Just to add - depends how you think about it. So for instance, a map of a continent, most people would say, is smaller than the continent. So surely it must have fewer points, you'd think?
(This answer was for original version of the question which said "amount" - which non mathematicians might interpret as either number or measure - now that it says "number" then this answer doesn't apply but leaving it in in case anyone is interested).
So for instance, most people would say that this map of Antarctica, on your computer screen:
Most people would say that this map is smaller than Antarctica.
But ask a mathematician which has most points, and he might say that the map has the same number of points as Antarctica - because he is thinking in terms of an ideal map, accurate in every detail.
So, then, to a pure mathematician, every point on the map corresponds to a point in Antarctica.
That's a bit like the way you can count pedestrians by drawing strokes on a sheet of paper, every time someone passes you. So long as you put one stroke on the paper for each pedestrian - that means you have the same number of strokes as you had pedestrians.
So in the same way so long as there is a point on the map corresponding to every point in the continent, then, looking at it that way, you must have the same number of points in the map as in the continent (or similarly in the country or city or whatever it is you have a map of).
So that's the sense in which there are as many numbers between 0 and 1 as there are larger than 1. By the simple mapping of x to 1/x, you can turn the section of the line from 0 to 1 into a kind of stretchy map of the section of the line from 1 to infinity.
Well, as it turns out, you can formalize this intuition that we have, that a map is smaller than a continent, by introducing rulers, basically. Some way of measuring the sizes of things.
That's not as easy as you might think though, because mathematicians want to talk about all sorts of wild collections of things.
If you just needed to measure line segments, that would be fine. Just measure the length of the line.
But what about, say, the collection of all decimals of the form
0.1011011011101....
any endless sequence of 0s and 1s - but this isn't in binary, this is a decimal number.
Or if you prefer, all decimals of the form
0.232332322323...
endless sequence of 2s and 3s, and no other numbers can be used.
How big is that set? Seems like there should be fewer points here than there are if you include everything - but how much smaller is it?
Also what about all the fractions between 0 and 1. You can find a ratio arbitrarily close to any given point (mathematicians say that the fractions are "dense"). So you might think intuitively perhaps that the total length of that set is 1.
But then, looking at it another way, an individual point has 0 size. And adding together finitely many 0s, you still have zero.
Then, turns out you can arrange all the rationals between 0 and 1 in a sequence, like this:
0/1, 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, ...
So adding together the total length of all of these and you get
0 + 0 + 0 + 0 + 0 + ...
And normally you would say that infinitely many 0s add together to make 0.
So, perhaps the total length for all the rationals between 0 and 1 is 0?
So, while struggling to deal with these ideas, mathematicians came up with "measure theory" which lets you work out a total length for these bizarre sets.
Surprisingly, many of them turned out to have zero total length according to most of the ideas. Not just the rationals, also all the numbers of the form
0.2323222323...
They also have zero total length.
That leads to the Cantor set - similar idea, but this time in ternary - that's like binary, but to base 3, so you have three digits, 0, 1 and 2 instead of just the usual two digits 0 and 1.
If you look at all numbers between 0 and 1 in ternary of the form
0.02022022002...
infinite ternary expansion using only 0 and 2, then turns out that it's quite easy to get an idea of the total length of this set.
You can show them like this:
There you've got a big gap in the middle, because there are no decimals that begin
0.1...
Then you have more gaps in the next row, because there are no sequences that begin
0.01..
or 0.21...
Turns out that with each new row, you subtract a third of what is left of the line.
So first row line is length 1.
Second row, total length of all the line segments is 2/3
Third row, total length is 4/3.
General formula is 2^n / 3^n
Is easy to see that that gets as small as you like as n gets larger and larger. Choose any number, say 0.00001, and we can find a row that has total length less than that.
So, mathematicians summarize that by saying, that, "in the limit" as you keep doing that endlessly, you have nothing left. The Cantor set - has length 0.
So - generalizing this idea, and being very careful in your reasoning, you come up with the idea of Measure theory
Which formalizes the intuitive idea that things have length and that a map can be smaller than a continent - but at the same time also permits the wild complex sets that mathematicians want to study.
However there is something a little strange about this "measure theory". Depends on whether you accept the axiom of choice - the idea that if you have a collection of infinitely many pairs of socks, that you can choose a single sock from each one. If you have infinitely many pairs of shoes, is easy to choose a single shoe from each pair, just choose the left shoe each time (say).
But if you have indistinguishable socks, then you can't do it so easily, is no mathematical procedure you can use to choose one of each.
So - mathematicians sometimes add in the "Axiom of choice" which lets you do that. Seems an innocuous axiom. But it leads to paradoxes in measure theory - it leads to sets that can't be measured.
For instance, you get the Banach Tarski theorem. With the axiom of choice, you can prove that you can take a sphere apart into finitely many pieces and put them together to make two identical spheres.
These are "rigid" pieces, but a bit strange as they can include collections of spread out points - but ones that you move around without changing their relative position.
I
So - measure theory is a bit mysterious. This is not really a logical paradox. Doesn't lead to a contradiction as such. But it is a paradox in measure theory, if you expect everything to be measurable - this shows that if you assume the axiom of choice, then you can't do it.
Obviously, there can't be any way to attach a consistent measure to all sets in 3D space, if it is possible to do a decomposition of a sphere like this.
So, either you drop the axiom of choice, or you drop the idea that you can measure everything. Mathematicians would really prefer to be able to keep both ideas - to be able to measure all sets in a measurable space like 3D space - and to be able to choose one of each from infinitely many pairs of indistinguishable socks - but you can't.
