Re: Null Value In Set Theory

On Jul 29, 2005 9:00PM, Brendan O'Sullivan wrote:

> I'd always take the null component to be a member of
> any set. By definition,
> an empty set is any set which has only the null
> component as its only
> member.

I haven't studied a lot of set theory, and I suppose it would all depend on what axioms you were using. However, I think that it's invalid to "take the null component to be a member of any set" (unless you take "null" to be the empty set and consider power sets).

Also, I think you have to be careful when you say "by definition, an empty set is any set which has only the null component as it's only member." (Again, this doesn't really make sense unless you define the "null compeonent" to be the empty set and consider power sets.)

>From what I have read about set theory there is only one empty set. If you were to have two empty sets it could be shown that they are actually subsets of each other and hence they would be equal. So there is really only *the* empty set.

Now consider the compliment of the empty set, sometimes called the universal set, it should contain EVERY possible element. However, if "null" is an element of the empty set (by your definition "null" is the only component of the empty set), then "null" cannot be an element of the universal set which doesn't make.

If this were the case then the empty set couldn't be a subset of the universal set, and this contradicts your first statement about the empty set. Also, it's well known that the empty set is a subset of any possible set.

Like I said though, I suppose it really depends on what axioms you use and how you actually define "null". I think the trouble arises from considering "null" to be a component (or element) of a set rather than an actual (empty) set itself.

Regards,
Kyle
.

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