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This seems to be a widely accepted theory on larger sets of infinite numbers, originally shown by Cantor.

After watching the video, I am trying to grasp in layman's terms why this is true. My understanding is the following, and I wondered if someone could confirm if this is correct;

The reason there are more real numbers between 0 and 1, than all the natural numbers, is because in this example, each real number can have a length of infinity.

Initially I thought that there is an infinite number of real numbers between 0 and 1, and an infinite number of natural numbers. This would allow for (to use the videos metaphore) a line to be drawn between every real and natural number. But if the real numbers are also infinite in length, there are "infinity to the power of infinity" real numbers, and just infinity natural numbers.

Have I understood this correctly? If not, could someone spell it out for me please?

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    I have not watched the video, but there as as many integers as there are fractions (i.e. the number of fractions are countable). However, there are uncountably many real numbers.2012-05-30
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    There are "countably infinite" natural numbers. There are "countably infinite" rational numbers. What does "countably infinite mean? A set A is said to be "countably infinite" if there is a bijection from N to A. The set of real numbers R is not countably infinite (we say that R is uncountably infinite). When we try to show that R is countably infinite, we fail due to a contradiction. You can use google and/or wiki for proofs of these facts. They are standard and accepted by most mathematicians.2012-05-30
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    Also, what do you mean by *length of infinity* ? Do you mean, infinite decimal places after the decimal point? If so, then yes, each real number can be written as an infinite string of numbers after the decimal point (but not uniquely, e.g. 0.99999... = 1)2012-05-30
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    If you look at all infinite sequences $x_1,x_2, x_3,\dots$ of *real* numbers, then it turns out that there are no more of these (in the sense of cardinality) than there are real numbers. So even though the explanation given is not *wrong*, there is detail to fill in before one can come up with a tight argument.2012-05-30

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