(This answer was for original version of the question which said "amount" - which non mathematicians might interpret as either number or measure - now that it says "number" then this answer doesn't apply but leaving it in in case anyone is interested).
So for instance, most people would say that this map of Antarctica, on your computer screen:
Most people would say that this map is smaller than Antarctica.
But ask a mathematician which has most points, and he might say that the map has the same number of points as Antarctica - because he is thinking in terms of an ideal map, accurate in every detail.
So, then, to a pure mathematician, every point on the map corresponds to a point in Antarctica.
That's a bit like the way you can count pedestrians by drawing strokes on a sheet of paper, every time someone passes you. So long as you put one stroke on the paper for each pedestrian - that means you have the same number of strokes as you had pedestrians.
So in the same way so long as there is a point on the map corresponding to every point in the continent, then, looking at it that way, you must have the same number of points in the map as in the continent (or similarly in the country or city or whatever it is you have a map of).
So that's the sense in which there are as many numbers between 0 and 1 as there are larger than 1. By the simple mapping of x to 1/x, you can turn the section of the line from 0 to 1 into a kind of stretchy map of the section of the line from 1 to infinity.
Well, as it turns out, you can formalize this intuition that we have, that a map is smaller than a continent, by introducing rulers, basically. Some way of measuring the sizes of things.
That's not as easy as you might think though, because mathematicians want to talk about all sorts of wild collections of things.
If you just needed to measure line segments, that would be fine. Just measure the length of the line.
But what about, say, the collection of all decimals of the form
0.1011011011101....
any endless sequence of 0s and 1s - but this isn't in binary, this is a decimal number.
Or if you prefer, all decimals of the form
0.232332322323...
endless sequence of 2s and 3s, and no other numbers can be used.
How big is that set? Seems like there should be fewer points here than there are if you include everything - but how much smaller is it?
Also what about all the fractions between 0 and 1. You can find a ratio arbitrarily close to any given point (mathematicians say that the fractions are "dense"). So you might think intuitively perhaps that the total length of that set is 1.
But then, looking at it another way, an individual point has 0 size. And adding together finitely many 0s, you still have zero.
Then, turns out you can arrange all the rationals between 0 and 1 in a sequence, like this:
0/1, 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, ...
So adding together the total length of all of these and you get
0 + 0 + 0 + 0 + 0 + ...
And normally you would say that infinitely many 0s add together to make 0.
So, perhaps the total length for all the rationals between 0 and 1 is 0?
So, while struggling to deal with these ideas, mathematicians came up with "measure theory" which lets you work out a total length for these bizarre sets.
Surprisingly, many of them turned out to have zero total length according to most of the ideas. Not just the rationals, also all the numbers of the form
0.2323222323...
They also have zero total length.
That leads to the Cantor set - similar idea, but this time in ternary - that's like binary, but to base 3, so you have three digits, 0, 1 and 2 instead of just the usual two digits 0 and 1.
If you look at all numbers between 0 and 1 in ternary of the form
0.02022022002...
infinite ternary expansion using only 0 and 2, then turns out that it's quite easy to get an idea of the total length of this set.
You can show them like this:
There you've got a big gap in the middle, because there are no decimals that begin
0.1...
Then you have more gaps in the next row, because there are no sequences that begin
0.01..
or 0.21...
Turns out that with each new row, you subtract a third of what is left of the line.
So first row line is length 1.
Second row, total length of all the line segments is 2/3
Third row, total length is 4/3.
General formula is 2^n / 3^n
Is easy to see that that gets as small as you like as n gets larger and larger. Choose any number, say 0.00001, and we can find a row that has total length less than that.
So, mathematicians summarize that by saying, that, "in the limit" as you keep doing that endlessly, you have nothing left. The Cantor set - has length 0.
So - generalizing this idea, and being very careful in your reasoning, you come up with the idea of Measure theory
Which formalizes the intuitive idea that things have length and that a map can be smaller than a continent - but at the same time also permits the wild complex sets that mathematicians want to study.
However there is something a little strange about this "measure theory". Depends on whether you accept the axiom of choice - the idea that if you have a collection of infinitely many pairs of socks, that you can choose a single sock from each one. If you have infinitely many pairs of shoes, is easy to choose a single shoe from each pair, just choose the left shoe each time (say).
But if you have indistinguishable socks, then you can't do it so easily, is no mathematical procedure you can use to choose one of each.
So - mathematicians sometimes add in the "Axiom of choice" which lets you do that. Seems an innocuous axiom. But it leads to paradoxes in measure theory - it leads to sets that can't be measured.
For instance, you get the Banach Tarski theorem. With the axiom of choice, you can prove that you can take a sphere apart into finitely many pieces and put them together to make two identical spheres.
These are "rigid" pieces, but a bit strange as they can include collections of spread out points - but ones that you move around without changing their relative position.
I
So - measure theory is a bit mysterious. This is not really a logical paradox. Doesn't lead to a contradiction as such. But it is a paradox in measure theory, if you expect everything to be measurable - this shows that if you assume the axiom of choice, then you can't do it.
Obviously, there can't be any way to attach a consistent measure to all sets in 3D space, if it is possible to do a decomposition of a sphere like this.
So, either you drop the axiom of choice, or you drop the idea that you can measure everything. Mathematicians would really prefer to be able to keep both ideas - to be able to measure all sets in a measurable space like 3D space - and to be able to choose one of each from infinitely many pairs of indistinguishable socks - but you can't.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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