28 Nov '14 23:59>
Originally posted by DeepThought"Talking about a decimal number with an infinite number of digits is perfectly sound. Writing out such a number is a supertask but that in itself isn't any great objection."
The proof I reproduced above is the standard one due to Cantor. If that is the one from the YouTube video then it is correct, but other proofs are possible. I'd regard a decimal number as a number written in base 10. Any decimal which can be written down with a finite number of digits is automatically rational. 0.123 for example is 123÷1000. I would ...[text shortened]... ht present the proof better than I did:
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
I beg to differ.
A question: Is the exact value of pi a decimal number? Is it possible to write down all decimals, infinitely many?
"You could just have a Turing machine calculate pi and have it write the first digit after ½ a second, the next digit ¼ of a second later and so on."
Sorry, such a Turing machine does not exist, and cannot ever be constructed. Time in itself is quantizised and a digit cannot be printed in an interval shorter than this smallest time interval, so the printer will stop there, well before you reach the very last digit in this infinite string.