the paradoxes of set theory seem to have been back-benched in the latter half of the 20th Century and yet the fact that they are not resolved poses a major epistemological challenge to the foundations of mathematics. i have some idea on how we might begin to resolve these paradoxes for once and for all but i need to discuss them with some interested parties to begin with let me propose that we can get rid of most of the paradoxes of set theory by falsifying Cantor's theorem, i.e the Cardinal number of a set is less than the cardinal number of its power set a good reason for doing this would be that it is premised on the existence of a power set. let us propose that the power set of a set cannot exist and therefore cantor's theorem is false - any objections?
i'm to scared to write my name! | (203.87.44.8) | Saturday, 7 April 2001 7:05:26 PM
e=mc2
snigdhayan | (61.11.73.30) | Monday, 7 January 2002 4:30:33 PM
Admitting my utmost ignorance on sets, I must say that power sets are used
very frequently. Discrete topology on a set. How will you address these issues?
Also do you have a proof that throwing out the power set axiom will make set
theory paradox free?
Garry Herrington | (202.37.23.2) | Tuesday, 21 May 2002 2:00:49 PM
I'm not clear about what definition of "set" is meant here - Cantor, Peano-Russell, ZFC, NBG, etc ? Give us a clue.
Garry Herrington | (202.37.23.2) | Tuesday, 21 May 2002 2:02:50 PM
I'm not clear about what definition of "set" is meant here - Cantor, Peano-Russell, ZFC, NBG, etc ? Give us a clue.
wamalwav | (66.185.84.209) | Wednesday, 31 July 2002 1:54:15 PM
Beginning with snigdhayan, the fact that set theory has wide-spread application does not make it contradiction free. This in fact is the problem at the foundation of mathematics - the entire structure of pure mathematics with all its wonderful applications is built on an inconsistent (or incomplete at the very least) system. The question is - can we build a consistent theory of sets that still finds widespread application?
The definition of a set that I will lean on is from the Zermelo-Fraenkel system. The interesting point is that we are compelled to accept the axiom scheme of comprehension as our definition of a set. To show this let me first show that the axiom scheme of specification leads to a contradiction.
In the axiom of specification (EZ)(Ax)((xey)iff(xez)& f(x)) the set z can be termed the specification set. Every instance of the axiom of specification specifies a different specification set because otherwise z would be the set of all sets.
We can collect all these specification sets z into a set - the set of all specification sets Z. Z is defined formally as (EZ)(Ax)((xeZ)iff (xek)& (x is a z)). By definition k itself is a specification set. Therefore keZ and letting x=k for some x we have (keZ) iff (kek) & (k is a z).
But the axiom of regularity implies that -(xex). Therefore the axiom scheme of specification contradicts the axiom of regularity.
The axiom of specification fares no better than the axiom of comprehension and we will be proposing that the latter be reinstated bec ause it is a less ad hoc axiom than specification.
| (195.229.241.229) | Friday, 29 November 2002 10:36:36 PM
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| (195.229.241.230) | Friday, 29 November 2002 10:36:55 PM
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leicarg | (202.81.160.28) | Monday, 24 February 2003 10:26:24 PM
hi, im just a passerby here!
just want to ask if you can send me the paradoxes of set theory. The persons behind it, the evolution and everything about it. PLEASE!
i really need it before thursday. send it to me on my email address which is spawn_068@yahoo.com
thanks a million!