take any real number in your set and read it right to left and it exists as a whole number, so they're the same.
My memory of his rationale may be hazy, but it's a question I've thought of from time to time over the intervening 27 years, with a bit of smugness that I've been right this whole time (it only matters cuz that prof is my dad). Any mathematicians out there able to set me straight?
My memory of his rationale may be hazy, but it's a question I've thought of from time to time over the intervening 27 years, with a bit of smugness that I've been right this whole time (it only matters cuz that prof is my dad). Any mathematicians out there able to set me straight?
Re: Maths question
math.stackexchange.com
Amazing. AMAZING. A.M.A.Z.I.N.G! site.
Edit: I reckon you must understand the difference between countably infinite and uncountably infinite to appreciate the difference between the two ways of counting to infinity.
Amazing. AMAZING. A.M.A.Z.I.N.G! site.
Edit: I reckon you must understand the difference between countably infinite and uncountably infinite to appreciate the difference between the two ways of counting to infinity.
Re: Maths question
I'm afraid the physics professor is correct. The integers are countably infinite while the real numbers are uncountably infinite. The classical proof is Cantor's diagonalization:
https://en.wikipedia.org/wiki/Cantor%27 ... l_argument
https://en.wikipedia.org/wiki/Cantor%27 ... l_argument

 Posts: 418
 Joined: Sat Oct 18, 2014 12:49 pm
Re: Maths question
DAMNIT!
This stays between us...
This stays between us...
Re: Maths question
The matter is also rendered more clear if you correspond the infinite set of whole numbers with an infinite set of symbols. Decimal notation is arbitrary so produces artifact. Something like that...
Re: Maths question
Here's a cool example. This is MY preferred system. YMMV.

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 Joined: Sat Mar 21, 2015 5:56 pm
Re: Maths question
Your rationale sounds good but 0.7, 0.07, 0.007,... would all be 7 under your approach.
That's what came to my mind.
Edit: oops. My mistake. You're reading right to left. So they would be 7, or 70 or 700. I was hoping to find a way to avoid grappling with Cantor's proof but I failed.
That's what came to my mind.
Edit: oops. My mistake. You're reading right to left. So they would be 7, or 70 or 700. I was hoping to find a way to avoid grappling with Cantor's proof but I failed.

 Posts: 418
 Joined: Sat Oct 18, 2014 12:49 pm
Re: Maths question
After four, five hours of research, I still cannot wrap my head around this problem.
After reading http://www.coopertoons.com/education/di ... ument.html and watching https://www.youtube.com/watch?v=mEEM_dLWY0g I still don't understand why Cantor's diagonal argument cannot be set up the other way round as well, i.e. in the example set up by coopertoons adding +1 to a natural number that previously has not been part of the natural matching set.
I think I don't understand the concept of different countabilities, which might be related to the axiom of choice (which I haven't dug into yet):
https://en.wikipedia.org/wiki/Uncountab ... _of_choice
Bluntly put, if you can add numbers via Cantor's diagonal argument by +1 on decimal places for the irrational numbers, why is the same not true for adding +1 to the set of natural numbers. Saying the +1 in the set of natural numbers should have been included before seems flawed, as the same could be said for any new irrational number we come up with. But then again, I seem to be missing something big.
Did anyone find themselves trapped in the same line of thinking and knows a way to escape this misleading thought?
After reading http://www.coopertoons.com/education/di ... ument.html and watching https://www.youtube.com/watch?v=mEEM_dLWY0g I still don't understand why Cantor's diagonal argument cannot be set up the other way round as well, i.e. in the example set up by coopertoons adding +1 to a natural number that previously has not been part of the natural matching set.
I think I don't understand the concept of different countabilities, which might be related to the axiom of choice (which I haven't dug into yet):
https://en.wikipedia.org/wiki/Uncountab ... _of_choice
Bluntly put, if you can add numbers via Cantor's diagonal argument by +1 on decimal places for the irrational numbers, why is the same not true for adding +1 to the set of natural numbers. Saying the +1 in the set of natural numbers should have been included before seems flawed, as the same could be said for any new irrational number we come up with. But then again, I seem to be missing something big.
Did anyone find themselves trapped in the same line of thinking and knows a way to escape this misleading thought?
Re: Maths question
Because if you increment a rational number digit, you create a duplicate number that appears somewhere else in the sequence, which was already accounted for. But when you increment a real number digit, you create a distinct number that wasn't already accounted for.
Another way of saying it is that there is a 11 mapping from steps of a loop that iterates through discrete steps, to the set of integers. But there is no such 11 mapping from loop steps to real numbers.
Another way of saying it is that there is a 11 mapping from steps of a loop that iterates through discrete steps, to the set of integers. But there is no such 11 mapping from loop steps to real numbers.