My favorite variation is:
A trolley is heading towards a line of people corresponding to every natural number. You can pull a lever and divert the trolley towards a line of people corresponding to every real number between 0 and 1. What do you do?
A trolley is heading towards a line of people corresponding to every natural number. You can pull a lever and divert the trolley towards a line of people corresponding to every real number between 0 and 1. What do you do?
But actually, if you choose "pull the lever" after having done the maths, then you are probably a misanthrope. Either that or you believe very strongly in the theory that if you do nothing you're not involved. But really, you're just a misanthrope.
Aren't they both just infinite? There's no such thing as a bigger inifinite?
thinking about it out loud here tho I would say that every natural number is a BIGGER number than every real number between 0-1 because, well, 2 is bigger than 0-1.
but then again since you said corresponding to every number.... 0.000000to infiniitiy -1 is a number in itself, so, you have infinite numbers, along an infinite number of 'places'.
Final Answer, you save lives by natural numbers, because real numbers have an infinite number of 'digits' per 'places' & those 'Places' are equal to all natural numbers.
Well, that is where the fun starts.
Take every natural number, assign a unique real between 0 and 1 to it. Put all of them under each other. Now take the 1st digit of the first number, 2nd of the 2nd and so on, set it to a different digit than the one you took.
Now you have a new, unique, real (Because it differs in at least one digit with every other real taken so far) but no more natural to assign it to.
"A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers."