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

loveMath
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Registered: 2015-02-18
Posts: 3

My question is concerned with set theory.

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)

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

varunvofficial
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Registered: 2015-04-10
Posts: 1

Re: My question is concerned with set theory.

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

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

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: My question is concerned with set theory.

{x} and {{x}} are different


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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

Bob
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Registered: 2010-06-20
Posts: 10,052

Re: My question is concerned with set theory.

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


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: My question is concerned with set theory.

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


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: My question is concerned with set theory.

Why is it important to define natural numbers?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: My question is concerned with set theory.

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


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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

Olinguito
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Registered: 2014-08-12
Posts: 649

Re: My question is concerned with set theory.


Bassaricyon neblina

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