VIDEO: http://video.google.com/videoplay?docid=-3...11448&hl=en Cantor was probably taught about the infinity 1,2,3,4,5.... onwards and upwards, but there is a "bigger" infinity still. There is the infinity of decimal numbers between 0 and 1 (technically called 'real' numbers). If you want to count every possible number that exists between 0 and 1, it gets very tricky very quickly, you cant really start from 0 and count up, because whats the smallest number after 0? any 0.0000...01 number you come up with you can squeeze another 0 in there and come up with a smaller number, the same problem happens counting down from 1. You can do it by halving, and then halving each half, and repeat forever, even though this is an infinite, you can still at least count them from 1. You end up with a picture like this 0..1 0..1/2 1/2..1 0..1/4 1/4..1/2 1/2..3/4 3/4..1 etc. You simply start at the top of the triangle and step down through each row, (0, 1, 1/2, 1/4, 3/4 etc.) you could put them in a big list and go from one to the next and keep a tally. But unfortunately there are gaps! A familiar example is the number pi, which has an infinite number of decimal places. Now what does an infinite number of decimal places mean? well it means that you cannot come up with a fraction that describes it, you have to write all the decimal places out otherwise you arent writing that actual number - you are just writing an approximation. And those are the gaps, no matter how you divide up the original 0..1, and no matter how you count up the fractions you create by dividing, you will still miss out on all those numbers that arent fractions. Cantor showed that these gaps actually contain an infinite number of numbers like pi, an infinite number of numbers that cant be described as fractions. And the biggy is there is no way of stepping into these gaps, as noted at the start, there is no specific number you can start from, that you could call the "first decimal number". So you cannot even count them! Now if you think about what counting is, its like putting stuff in a big list and labelling each thing in the list with the numbers 1,2,3,4,5... So if you cannot count them, then you cannot list them, which means these infinite gaps are bigger than the infinite list of numbers 1,2,3,4,5... Note: This isnt meant to be a proof by any means, just a simplistic explanation of countability and infinite sets.
The video is an hour and a half long...but this is cool nontheless. It wouldn't hurt for the BBC people to put some cool effects in the vid though.
n this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan ... all » Turing - whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.
my teacher did something like this in mathsB. he said that if he had a certain distance (say from one side of the room to the next) and he walked half way. then half that distance, then half that distance....and so on and so forth you could keep doing that for ever. but you would never reach the other side of the room. Therefore it has an Infinite number of variables. got to do with graphs and ----....yeah i hate maths. but am i on the right track?
yeah somewhat. It's like finding the amount of numbers between 0.1 and 0.2 There's 0.1000000000000000000000000000000000000000000000000000000000000001 and even smaller numbers just by adding zeros. You can add an infinite amount of zeroes, and between that number and the one closest to it, just add 1 more zero and it gets even crazier. The law of infinite infinites
