Does a circle have a countable infinite number of points?

Steven G

Steven G

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I watched this great video on the Banach-Tarski Paradox and have a question about something the author said. It starts at 8:02 in the attached video. He says that the number of points around the circumference of a circle is countable. Can this be shown? Just because he starts to number the points does not mean that he will hit every point on the circumference. After all, we can start to count the reals between 0 and 1 but will miss some (actually many).
Thanks!!!
 
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That can't be correct. There's clearly a bijection between points on the circumference and the interval [MATH][0,2\pi)[/MATH]That interval is not countable.
 
Wait a minute. You can map points onto the circumference to the number from 0 to 2pi*r which is uncountable!
 
Romsek said:
That can't be correct. There's clearly a bijection between points on the circumference and the interval [MATH][0,2\pi)[/MATH]That interval is not countable.
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I just thought about that just before I saw your post!
So that part of the video is nonsense?
 
Actually he did not say the points were countable. He said the set was infinite.
It is true that every infinite set contains a countably infinite subset.
 
isn't an unmeasurable set at the heart of the paradox?

Topology isn't my forte but I thought that the set of pieces they use to assemble the multiple copies is unmeasurable which is why they can get away with it.
 
Romsek said:
isn't an unmeasurable set at the heart of the paradox?

Topology isn't my forte but I thought that the set of pieces they use to assemble the multiple copies is unmeasurable which is why they can get away with it.
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Yes, that is why they can get away with it.
pka said:
Actually he did not the points were countable.He set the set was infinite.
It is true that every infinite set contains a countably infinite subset.
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Yes, he did say what you said and boy does that make a big difference. Thanks for pointing this out!
 
Jomo said:
Starting at 11:13 he starts to talk about labelling points on the surface of a sphere. He says from a fixed point on the surface that he can get to points on the surface and without allowing backtracking (no UD, RL ...combinations) never arrive at the same point twice. This confuses me a bit. What if you keep going right (or up, down, left)? Would you not arrive at the starting point eventually?
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This is just spherical coordinates with a fixed radius.
 
Next question.

Starting at 11:13 he starts to talk about labelling points on the surface of a sphere. He says from a fixed point on the surface that he can get to points on the surface and without allowing backtracking (no UD, RL ...combinations) never arrive at the same point twice. This confuses me a bit. What if you keep going right (or up, down, left)? Would you not arrive at the starting point eventually?
 
Romsek said:
This is just spherical coordinates with a fixed radius.
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That response flew right over my head. Once I learn something I have it down nicely but it has to be explained to me s l o w l y.
Seriously, I did not understand your response, so can you please try again. Thanks
 
spherical coordinates is just travel down an arc a given arc length \(\displaystyle \theta\) and then travel along the latitude line if you will a given amount \(\displaystyle \phi\)

that gets you to whatever point on the sphere you like.
 
Counting infinity...this is why everybody thinks that mathematicians are crazy.
 
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