Yes. By a long long way.
What
specifically was wrong with the proofs presented by
@Rapscallion in
post #489 and
post #491above?
Indeed. In maths, being right is governed by what is actually right. Something which you still don't seem to understand.
So tell me where I am wrong,
1-1+1-1+1-1+1...........to infinite, is not 1/2 there is no answer to it as it runs to infinite. You can't say it's neither one or the other so it must be 1/2, that is just a cop out to appease yourself. It is simply neither one nor the other. End of.
The hotel paradox. A hotel with an infinite number of rooms, yet we are told that all rooms are occupied. There is only one way we can know all rooms are occupied and that is by knowing how many rooms there are, so there isn't an infinite number of rooms. With over 7 billion people as the earths population, you can accommodate them all if you truly have an infinite number of rooms and no -one needs to start swapping rooms to vacate a room or rooms. If you wanted to get the whole lot to swap rooms at the rate of 1 a second by the time the last person was re-accommodated, it would take over 200yrs. So to start changing rooms serves no other purpose than p*** off the customers, it's pointless and there is no paradox.
Gabriel's Trumpet (Horn) We are told that the trumpet is of infinite length yet the volume inside is finite. The only way the volume can be finite is if it tapers down to nothing but if the inside tapers down to nothing the horn itself must also taper down to nothing, hence the length of the horn isn't infinite, it will be of a finite length the same or very close to that of the depth of the whole. The only way the length of the horn can be infinite is if it stops tapering down and runs parallel. If it runs parallel then the hole can also run parallel and it's depth is also infinite. As I said before if you truly believe the volume inside the horn to be finite as we are informed it isn't a horn or trumpet it is just a shaped piece of metal.
If you want a true paradox, stand 10 metres from a wall and throw a ball at it. At 5 metres from the wall, it's half way there, at 2.5 metres from the wall, it's halved the distance again, at 1.25 metres from the wall it's halved the distance again and so on and so on, it keeps halving it's distance from the wall. As the ball can continue to keep halving the distance no matter how small the number it can still be halved, the ball will never actually reach the wall.
