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vfilipch wrote:
> Rupert wrote:
> > vfilipch wrote:
> > > 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."
> > >
> >
> > I'm not really following you here. Perhaps you could clarify which set
> > theory you're talking about. Are you talking about ZFC, in which all
> > the sets are pure? Or are you talking about ZFC with atoms?
> >
> > > Is there something in Set Theory which would prevent situations like
> > > that? Is there any restrictions on elements in the Set Theory ?
> > >
> >
> > Given any theory T, we can construct an extension of that theory
> > ZFCA[T] in which the axioms of T are true relativized to the
> > individuals, and also all the axioms of ZFCA are true. Assuming the
> > axiom of inaccessible cardinals, if T is consistent then ZFCA[T] will
> > be consistent.
> >
> > > BTW, I couldn't find anywhere a definition of "element" in the Set
> > > Theory.
> >
> > The notions of "set" and "membership" are undefined. They are primitive
> > notions of the theory.
> >
> > > Is there a such definition besides "member of set", which
> > > sounds to me like a tautaulogy.
> > >
> > > I hope I am clear in my questions :) and hope to read more replies.
> > >
> > > Thanks for your thoughts.
>
> Thank you for a reply.
>
> When I stared to look at the problem I mentioned in the beginning I
> looked at Set Theory described in
> http://www.mtnmath.com/book/node53.html#SECTION001420000000000000000
>
> You say, "The notions of "set" and "membership" are undefined. They are
> primitive notions of the theory."
>
> This is a surprising statement. If such basics are not defined, then
> HOW they are understood, and HOW can they be used for building a Set
> Theory?
>
> For example, in simple Euclid's geometry we cannot advnace utill there
> is a common understanding of things like point, line, direction, etc.
> This basics concepts I see as abstarctions of objects of reality, I
> mean I don't know any other way to teach a child what "point" is
> without simply showing them examples of "point" in real world. In other
> words basic concepts of geometry are feferencing to qualities of real
> objects, we can say that point is referencing to location and nothing
> else.
>
It can actually be quite helpful not to know what points and lines are.
That way you won't think that things follow from the axioms when they
don't, you won't be misled by your preconception of what points and
lines should be like. Hilbert said "one should always be able to say,
not points, planes and planes - but also tables, chairs, beermugs." The
idea is that the theorems hold for *any* structure of which the axioms
are true.
But you may have in mind a particular structure of which the axioms
hold, and you may be able to explain in some way what structure you
have in mind.
Similarly with set and membership. Mathematicians have a particular
notion of set and membership in mind, and they can explain roughly what
they mean. But they can't give a precise definition, unless they use a
different foundation. The notions of "set" the "membership" are the
basic notions in terms of which everything else is defined. Definitions
have to come to an end somewhere.
> But what about Set Theory? How "element", "set" and "membership" are
> understood?
> Or in other words, to what exactly they are referencing?
>
> Thanks for your thoughts.
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