Emptiness and Set Theory

Thanks to a good friend of mine, I’m finally learning set theory. I’m going through a textbook; he checks my work and helps me out if I’m not following.

Set theory seems, at first glance, to be incredibly simple. Sets are just collections of other sets. So, if I have set A and set B, I might then be able to create set C, the set that has A and B in it. Or I can create set D, the set that has only set A in it. Or set E, the set that has set A and another set – call it F – which in turn has B and C in it. And so, from this simple operation of “set membership”, one can create systems so complex that all of arithmetic can be recreated in terms of sets.

You might ask – how did I get sets A and B in the first place? If sets only have other sets in them, do we have a “starting set” from which to build? Indeed, we do: the empty set. The empty set has no sets in it. But we can create a set of the empty set. That has one set in it – namely, the empty set. And we can create the set that has the empty set and the set of the empty set. And off we go.

All of this means that the gargantuan, entangled edifice that is mathematics is built on a single object – which we might poetically call nothing – and a single basic operation: “is a member of”.…

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Zeno’s Arrow

I wrote about Zeno’s Achilles paradox – along with the Dichotomy (a.k.a. Racetrack) and the lesser know Plurality – here. You should read that before reading this. In that post, I didn’t mention Zeno’s other very famous paradox: the Arrow. Though a huge majority of philosophers think Zeno’s other paradoxes were cleared up by calculus and set theory, many consider the Arrow still unsolved. I’ll be arguing that the Arrow, though it appears to deal with separate issues, is really the same Achilles paradox in a different form.

Consider an arrow traveling down a trajectory. Take a snapshot at one static instant. In this one duration-less instant, the arrow is at rest. It is not moving, insofar as motion requires movement from one point to another, and this arrow is, in this one instant, in only one spot. So the arrow is at rest in one instant, at rest again in the next, at rest again in the next. If the arrow is at rest at every instant along its journey, how can we say that the arrow is in motion? When exactly does this arrow travel?

One aspect of the Arrow is simple enough to dissolve: being at rest, defined as “not being in motion,” only makes sense over some duration of time. Motion is defined as distance traveled over some period of time, so it makes no sense to even inquire as to whether motion is happening or not happening when we’re limited to a time range of 0.…

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Zeno’s Achilles Has Not Been Solved

Zeno’s paradoxes of motion – particularly “Achilles and the Tortoise” and the “Dichotomy” (aka the “Racetrack”) – are maybe the most well known and historically popular paradoxes in western philosophy. The consensus among philosophers is that they were solved by the insights of calculus. Even those who see a problem beyond what calculus resolves tend to think that set theory takes care of the loose ends. I will argue that neither calculus nor set theory satisfactorily deal with all of the problems involved in Achilles and the Dichotomy, and that the paradoxes have yet to be fully solved. I will further argue that a lesser known of Zeno’s paradoxes – the “Plurality” – also remains unsolved.

I won’t assume prior knowledge of either the paradoxes, calculus, or set theory, and will take care to introduce relevant concepts as they arise. I must also acknowledge that I am myself neither a professional philosopher or mathematician. Despite my attempt to thoroughly research the topic, it is more than likely that I overlook important points (I’ll tackle the paradox of the Preface some other day). Anyone who thinks they see an error, or something that I may be missing, is encouraged to message me.

The paradoxes and the calculus solution
“Achilles and the Tortoise” – they are having a race. Achilles knows he is twice as fast as the Tortoise, and so, wanting to spare the Tortoise’s feelings, decides to give it a head start of 10 meters. It takes Achilles a certain amount of time – let’s say 10 seconds – to catch up to where the Tortoise started; in this time the Tortoise has advanced, albeit, his overall advantage has halved (he’s now at the 15-meter mark).…

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