And there in lies the problem.  You are operating under what is called (I swear, this is the actual term, I'm not trying to be insulting), "naive set theory".  It is the concept that any collection can be a set.  Naive set theory is called so because it is, well, naive.  It leads to contradictions and paradoxes.

On the other hand, I am going off of ZF set theory, which is the just about the standard when it comes to set theory in mathematics (although there are others).  In ZF set theory, you can't just claim something is a set.  The set must be constructed through the use of existing sets.  The only set that is guaranteed to exist is the null set (which I will be denoting θ simply because of laziness).  But through precise axioms, we can take the null set and form sets such as {θ} and {{θ}} or {θ, {{θ}}}.  In ZF set theory, the Axiom of Union is used to define what we typically think of when we see union (though said in a completely different form).

So you need union to state that your set {1, 2, 3, 4, 5, 6} exists before you can ever even talk about it being a set.  And this is the reason why we can never have union be an operator.  Because the sets need to exist before we can call it an operator, but we need union to exist before we can say the sets exist.

Guys, this is such a great discussion, but it is in the Formulas section, so due for deletion at some point!

Ricky wrote:

Sekky, we're starting to beat around the bush here.  Please come up with a binary operator definition for the union of two (possibly different) sets.

Hello? The definition is right there in my last post!

Last edited by Sekky (2007-02-27 01:12:22)

Ricky wrote:

I'm sorry, but that is not a binary operation.  A binary operation is a mapping from AxA to A.

Hello!? A binary operation is an operation whose arity is two

Which is exactly the same as saying a mapping from AxA to A

It maps a power set cross power set to the power set, it will always be closed.

You're arguing me back down the exact point I'm putting forward. Here is an extremely simple analogy.

Last edited by Sekky (2007-02-27 06:05:06)

defining union from AxA to A?

Last edited by kylekatarn (2007-02-27 10:29:29)

I see your point, the definition doesn't seem correct after all.

if X,Y are subsets, then 'U' cant be defined in AxA. , but in P[A]xP[A], cartesian product of the power sets of A, right ?

darn: (

Remove this after cleanup:
Something from wikipedia....
Some philosophers take a more radical approach, holding that some contradictions are true, and thus a theory's being inconsistent is not always an indication that it is incorrect. This view, known as dialetheism, is motivated by several considerations, most notably an inclination to take certain paradoxes such as the Liar and Russell's paradox at face value.
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What if you guys just make two or more different threads on different kinds of set theory?

Last edited by John E. Franklin (2007-02-27 12:22:03)

Yet again, you post an example.  What I am asking for is a general definition.  For example, a general definition for addition on the natural numbers is:

Ricky wrote:

Yet again, you post an example.  What I am asking for is a general definition.  For example, a general definition for addition on the natural numbers is:

Where S(n) is the successor function of n.

Ok, define addition for the set {{},{1},{2},{1,2}}

You can't, it makes no sense.

Define addition for the set {fish}

Define the union operation for the natural numbers.

The natural numbers are a set, no more or less specific than the set {{},{1},{2},{1,2}}

I've defined union on my set, you've defined addition on your set. There is nothing more or less specific or general about either of them. You have generalised nothing. You have defined one operation specific to the set. I have defined an operation specific to my set. My operation just happens to be applicable to all power sets.

Thanks, medivijaysagar!
It is a combined effort of all the members of this forum.
Welcome to the forum!!!