Gravity's Rainbow in depth on Studio 360

[David Morris]

http://boingboing.net/2012/08/07/what-do-christian-fundamentali.html#more-175190

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"Unlike the "modern math" theorists, who believe that mathematics is a
creation of man and thus arbitrary and relative, A Beka Book teaches
that the laws of mathematics are a creation of God and thus
absolute....A Beka Book provides attractive, legible, and workable
traditional mathematics texts that are not burdened with modern
theories such as set theory." — ABeka.com
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[...]

Sets are exactly what you think they are—groups of things. Prime
numbers, unicorns, cats, whatever ... you can make a set of it. Set
theory is just a way of talking about what sets do and what they are
like.

On the surface, this sounds pretty simple. For instance, most of what
I learned about set theory when I was in college came through classes
in anthropological linguistics. That's because sets, being made of
anything you damn well please, have applications outside of pure math.
Ted Sider, a professor of philosophy at Cornell University has some
good examples of this in a set theory primer he's written:

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In linguistics, for example, one can think of the meaning of a
predicate, ‘is red’ for instance, as a set — the set of all red
things. Here’s one hint of why sets are so useful: we can apply this
idea to predicates that apply to other predicates. For instance, think
of ‘red is a color’. ‘Is a color’ is a predicate, and it looks like it
holds of things like the meaning of ‘is red’. So we can think of ‘is a
color’ as meaning a set containing all the colors. And these colors in
turn are sets: the set of all red things, the set of all green things,
etc. This works because i) a set collects many things into one, and
ii) many sets themselves can be put together to form a new set.
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But sets and set theory can also be a lot more complicated. For
instance, You can make up sets that contradict themselves. Here's an
example: Furry creatures are not cute and this set is made up of cute
furry creatures. Oops. The classic example is a set made up of barbers
who shave everyone and who shave people who don't shave themselves.
Another problem: Sets that are too broadly defined, so you don't know
if you're actually putting the right stuff in there. A set made up of
the favorite things of a tall person, say. Paradoxes like this are
what really drive set theory, much of which centers on defining rules
for sets and how they work so that we don't just go around assuming
certain sets exist when they clearly can't—and so that we can still
use the valuable logic and math of sets even when we can't prove that
the stuff we're sticking into a set actually exists in the real world.
Basically, set theory has a lot to do with creating rules and helping
us apply a rule-based system in weird, hypothetical situations.

All of which turns out to be really important when you want to talk
about the idea of infinity. Set theory actually has its origins in
attempts to define infinity and deal with it in a concrete way in
mathematics. Checking Wikipedia, you'll learn that this "modern"
theory was actually established in 1874. Why 1874? Because that was
when a guy named Georg Cantor proved that there are different
infinities and that not all infinities are created equal.

This is really where set theory starts to sound like something you
thought up while high and later forgot about.

You can have an infinite set of numbers, right? That makes sense. But,
Cantor figured out that an infinite set of, say, whole numbers, is
smaller than an infinite set of decimal numbers. They're both
infinite. But they're not the same. This TEDEd video explains it
really, really well:

http://www.youtube.com/watch?feature=player_embedded&v=UPA3bwVVzGI