Axiomatic set theory builds up set theory from a set of fundamental initial rules. The most common axiomatization, which we’ll be used, is the ZFC system: Zermelo-Fraenkel with choice set theory. The ZFC axiomatization consists of 8 basic rules which are pretty much universally accepted, and two rules that are somewhat controversial – most particularly the last rule, called the axiom of choice.
Naive set theory is fun, and as we saw with Cantor’s diagonalization, it can produce some incredibly beautiful results. But as we’ve seen before, in the simple world of naive set theory, it’s easy to run into trouble, in the form of Russell’s paradox and a variety of related problems.
For the sake of completeness, I’ll remind you that Russell’s paradox concerns the set R={ s | s ∉ s}. Is R∈R? If R∈R, then by the definition of R∉R. But by definition, if R∉R, then R∈R. So R is clearly not a well-defined set. But there’s nothing about the form of its definition which is prohibited by naive set theory!
Mathematicians, being the annoying buggers that they are, weren’t willing to just give up on set theory over Russell’s paradox. It’s too beautiful, too useful an abstraction, to just give up on it over the self-reference problems. So they went searching for a way of building up set theory axiomatically in a way that would avoid problems by making it impossible to even formulate the problematic statements.
I’ve been seeing articles popping up all over the place about a recent
PLOS article called Order in Spontaneous Behavior. The majority of
the articles seem to have been following the lead of the Discovery Institute, which claims that the article demonstrates the existence of free will, which they argue is inconsistent with naturalism and darwinism.
So, what’s set theory really about?
We’ll start off, for intuition’s sake, by talking a little bit about what’s now called naive set theory, before moving into the formality of axiomatic set theory. Most of this post might be a bit boring for a lot of you, but it’s worth
being a bit on the pedantic side to make sure that we’re starting from a clear basis.
A set is a collection of things. What it means to be a member of a set S is
that there’s some predicate PS – that is, some way of describing things via logic – which is true only for members of S. To be a tad more formal, that means that for any possible object x, PS(x) is true if and only if
x is a member of S. In general, I’ll just write S for both the set and the predicate that defines the set, unless there’s some reason that that would be confusing and/or ambiguous. (Another way of saying that is that a set S is a collection of things that all share
some property, which is the defining property of the set. When you work through
the formality of what a property means, that’s just another way of saying that there’s a
predicate.)
Have you ever wondered about the real reason why math education in our schools is so awful? Why despite the best efforts of large numbers of parents, the schools seem to be incapable of figuring out why they’re so dreadfully bad at recognizing the difference between a halfway decent math curriculum and a trendy piece of garbage?
Read below the fold for a perfect example of why. The short version: the people who are involved in running education in America consider it perfectly acceptable to be idiots when it comes to math.
While I’ve been writing about the Surreal numbers lately, it reminded me of some of the fun of Set theory. As a result, I’ve been going back to look at some old books. Since I’ve been enjoying it, I thought you folks would as well.
Set theory, along with its cousin, first order predicate logic, is pretty much the
foundation of nearly all modern math. You can construct math from a lot of
different foundations, but axiomatic set theory is currently pretty much the dominant approach. (Although Topoi seem to be making some headway…)
There’s a reason for that. Set theory starts with some of the simplest ideas, and extends in a reasonably straightforward way to encompass the most astonishingly complicated one. It’s truly remarkable in that – none of the competitors to set theory
as a foundation can approach the intuitive simplicity of set theory.
So I’m going to write a bit about set theory as I explore my old books. And I thought that a good place to start was Cantor’s diagonalization. Cantor is the inventor of set theory, and the diagonalization is an example of one of the first major results that Cantor published. It’s also a good excuse for talking a little bit about where set theory came from, which is not what most people expect. Set theory was originally created as a tool for talking about the relative sizes of different infinities.
I debated about whether or not I should write this post. But as you can see, in the
end, I overcame my better judgement, and so he we are.
Over the weekend, PZ wrote a Pharyngula post about the reaction people have had to Mitt
Romney’s statement about evolution. He was pissed. And I agree with his initial reaction.
What we have is a politician basically saying “Yes, I agree with the facts”. And somehow,
that’s been taken by a seemingly huge number of people as something brave and bold,
something that should impress us. Nope, sorry folks: acknowledging that facts are facts is
not brave. I’m no more impressed with him for saying that evolution is true than
I’d be if he got up and said “I admit it: I believe that 2+2=4.”
