### Appendix 4

## 1 Zeno's Paradoxes

To examine the properties of quantized spacetime, and show its practicality in resolving spacetime paradoxes, consider how it deals with a paradox posed by Zeno.

Zeno’s paradoxes, put forward around 400 BCE, included a race between Achilles and a tortoise. The contestants start at the same time, but the tortoise begins ahead of Achilles. By the time Achilles covers that lead, the tortoise has moved ahead a small distance. When Achilles covers that distance, the tortoise has moved ahead a still smaller distance . . . With every advance of Achilles, the tortoise is always slightly ahead. Achilles cannot pass him to win.

The paradox involves a conflict between the assumption of a continuous, infinitely divisible spacetime and our practice of measuring length in fixed units like inches or feet. The mathematics of an infinite series of numbers offers one resolution, for those happy with math. But the spacetime quantum as a limit to the division of spacetime provides a simpler approach, a faster solution, and requires no recourse to infinity.

However, as measurement of the speed of light suggests our spacetime quantum must be about a trillionth, trillionth, trillionth of a yard, it is easier to understand what is happening by imagining spacetime quantized on a much larger scale, more suited to our surroundings and Zeno’s paradox. This will also show how the relationship between space and time influences our view of the universe.

## Quantizing Zeno

Suppose Zenovian spacetime allows no length shorter than one thousandth (0.001) of a foot and no time shorter than one hundredth-thousandth (0.00001) of a second. The ratio is important. It gives us familiar race track speeds. If you travel this minimum length in this minimum time, you move at 100 fps (feet per second). Homer’s swift-footed Achilles could probably run 100 yards in 10 seconds, or about 30 fps. Magnifico, Zeno’s racing tortoise, might be one hundred times slower, at 0.3 fps. Let’s give it a 30 ft lead and allow both contestants a running start.

In the first second, Achilles runs 30 ft to arrive at 30 ft from the start. Magnifico runs 0.3 ft, to reach 30.3 ft.

Not even pausing for breath, in the next 0.01 second Achilles runs 0.3 ft to arrive at that 30.3 ft. Magnifico runs 0.003 ft to reach 30.303 ft. Still ahead. Go Magnifico, go!

According to Zeno, in the next 0.0001 second, Achilles would run 0.003 ft to reach 30.303 ft while Magnifico would run 0.00003 ft to reach 30.30303 ft. Still ahead. This is possible in continuous spacetime. It is impossible in quantized spacetime.

A move of 0.00003 ft is less than 0.001 ft, the shortest possible length. To make the smallest permissible move, Magnifico must use all of 0.0033 second to cover that 0.001 ft, reaching 30.304 feet.

But Achilles continues at his own speed and covers 0.099 ft in that 0.0033 second. He reaches 30.402 ft and wins! Spectator cheers. Paradox resolved!

Let's move on to the true importance of the ratio of minimum length to minimum time.

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. . .cadence in words lucent as the
moon over waters of. . . *