Infinity 24: Last class: review and final explication of Zeno

Last class: a review of the irrationality of the square root of 2; and of Cantor's diagonalization proof.  How the difference between infinity by addition and by division corresponds to the difference between the infinity of natural numbers (which are all under the rubric of infinity by addition) and the reals between 0 and 1, all of which may be ranged under the rubric of infinity by division.  What makes one set "larger" than another.  The idea of a list as involving the concept of "next."  The non-denumerability of the reals means that the concept of next doesn't apply to them.  Zeno's paradoxes rely on the idea of the next: the next point on space, the next time slice.  The larger order of infinity that comprises the reals means that Achilles passes the tortoise at a point between two rationals, which are the only points Zeno considered, in considering the next rational point the tortoise gets to, while Achilles is still at a previous rational point.  A similar intuition applies to trying to come up with a commensurate scale for measuring hypotenuse and leg of an isosceles right triangle.  Whatever units you divide one line segment up into, there'll never be a point which is the exact passing point, so to speak, as you go from fewer of those units than you need to measure the other line segment to more of those units than you need that you'll have exactly the right number -- no point, you could say, where one line segment is passing another at a rationally measurable distance from its other endpoint.  And so farewell to this class. (I'm actually not sure why there are 24 and not 25 classes.  I may have miscounted somewhere)