Re: The nature of the mathematical set

On Aug 14, 3:28 pm, John Jones <jonescard...@xxxxxxx> wrote:
So I think that you make three distinctions, while I make two. You have:
1) A list of elements (attributes in this case), comprised of 'unnamed'
elements (I presume this means elements that are all of one type, and
which can only be numbered)

Yes, relation attributes have the same type; they can be numbered or
named.

2) A set of elements (attributes) that is distinct from -
3) A set of elements (attributes) that is a bouquet.

I would say that 2) is not a set, but is still only a list like 1).

What do you mena by putting "attributes" in parenthesis next to
"elements"? Are they supposed to refer to the same thing?

Granted, a relation defined as as a set of tuples, each tuple having
the same signature which is abstracted as relation header is not
really simpler concept than (mathematical) set. However, this is just
a model. It honors some system of axioms (which differs a little from
boolean algebra). Therefore relations can be defined as objects that
satisfy these axioms, not as objects structured in the way I described
(tuples, header, etc).

... A particular set must be distinguished from its elements.

Turn off your crank-o-meter, because I'm going to redline it with the
following statement. It is the concept of set membership that makes
set theory axiomatization incomprehensible. All other operations
(union, intersection, complement) and relations (set inclusion) fit
into a nice algebraic model of boolean algebra. Can we please not to
talk about set elements?

I first thought that you were right. But then I saw no reason why such a
union is not allowable, it is just that in this case there are no common
elements? But we can have a mixed bouquet of course.

Definig a union of two bouquets is not a problem as they have the same
set of attributes. It is more of a problem to define a union of
relations with different signatures. Consider cars unioned with people
for example. A car has different set of attributes than human, of
course, but perhaps we'll find few attributes in common. When defining
a union of relations one has first to decide what attribute set of the
result is desirable: should we take their union, or intersection (or
maybe even symmetric difference)?

Yes, and to expand: First I need to be able to associate a set with
cars, and to associate a set with people. I can't just announce a set as
a "set of cars", or a "set of people", for no set is distinguished in
this format - I would have to distinguish a particular set. So, for
example, particular sets could be "a crowd (set) of people", and a
"convoy (set) of cars". Then, if I take the union of a crowd and a
convoy, I am not sure if anything meaningful emerges because, as you
say, they have different signatures. But what is certain is that there
is no set associated with a crowd and a convoy, whether or not their
members are the same.

Well, people had the exact same problem every time they extended
mathematical concept beyond its well established boundaries. Some
people still question legitimacy of real numbers!

In our case, we didn't even define this "generalized union" operation
yet, so speaking of it's practicality is little premature. I can
expand on this idea if anybody interested.

I think what Boole is getting at in this passage is that universes of
discourses are i) unique and ii) descriptively exhaustive concerning
their elements. Which is also what I was saying about sets - that they
are unique and cannot enter into relationship with other sets, because
entering into a relationship means that they are not descriptively
exhaustive on their own terms. Godel would disagree I think. From my
limited understanding, Godel conflates unique entities with incomplete
descriptions of their elements

There is little point to discuss a paragraph taken out of context. One
have to read the whole book, or a chapter, at least. Same applies to
usenet exchanges. Do you seriously believe that you can convey a
substantial idea in a series of usenet messages?

The above exerpt from the Boole's work seems pretty unimportant to me.
It doesn't matter how he arrived to his theory. What matters is the
beauty and symmetry of the laws in his algebra. What matters even more
is that his work was a stepping stone for the followers to greatly
expand his system.
.

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