But just because there still seems to be very little consenus on the airplane problem… THE AUTHOR IS CORRECT!!!!!

The engines are producing a force that is completely independent of what is happening with the wheels. Consider this: Imagine that you had two mechanical winches (like the ones on the front of some pickup trucks.. only bigger). Now, imagine that those two winches were mounted onto some really strong object in front of the plane, right at wing level, right in front of each wing of the airplane. Next, extend the cables from the winches and hook them onto both wings of the plane, and then turn on the winches…

Even if there is a treadmill under the plane running backwards at any speed, by God, our plane is still going to be pulled forward by the cables that are attached to the wings. This is NOT different from just turning on the engines of the plane. It simply doesn’t matter what the wheels are doing — the forward force is coming from the engines mounted on the wings, and the engines produce forward momentum regardless of what’s happening down below.

0.124 = 1*10^-1 + 2*10^-2 + 4+10^4

but for simplicity we just write 0.124.

more general a is from 0 to 1

a = a1*10^-1 + a2*10^-2 + …

where a1,a2.. are numbers 0..9 and we simply write 0,a1a2a3…
So if you write 0.999.. You should understand it as 9*10^-1 + 9*10^-2 + … and this sum converges to one.

as you probably know infinite sum

1/2 + 1/4 + 1/8 + … + 1/2^n + … = 1

if you write 1/2, 1/4, 1/8 in binary you get 0.1, 0.01, 0.001 etc.

so 0.11111.. = 1 in binary

Ok. lets suppose a game. There are two players and target( a square [0,1]x[0,1]) they are shooting at.
Lets suppose continuum hypothesis and map the target(a continuum) to the alef one(the first uncountable ordinal,a set of all countable ordinals). And and who hits the higher ordinal wins. So…
Lets the first player begin. He hits a point it the target which corresponds to some ordinal, a countable ordinal!!!. So there are only countable many ordinals smaller than that ordinal the first player hit. Countable set has measure zero. So the probability that second player hits smaller ordinal than the first player is zero.
So the probability that the second player wins is ONE!!! Isn’t it weird that the second player always wins??
Now lets make it even weirder. Lets players one and two shoot at the same time. And figure out who has higher probability of winning?

Game over.