Synopsis:
- Set Theory: emancipates Mathematics from its traditional domain (numbers)
- Transfinite numbers
- Zeno's Paradoxes: "if space is infinitely divisible in finite points, then_"
- Solutions to Zeno's Paradoxes
- Hume: space and time are composed of indivisible units having magnitude
- Kant: contradictions are immanent in our conceptions of space and time, so space and time are not real
- Hegel: all reasoning leads to contradictions which can only be reconciled in a higher unity
- Cantor's solution to Zeno's Paradoxes
- A one-dimensional line is not a sum of an infinite number of infinitely small points, but a set-theoretic union of an infinite number of unit-sets of zero-dimensional points
- What Zeno proved is a general property of space...
- There is no point next to any other point: between any two points there is always an infinite number of points
- The non-denumerable infinity of points in space and of events in time is much larger than the merely denumerable infinity of integers.
- An infinite series of numbers can have a finite sum
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