So far, we’ve been talking mainly about the ZFC axiomatization of set theory, but in fact, when I’ve talked about classes, I’ve really been talking about the von Newmann-Bernays-Gödel definition of classes. (For example, the proof I showed the other day that the ordinals are a proper class is an NBG proof.) NBG is an alternate formulation of set theory which has the same proof power as ZFC, but does it with a finite set of axioms. (If you recall, several of the axioms of ZFC are actually axiom schemas, which need to be distinctly instantiated for all possible predicates.) NBG uses one axiom scheme, but it’s possible to show that that schema only expands into a finite number of distinct axioms.
Part two of our crackpot’s babblings are actually more interesting in their way, because they touch on a fascinating mathematical issue, which, unfortunately, Mr. Brookfield is compeletely unable to understand: the Poincare recurrence theorem.
Brookfield argues that the second law of thermodynamics in not really a law, since it’s statistical, and that there must therefore be some real law underlying the statistical behavior normally explained by the second law. Here’s his version – be prepared to giggle:
“The second law of thermodynamics has a rather different status than that of other laws
of science, such as Newton’s law of gravity, for example, because it does not hold
always, just in the vast majority of cases.”
Well, if it is a “law” then it must hold always by definition. If it “does not hold always”
then it is not a law, period. If it is a “pseudo law” then that is fine for pseudo science, but
I am not interested in doing pseudo science. Hawking says that the thermodynamic arrow
is reversible because..
“…The probability of all the gas molecules in a box being found in one half of the box at
a later time is many millions of millions to one, but it can happen.”
The type of event that Hawking is referring to here is known as “Poincaré Recurrence”–
named after the French mathematician Henri Poincaré. The result of any such occurrence
will indeed reverse the thermal characteristics of the box contents, violating the internal
thermodynamic arrow. This internal reversal however will not (in my opinion) reverse
the real arrow — the unrelenting order to disorder movement of the total physical system.
Yes, folks – Brookfield is a real scientist, doing real science; Steven Hawkings and his ilk are all just pseudo-scientists studying psuedo-laws; real scientists like Brookfield throw out hopeless pseudo-laws like the second law of thermodynamics in favor of Murphy’s law. And yes, that Murphy’s law. Brookfield really tries to argue for the use of Murphy’s law as a better statement of the principle of the second law. But we’ll get to that later.
As several of my fellow science-bloggers pointed out, William Dembski has written a post at Uncommon Descent extolling an “international coalition of non-religious ID scientists“, and wondering how us nasty darwinists are going to deal with them.
Alas for poor Bill. I’m forced to wonder: is there any purported ID scholar so stupid that Bill won’t endorse them? In his eagerness to embrace anyone who supports ID, he didn’t both to actually check who or what he was referencing. This “international coalition” turns out to be a lone uneducated crackpot from Canada who uses his ID beliefs as a justification for running on online sex-toys shop! Several people have written about the organization; I decided to take a look at the “science” that it/he published, in the form of a sloppy paper called “In Search of a Cosmic Super-Law: The Supreme “Second Law” of Devolution“.
With ordinals, we use exponents to create really big numbers. The idea is that we can define ever-larger families of transfinite
ordinals using exponentiation. Exponentiation is defined in terms of
repeated multiplication, but it allows us to represent numbers that we
can’t express in terms of any finite sequence of multiplications.
I’ll continue my explanation of the ordinal numbers, starting with a nifty trick. Yesterday, I said that the collection of all ordinals is *not* a set, but rather a proper class. There’s another really neat way to show that.
I’ve talked about the idea of the size of a set; and I’ve talked about the well-ordering theorem, that there’s a well-ordering (or total ordering) definable for any set, including infinite ones. That leaves a fairly obvious gap: we know how big a set, even an infinite one is; we know that the elements of a set can be put in order, even if it’s infinite: how do we talk about *where* an element occurs in a well-ordering of an infinite set?
For doing this, there’s a construction similar to the cardinal numbers called the *ordinal numbers*. Just like the cardinals provide a way of talking about the *size* of infinitely large things, ordinals provide a way of talking about *position* within infinitely large things.
This is a short post, in which I attempt to cover up for the fact that I forgot to include some important stuff in my last post.
As I said in the last post, the cardinal numbers are an extension of the natural numbers, which are used for measuring the size of sets. The extended part is the transfinite numbers, which form a sequence of ever-larger infinities.
One major problem with adding the transfinite numbers is that natural number arithmetic doesn’t work anymore with the cardinals. It still works for the natural number subset of the cardinals, but not for the transfinites.
But we *do* want to be able to talk about at least certain kinds of arithmetic on the full set of cardinals. So we need to figure out what arithmetic means for this strange sort of number.
One of the strangest, and yet one of the most important ideas that grew out of set theory is the idea of cardinality, and the cardinal numbers.
Cardinality is a measure of the size of a set. For finite sets, that’s a remarkably easy concept: count up the number of elements in the set, and that’s its cardinality. But there are interesting questions that we can ask about the relative size of different sets, even when those sets have an infinite number of elements. And that’s where things get really fun.
One of the reasons that the axiom of choice is so important, and so necessary, is that there are a lot of important facts from other fields of mathematics that we’d like to define in terms of set theory, but which either require the AC, or are equivalent to the AC.
The most well-known of these is called the well-ordering theorem, which is fully equivalent to the axiom of choice. What it says is that every set has a well-ordering. Which doesn’t say much until we define what well-ordering means. The reason that it’s so important is that the well-ordering theorem means that a form of inductive proof, called transfinite induction can be used on all sets.
One of the things that we always say is that we can recreate all of mathematics using set theory as a basis. What does that mean? Basically, it means that given some other branch of math, which works with some class of objects O using some set of axioms A, we can define a set-based construction of the objects of S(O), and them prove the axioms A about S(O) using the axioms of ZFC.