(Part 2) Reddit mentions: The best combinatorics books

> The distinction is that in math, all foundational meta-theories are require to get the right answers on simple object-level questions like "What's 1 + 1?". If your mathematical metatheory answers, "-3.7" rather than "2", then it is not "different", it is simply wrong. We can thus say that Foundations of Mathematics is always done with a realist view.

The natural numbers are an interesting example, which goes back to ADefiniteDescription's point about a privileged model. The basic axioms of arithmetic are categorical, i.e. have only one model, up to isomorphism. Not all theories have this property, though.

If it could be shown that some moral theory similarly has only one correct interpretation - that all alternative interpretations end up being isomorphic - then that could support a kind of realism, at least in the context of that theory. A lot would depend on the nature and scope of the theory in question, and its interpretation.

So perhaps Parfit's position would be better captured by saying that he believes there are unique true answers to moral questions, as there are for questions in categorical mathematical theories.

> What's a good textbook for that field, anyway?

The books I studied are quite outdated now, but a classic modern text is Model Theory by Chang & Keisler. That might be more comprehensive than you're looking for. You could try Model Theory: An Introduction - its first chapter is quite a concise basic intro. There's also A Shorter Model Theory.

This is probably what you're looking for: A Tour Through Mathematical Logic by Robert S. Wolf.

It's a great textbook and takes you through these topics:

) Predicate logic

) Axiomatic set theory

) Recursion theory and computability

) Godel's incompleteness theorems

) Model theory

) Contemporary set theory

) Nonstandard analysis

) Constructive mathematics

I always liked C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, 1968. It's old but i think still one of the best introductions on that subject.

Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994 is imho also decent. More information on the book is available here.

I recommend this: http://www.amazon.com/Problems-Theorems-Classical-Problem-Mathematics/dp/1441921400/ref=sr_1_4?s=books&ie=UTF8&qid=1332582305&sr=1-4

You prove essentially everything yourself. It's great fun.

You should check out the books published in the New Mathematical Library series.

Here are a few I think might be really awesome:

Geometric Inequalities by Kazarinoff

Invitation to Number Theory by Ore

Numbers: Rational and Irrational by Niven

Mathematics of Choice: Or, How to Count Without Counting by Niven

Episodes from the Early History of Mathematics by Aaboe

Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Honsberger

I wish I knew about these books when I was in high school.

What's your previous math experience?

Personally, I think Van Lint and Wilson would be excellent for someone who already has several undergraduate math courses under their belt (e.g. calculus and linear algebra). However, if you've taken fewer courses than that or are a bit earlier in your education, other books might be better.

link to that book on Amazon

Ah, the "least read great book of modern science."

Laugh if you will, but when I was in high school I nearly blew a substantial chunk of my savings on the (rare and monstrously overpriced) Principia Mathematica. Fortunately, I found and bought the abridged edition, which cured me of the temptation forever.

yes, but ∅ is not contained in {}, i.e. ∅∉∅. So your thinking isn't quite right yet.

As for "Russell's Paradox does not exist in the real world"... I suggest reading Vicious Circles or The Liar: an Essay on Truth and Circularity. The theory of non-wellfounded sets is closely related to the topic of circular programming, self-referential data structures, and value recursion, such as my paper on corecursive queues. The bibliography includes a few references to a few of the classic works in this area.

glad i'm not the only one, damn you machover

A three-volume set (one, two, three) appears to be available on Amazon.com for $25 per volume.

I took a college class that spent about half of the semester answering this very question. The class was "Metatheory of Propositional Logic", and the textbook was Set Theory, Logic, and Their Limitations. As an engineer, I found it grueling and unpleasant.

First we had to establish what "1" was, and we decided it was "the set of all sets which contain only 1 item". Then we had to decide what "plus" was, and of course it was a set union. Then we had to show that the cardinality of the set 1 union 1 was the same size as the set of all sets that contain 2 items. I think. It's been a while.