Great job so far, but I’m not sure what you mean here–any set can be ordered. In fact, I think probably the easiest statement of the axiom of choice is the following: Every set can be well ordered (A well ordered set is a set with an order such that if you take any nonempty subset, it’s guaranteed to have a smallest element). So, for example, the real numbers aren’t well ordered by the usual order (there is no smallest element of the reals), but there is a well ordering of the reals, since any set can be well ordered. Did you perhaps mean to say that the set of solutions of polynomial equations does not form an ordered field?
I was being a little loose with my terminology, but what I meant was, “ordered consistently with the axioms of addition and multiplication”. Sorry about that, chief.
Cool. Sorry, didn’t mean to nitpick, I figured that was basically what you meant, just wanted to make sure.
I’m not really familiar with the construction of the surreal numbers, so I’ll be interested to read what you have here when I get the time. Carry on! 
Chas.E wrote
Why Chas.E, you made a joke of a humble and self-deprecating nature. And it was funny. I’m impressed.
Are all fields of mathematics open to discussion here?
I have what should be a basic question in complexity theory. I’ve studied it randomly, and am familiar with many of the concepts. I understand orders of magnitude, some of the related graph theory, TSP and other standard problems, turing machines, etc… One of the basic tenets that comes up again and again I just don’t get. Please explain the meaning of NP-Complete.
It’s a bit of a hijack, but it’s something I happen to know about. Gimme just a couple minutes.
MPSIMS now stands for “Mathematical/Philosophical Stuff I Must Share”. Check this thread for a discussion of NP-completeness.
Thanks for fielding that one ultrafilter. Although I have an idea in my head what NP-complete means, I would have made a total hash of explaining it.
No problem.
Previously I’d only had a vague familiarity with the foundations of mathematics, so it is easier for me to assume I know nothing and so far work only with what’s been laid out excellently by MrDeath. Although more book recommendations wouldn’t hurt. In fact, it might not be bad to start a thread of math or other technical book recommendations here. (Were there any previous? I’ll search, and if not I’ll start it up soon.)
Now I’m just going to throw out some thoughts on the following. Probably there’s a simple answer, but I hope to learn from my attempts anyway :
*From MrDeath : *
Exercise: Show that the class {X | X is a set} is not a set.
We have to show that it violates the axioms or is inconsistent (although I don’t really know what consistency is, I’m assuming it means ‘makes sense logically’ for now).
I’ll call this class K. If I read this right, K is the class of all sets (is that proper usage of the term?).
Now, it seems strange that K would have to be a member of itself, if it were a set, but I don’t see how that really violates anything.
I had thought that there might be a set which is defined as “not a member of K”, but I would think that such a set could not be defined, since K’s definition would not allow it. Or is the ability to think of such a set indicate that K itself cannot be a set?
Feel free to give hints if you’d rather I work on it some more.
Hint alert!
Consider the axiom of specification.
Hint alert!
I didn’t find any threads like that, so I went ahead and started one in IMHO.
One request to MrDeath: Could you say a few words about supernatural numbers?
Hm, supernatural numbers. First time I ever heard of them was, oh, about ten minutes ago, in this very thread. The only thing I can find on the Web is a link to Knuth’s book The Mathematical Gardener. I’m guessing that it’s just Knuth doing a too-clever-by-half job of describing the ordinals (which are a superset, and in a sense the natural extension of, the natural numbers). There, that’s a few words. You want more, I’m going to have to hunt through my university’s library.
Again, I’m skipping a day; an every-other-day pace feels about right to me, and gives plenty of time for pleasant diversions So come back tomorrow and I’ll go a long ways further on numbers.
I don’t know if they’re the invention of Knuth, but the first place I heard of them was in Godel, Escher, Bach. They are described as the result of making an axiom of the negation of the Godel sentence for standard number theory. But if you don’t know anything, don’t worry about it.
As is my habit, I am going to hijack your thread. Primarily because I’m a grammar geek, and numbers have special grammatical properties, but secondarily because I hit the wall at Calc 2, so I cannot really contribute to a math theory thread, but I hate being left out. So, here is a short primer on numbers as words.
Cardinal numbers:
- When used as noun modifiers, they are in form adjectives;
The four girls walked to school together.
- When they modify a noun without “the”, they belong to a special function class called determiners:
I have twenty cats.
(Other determiners are articles [a, an, the] and posessive nouns and pronouns)
- When the nouns that they modify are deleted, they can function as nouns:
Those two are crazy.
- “One”, when used by itself, is a gender-neutral indeterminate personal pronoun. When used as a noun modifier or a posessive it is a determiner.
One should always brush one’s teeth.
I only have one car.
-
When used to order a list, they are adverbs. The numerals at the beginning of each item on this list are all adverbs.
-
Up to twenty, numbers should generally be spelled in writing; from 21 and up, it is kosher to use numerals.
-
Other than in rule #4, numbers written as words and written as numerals are treated the same.
-
All numbers and mathematical symbols are alphabetized as if they were spelled out. Computers don’t understand this, which is why they alphabetize numerals as if 0-9 were the first ten letters of the alphabet.
Numerical Order: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
True ABC Order: 8, 11, 5, 4, 9, 1, 7, 6, 10, 3, 12, 2, 0
(Interesting factoid–no matter how many numbers on your list, eight is always first, and zero is always last alphabetically)
Computer ABC Order: 0, 1, 11, 12, 2, 3, 4, 5, 6, 7, 8, 9
Ordinal Numbers
- When used as noun modifiers, are adjectives.
We’ll take the tenth caller.
- When used to order a list, are adverbs.
First, I opened the package. Second, I checked the parts. Third, I read the instructions. Etc.
-
Up to ten, they can be written in word or in numerical form; above ten, they are generally written in numerical form. The two forms are treated the same.
-
Can be used with or without the -ly suffix in writing. It is acceptable to use -ly starting with either first or second. Once the -ly suffix has been attached to either first or second, it must be attached to every subsequent member of the list. All of the following are acceptable; the first is the preferred form.
First, I never said that. Second, you misquoted me. Third, piss off.
Firstly, I never said that. Secondly, you misquoted me. Thirdly, piss off.
First, I never said that. Secondly, you misquoted me. Thirdly, piss off.
- Can be used as determiners.
First children are usually most confident.
I now return you to your regularly scheduled mathematics discussion.
[major hijack, but no better place for it]
Crispin Sartwell, a professor of philosophy at Penn State, has posted the following challenge on his Web site:
The contest seems to have stemmed from his happening across the definition of the word ‘seven’ in a dictionary; as he discusses here, he was bowled over by the lack of rigor. (Wow! Dictionaries aren’t mathematically rigorous! And in other breaking news, elementary school science textbooks don’t give a rigorous defense of the theory of evolution!) As he says:
The previous entries in his contest are here. Thought that if anyone here wanted to take on this poseur, they’d probably be reading this thread.
[/hijack]