Ted Hoffman

I have done maths up to integration (calculus) - cant remember the rest.
I would like to understand sets (I dont know how they relate to topology?) with regard to the (bijections in) morphisms (homo, Iso, endo, auto etc) and transformations. In the end I hope to understand spaces like Hilbert space, perhaps riemmanian space and their relation to spacetime (manifold) etc, but first, sets.

Anyone knows of an online tutorial or resource or even book that can introduce me to sets - or do I have to fo to school?

Feather

You might start here: http://mathworld.wolfram.com/


I'm not a mathematician, so that's all the help I can give, but the site seems to list good references.

wade-w

Ahhh... Set Theory... my favorite branch of mathematics. Its also the foundation of all modern mathematics.

In "naive" set theory, a set is simply any collection of objects that can be described. You can either explicitly list the elements, or define some property P and say that your set consists of all objects that satisfy P. This is what most people think of when talking about sets. However, allowing a set to be formed by any property that can be articulated leads to serious problems such as Russell's Paradox, first discovered by Bertrand Russell.

This led to a more formal system called axiomatic set theory in which the idea of a set and set membership are primitives, and not formally defined, much as point, line and plane are not formally defined in geometry. The most common set of axioms for set theory are called the ZF axioms.

Here is a list of links on Set Theory.

This link provides a good overview, and a hypertext bibliography.

Axiomatic Set Theory is discussed on this link, with a formal statement of the ZF axioms.

Undercurrent

Michael's Book Reccoomenations:

The Joy of Sets

A good introduction to axiomatic set theory.

Introduction to Topology

A cheapie book that's real easy to read on topology taken from the axia. Probably the best book if you're interested in how set theory relates to topology.

A Course in Functional Analysis

Mmmm. Hilbert space. Will put hair on your chest though.

m.

Ted Hoffman

Thanks guys I really found the links useful. Especially the plato.stanford.edu one. Aah, just beautiful.

The Admiral

If you take the set of all SETI crunchers and apply the formal rules of SETI you will see that no individual or sub-set of SETI crunchers will ever acheive 100%

The Admiral

Ernest Sparks

Since nobody has mentioned it, check out abstract algebra too. You will ultimately run into it anyway, even in Topology.

abstract algebra

Peter Soderqvist

Classical syllogism consists of two real or imagined premises, and a conclusion!
A classical proposition; all philosophers are mortals, Socrates was a philosopher, therefore; Socrates was a mortal!

All Cretans are liars, I am a Cretan, therefore; I am a liar!
I have told you the truth that I am a liar; therefore I don't belong to the set of liars!

Skeptics don't know anything, you are a skeptic, and therefore you don't know anything! You know that you don't know anything, therefore; you don't belong to the set of un-knower!

Intensity if you are alleging that the numbers of propositions or sets are computable units, you need to know if all propositions or all sets are a member of it self!

Are all propositions, or all sets a member of itself?

A simple explanation with few explanation grounds is to prefer, except when you need to hide your flaws!

wade-w

Flaws? As I thought I explained in my earlier post, that is a problem in naive set theory. And your sources explanation of Russell's Paradox in terms of set theory is is rather incoherent. The set of all sets doesn't have anything to do with it. It goes like this: Consider the set S of all sets that do not contain themselves. Then S is a member of S if and only if S is not a member of S. The problem comes from allowing self referential definitions. This is well known, and there is no attempt to "hide" any flaws.

Peter Soderqvist

TO WADE-W

Soderqvist1: My source has referred to Bertrand Russell, and Alfred whitehead's Principia Mathematica! This is what I found on Internet regarding Principia Mathematica, the theory of logical types!

Fuzzy logic
The set of all sets is a set, and so it is a member of itself. Yet the set of all apples is not a member of itself because its members are apples and not sets. Perceiving the underlying contradiction, Russell then asked, "Is the set of all sets that are not members of themselves a member of itself ?" If it is, it isn't; if it isn't, it is.
http://www.fortunecity.com/emachines.../fuzzylog.html

Stanford Encyclopedia of Philosophy Russell's Paradox
Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves.
http://plato.stanford.edu/entries/russell-paradox/

A history of set theory
In 1899 Cantor discovered another paradox which arises from the set of all sets. What is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal yet the cardinal of the set of all subsets of a set always has a greater cardinal than the set itself.
http://www-groups.dcs.st-and.ac.uk/~...et_theory.html

Principia Mathematica by Bertrand Russell
February. Cantor had a proof that there is no greatest number, and it seemed to me that the number of all the things in the world ought to be the greatest possible. Accordingly, I examined his proof with some minuteness, and endeavored to apply it to the class of all the things there are. This led me to consider those classes which are not members of themselves, and to ask whether the class of such classes is or is not a member of itself. I found that either answer implies its contradictory.
http://www.cut-the-knot.com/selfreference/russell.shtml

Soderqvist1: The set of all sets, which are not members of them themselves, or the set of all sets of all the things there are, appears at least to me as complementary contradictions, as my earlier link has said!