But typically, when discussing anything that involves religion, PZ went overboard. And
it ticked me off.
Sorry for the slow pace of things around here lately; life interferes sometimes. I’ve mentioned my fathers illness before; things took a drastic change for the worse a week ago, which made things more than a little bit crazy. Of course, life wouldn’t be life in things happened one at a time; so naturally, my asthma picked now to act up as well, so I’ve been coughing my lungs out. It’s been a fun week, really.
Anyway, I’ve been wanting to look a bit more at Pica. I’ve been thinking about how a named process definition looks, and what it does. The type part of a definition is relatively straightforward: a process is defined by a set of externally visible channels; each channel is defined by a type. In a named definition, there are two types of channels to think about. One is a set of channels supplied to the process when it’s created; the other is a set of channels created by the process, but which are ‘exported’: that is, made visible to other processes. The type of a process is a tuple of all of the channels which the process exports the outside world.
This isn’t math, but I felt like commenting anyway. That shining example of
an Intelligent Design advocate, Dr. Michael Egnor, is back once again. And this time,
his point, such as it is, is to basically fling insults at PZ Myers. What did PZ do to bring on his ire?
Well, PZ was annoyed with Time magazine, because for their “Time 100” list, they
had Michael Behe write the entry about Richard Dawkins. The passage which Engor took such offense at was the following:
The incompetence is stunning. Richard Dawkins makes the Time 100 list, and who do they commission to write up his profile?
Michael F**king Behe.
That’s not just stupid, it’s a slap in the face. It would have been no problem to find a smart biologist, even one who might be critical of Dawkins’ message, to write something that expressed some measure of respect from the editorial staff. But to dig up a pseudoscientific fraud whose sole claim to fame is that he has led the charge to corrupt American science education for over a decade is shameful.
Now, what’s wrong with PZ’s reaction, according to Egnor?
One word.
Yes. PZ used the word “Fucking” to refer to Behe.
I added the asterisks. Both Behe and Myers are college biology professors who teach young biologists and biochemists the methods of scientific inquiry and, by example, teach students the appropriate standards of scientific discourse.
Which professor is shamefully corrupting American science education?
Yup. That’s it. It’s the one word. In the mind of Michael Egnor, throwing off a nasty word in the heat of the moment is a grave offense, far worse than spending more than a decade as a professional liar, far worse an offense for a scientist than, say, pretending to know the content of several dozen papers that you’ve never read.
Now, I’m not exactly PZs biggest fan. I think he’s abrasive and arrogant. But I also know, from the experience of reading his writings, that he’s an intelligent, passionate advocate for science; he’s a teacher who works hard at teaching his students real science, critical thinking, and the scientific method.
And I agree with him about the incredible stupidity of asking Behe to write the piece on Dawkins. Behe is not a legitimate scientist. Behe is a dreadful hack who’s spent most of the last two decades hard at work on a program to deliberately and dishonestly misrepresent science as part of a political agenda. He’s been caught lying repeatedly; his ideas, such as they are, have been discredited. He even made an ass of himself in the Kitmiller trial, by handwaving away a stack of papers that he’d never seen before, because even without reading them, he just knew that they contained
nothing of any relevance. Choosing him to write a profile of Richard Dawkins is just plain offensive. It’s sort of like asking PZ to write a profile about the Pope’s influence as a theologist.
So, Dr. Egnor. Which professor is shamefully corrupting American science education? The professor who is a passionate (if arrogant) teacher of real science? Or the dishonest (and arrogant) professor who is an advocate of introducing misrepresentations into the science curriculum in order to discredit scientific theories that disagree with his religious beliefs?
Let’s try looking at one other little quote, which I think is pretty illustrative of the difference between the two as educators. PZ is a tireless advocate for the teaching of real science to all science students. Behe, on the other hand, at the end of his interview with the director of the movie “Flock of Dodos”, said “Why should I care what gets taught in public school? My kids don’t go to public schools.”
Who’s the teacher who really cares about science education? And who’s the one corrupting it? Look at the facts, and it’s pretty obvious.
Of course, Dr. Egnor won’t do that. Because as he’s demonstrated in the past, he
doesn’t need to waste his time looking at petty little things like facts, because he already knows the truth. Just like Professor Behe.