This stuff is part of an area of mathematics called "set theory" which, despite being a real headache, is pretty interesting and also has philosophical implications etc.
You can't really apply the notion of "size" or the adjectives "larger" or "smaller" in the conventional sense of the term when talking about infinite sets. That is why mathematicians have come up with a concept called "cardinality" to generalize the notion of size to infinite sets.
Basically, the cardinality of two sets is the same if there exists a bijective function mapping the sets onto each other.
What the guys in the videos really are saying is that the cardinality of R (the set of all real numbers) is strictly greater than the cardinality of N (the set of all natural numbers). Additionally:
- the cardinality of R is the next bigger cardinality after the cardinality of N. This means that you cannot find any set the cardinality of which is between that of N and that of R. (interestingly, this one cannot be proven or disproven)
- the cardinality of N x N (all pairs of natural numbers) is the same as the cardinality of N. Same for NxNxN or NxNxNxN etc.
- the cardinality of Z (set of all integer numbers) is the same as the cardinality of N.
- the cardinality of Q (set of all rational numbers) is the same as the cardinality of N.
- the cardinality of R can be shown to be equal to 2^N0, where N0 represents the cardinality of N.
- the cardinality of any interval [a,b[ or [a,b] of real numbers where b > a is the same as the cardinality of R
- the cardinality of RxR, RxRxR, etc is the same as the cardinality of R
- what set has a cardinality greater than the cardinality of R then? It can be shown for example that the set of all subsets of real numbers has a cardinality greater than that of R.
So in the end, when you say that an infinite set "is bigger" than another, what you really mean is that its cardinality is greater than that of the other set.
Fun fact: intuitively, you might think that the set of all real numbers in the interval [1, 3] is "bigger" than the the set of all real numbers in the interval [1, 2] because [1, 3] includes [1, 2] and you can easily find extra numbers in [1, 3] that are not in [1, 2]. However this is wrong in the mathematical sense: both intervals have the same cardinality: that of R.
This can get really complicated for non mathematicians... here is a simple article about it:
http://www.sciencenews.org/view/gene...y_Big_Infinity
And some wikipedia and other references which require more math:
http://en.wikipedia.org/wiki/Cardinality
http://en.wikipedia.org/wiki/Cardina..._the_continuum
http://en.wikipedia.org/wiki/Cantor%...gonal_argument
http://www.ma.utexas.edu/users/mwill...ardinality.pdf
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