My question is concerned with set theory.

loveMath · 2015-02-18 21:23:05

I have read that math is under a crisis due to be heavily based on set theory, and in its turn set theory has unanswered questions. In my opnion, set theory the way I learned is problematic even from the very beginning. Some may not agree with this statement, but I will try to give a start: if possible, can anybody help me with this question : given the two sets A={x} and B={{x}} are they equal?
Well, if we forget for while the rules of set theory and the way of thinking (stablished up to now) and behave like a child and ask: if A = B then if we do operations on left hand side and/or right hand side, and if in any circunstance we find the same set of symbols, representing LHS and RHS, then A = B.

So following this the LHS = A and RHS = B. Now just ask the contents of the RHS, or what are/is the element(s) of set B, you will find {x}, which is A, then for this analogy A=B (again forgeting rules of set theory for while).

Another question is: could we devolope a set of rules for set theory where this kind of situation is ruled out?

For instance if we think about the expressions A = t and B=(3-2) t^(5-4), in order to find out that A=B we just need to develop the RHS, as before said, and it is enough!

After tiping I realize that according to authors, B is not a set, but a family of set !!

Another question would be: is a family of set a set? That it is to say: can a family of set be considered a set or not?

Last edited by loveMath (2015-02-18 21:41:49)

varunvofficial · 2015-04-10 22:57:18

i think yes. Because the family of sets are also a set but the members in that set is some other sets. I dont remember the exact explanations but it will be explained in the book of RD SHARMA

Agnishom · 2015-04-11 00:47:59

{x} and {{x}} are different

Bob · 2015-04-11 01:00:44

A={x} and B={{x}}

Are they the same? No. A ={x} and B = {A}. These are not the same any more than sin(sin(x)) is the same as sin(x).

As far as I am aware, set theory was only thought to create a difficulty because of this paradox:

Let A = {the set of all sets that do not contain themselves}

Is A ∈ A ?

Well, A is a set so that's a good start.

Let's say it does contain itself.

Then it is in the set of all sets that don't contain themselves. =><= ie. a contradiction.

So, alternatively, let's say that A doesn't contain itself.

Then it should be placed in A, by the definition of A. And we have a contradiction again!

The paradox is avoided if you simply require that sets must be 'well defined'. That means, you must be able to decide whether an item is a member or not. So sets like A are excluded from set theory ... paradox avoided.

http://en.wikipedia.org/wiki/Russell's_paradox

Bob

anonimnystefy · 2015-04-11 07:43:45

What's more, the definition of natural numbers hinges on the fact that {} is different from {{}}.

Agnishom · 2015-04-11 22:17:44

Why is it important to define natural numbers?

anonimnystefy · 2015-04-12 01:56:05

It's the number set we are the most familiar with. If we do not define it, we might as well not have mathematics.

Olinguito · 2015-04-12 02:37:35

https://www.youtube.com/watch?v=BE657F9sqyQ

Math Is Fun Forum

#1 2015-02-18 21:23:05

My question is concerned with set theory.

#2 2015-04-10 22:57:18

Re: My question is concerned with set theory.

#3 2015-04-11 00:47:59

Re: My question is concerned with set theory.

#4 2015-04-11 01:00:44

Re: My question is concerned with set theory.

#5 2015-04-11 07:43:45

Re: My question is concerned with set theory.

#6 2015-04-11 22:17:44

Re: My question is concerned with set theory.

#7 2015-04-12 01:56:05

Re: My question is concerned with set theory.

#8 2015-04-12 02:37:35

Re: My question is concerned with set theory.

Board footer