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Assumption:
Many thing have been Categorised.

Assumption:
There are more than many things.

Conclusion:
There are some things which have not been categorised.

Problem 1:
I just categorised them, up there, in that sentence.

Problem 2:
MoonShadow is running around categorising everything.

Solution: This solution cannot be Categorised.

Problem 3:
A category of all solutions which cannot be categorised.  Ah me.  SetTheory? is fun.

Suggestion:
A category may have a different bunch of members depending upon how you look at it.

Proposal:
Wait until you get to PartIIIMaths, or alternatively study CategoryTheory in your spare time.  Then define the category of all Categories, and all links between them.  This allows Categories to be categorised.  Since we already know categories can be Categorised, we are done. []

Problem:
1) PartIIIMaths maths dissolves your brain.  2) Out of interest, what do you _call_ the 'category of all things not in any category' ?  I can see how you would use it as a 0 in some particularly twisted groups, but not what you would call it - since common sense says it can't exist in any sane universe.
Why isn't it just the EmptyCategory??  -- Senji
Because there are assumed to be elements in the universe which have not been categorosed - though MoonShadow is attempted to rectify that.

Solution:
1) Yes, it does.  Is there a problem with this?
2) In sufficiently sensible SetTheory?, such a phrase does not describe a set.  There are certain requirements about the definitions of sets (which currently escape me) to ensure that such ill-behaved "sets" aren't actually sets.

This is true - the simpler definition I think is usually given as "A set may not have a member which includes a member which is not a member of the set" - though that is by far the least elegant or MathMinded? solution.  I was under the impresison that a category was intended to let you define things that sets couldn't.  (Since, after all, it would be useful to talk about such things and be able to do things with them)  I guess you're hitting the DecidabilityProblem? though.

Ahh, no.  You're thinking of a Class.  The capital letter is very important - there's no class of things as described above, but there may be a Class of them.  I say "may be" because something as logically inconsistent as "All objects which don't contain themselves" might defy even Classship.

Hi all - for all those out there reading this who want to see how things fit together, have a look at
http://mathworld.wolfram.com/ClassSet.html
The three interesting terms mentioned here, namely Set, Class and Category, are fairly clearly stated, even for those who have not gone through the great ritual-of-mental-&-physical-masochism that is Part III maths.

Physical masochism?  OTOH, forget I asked, I don't think I want to know... :-)  --Jumlian
Nine o'clock lectures?
I quote, from the CULES members webspace:
"Paul, I didn't know you were into whips."
"He's a MathMo."
- Catherine and Sally



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