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Re: Normalisation

From: VC <boston103_at_hotmail.com>
Date: Sun, 10 Jul 2005 20:38:43 -0400
Message-ID: <j5-dnQAOB6YMIkzfRVn-3Q@comcast.com>


Hi,

Now that we are done with conceptual objects and stuff, let's take a look at sets ;)

"Jan Hidders" <jan.hidders_at_REMOVETHIS.pandora.be> wrote in message news:7DCze.140558$Jn.7313539_at_phobos.telenet-ops.be...
> Jon Heggland wrote:

>> In article <Ezeze.139544$p21.7384199_at_phobos.telenet-ops.be>, 
>> jan.hidders_at_REMOVETHIS.pandora.be says...
>>
>>>>>Ah, but now you are using the domain or relations, right? There is a 
>>>>>problem with that domain. It doesn't exist. The collection of all 
>>>>>relations is a proper class, and not a set, but domains have to be 
>>>>>sets.
>>>>
>>>>You'll have to educate me on the difference between "proper class" and 
>>>>"domain", I'm afraid. The term "class" is used for so many slightly 
>>>>different things.
>>>
>>>http://en.wikipedia.org/wiki/Class_(set_theory)
>>
>> Call me stupid, but you still have to explain to me why the "collection" 
>> of relations is not a set. I can't figure out from Wikipedia what it is 
>> that disqualifies something from being a set.
>

> That's not so easy to explain, but I'll give a hint. Do you know about
> Russel's paradox? The contradiction that appears when you assume the
> existense of the set of all sets?

So far so good.

> A solution to that paradox is to accept that it is actually not a set, at
> least not in the sense that all the usual axioms of set theory apply to
> it.

That would not be what's considered the "standard" solution (ZF set theory). The notion of "proper set" simply does not exist in ZF. Everything is a set and its elements.

> In some sense you might say that is is "too large" to be a set.

That's a possible approach but not necessary with the relational model where ZF is quite sufficient, thank you very much.

>> Are you saying that the "domain of sets" is not a domain either?
>

> Indeed.

>

Set of *all* sets does not exist in ZF, and if you define *your* set of sets so that it would satisfy the ZF axioms, you are OK (e.g. do not include in your set of sets the set of sets itself or any other set where your set of sets would be a member). The Z (Zermelo) axioms can be derived from the "iterative" conception of sets which can be briefly described so: at stage zero let's form all possible collections of elements, if no elements available, form an ampty set; at stage one let's form all possible collections using individuals *and* collections fom stage zero; ... ; at stage three let's form all possible collections using stuff from stage zero, one and two; and so on. It's easy to see that with the iterative conception, no set belongs to itself, and therefore, there is no set of all sets. So, if you take care to form your set of sets so that it would contain only sets from the previous stages (easy to do), you should be cool.

>>>>Should I be forbidden from treating "relation" as a (generic) domain 
>>>>when defining this operator? Why?
>>>
>>>Because by definition it isn't, and redefining the notion of domain such 
>>>that it is, is not that easy without either running into paradoxes or 
>>>getting a notion which it is almost impossible to reason about.
>>

Well, a set of relations would most certainly be a set if you take necessary precautions (see the iterative conception). I cannot imagine how one can describe a set of relations (in the RM) so that one would run into the Russell paradox. Can you ? ;)

>> Can you give me any examples of trouble arising from this? And an 
>> explanation why the relational operators do not run into paradoxes? Or do 
>> they?
>

> They are not functions defined over domains

The relational operators are of course ordinary functions, nothing fancy about them, of the kind F: R->R where R is a *set* of relations.

> Note that my remark that started this was that the nest operation cannot
> be defined as a function *over* *domains*.

But it can, why not ?

vc Received on Sun Jul 10 2005 - 19:38:43 CDT

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