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"vfilipch" <vfilipch@xxxxxxxxx> wrote in message
news:1154654905.906049.75650@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> Rupert wrote:
>> vfilipch wrote:
>> > I would like to hear a opinion of prfessionals (because I am not one)
>> > on the following question:
>> >
>> > Is it possible to apply Set Theory for an elment which I would like to
>> > define as an "unobserved object"? In other words, for an element for
>> > which there is no way to know ANY properties.
>> >
>> > Thanks in advance.
>>
>> Set theory usually doesn't care much about the nature of the objects.
>> There are versions of set theory in which everything is a set, and
>> there are versions in which there are objects which are not sets (known
>> as individuals). In the latter case, we usually don't care much about
>> the properties of the individuals. We just study the sets of them, the
>> sets of sets of them, and so on.
>
> Thank you very much for a reply!
>
> I asked this question because I was thinking about a following
> possibility:
>
> Since an "unobserved object" may have (I can imagine) a specific
> "property" which would make all axioms of Set Theory to be invalid, I
> would guess that it would be improper to use Set Theory on such wierdly
> defined elements as "unobserved object."
>
Set theory deals with the properties of sets, not elements of sets. The only
propert that elements have is set membership.
I can easily define and use the set of "All odd perfect numbers", without
knowing if this is the empty set, a finite set, or an infinite set.
I can even define a set of "all statements that are provable but untrue in
ZFC" without any problems.
> Is there something in Set Theory which would prevent situations like
> that? Is there any restrictions on elements in the Set Theory ?
In ZFC, sets are constructed out of nothing but the null set and sets which
can be formed from the null set using the Axioms of ZFC (the axiom of
infinity says that an infinite set can be formed in this manner; I don't
know if you would consider this an exception).
If by "set theory" you mean something other than ZFC, you will have to say
exactly what you mean by "Set Theory".
>
> BTW, I couldn't find anywhere a definition of "element" in the Set
> Theory. Is there a such definition besides "member of set", which
> sounds to me like a tautaulogy.
>
Its more a definition of what a set is, than a definition of what an element
of a set is.
In ZFC the elements of a set are either nothing (for the empty set, defined
as existing) or other sets constructed using the Axioms of ZFC. Feel free to
add additional sets if you want making ZFCvfilpch.
> I hope I am clear in my questions :) and hope to read more replies.
>
> Thanks for your thoughts.
>
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