Re: Is {} and element of all sets?

*** T. Winter wrote:

In article <keGdnWh68oVpNTDZnZ2dnUVZ_o2dnZ2d@xxxxxxxxxxxxx> Hatto von
Aquitanien <abbot@xxxxxxxxxxxxxx> writes: ...
> ORIGINAL QUESTION:
>
> "Is there some inconsistency between common interpretations of
> the empty set as a member of other sets?"

This is a different question. The answer to this question is NO.

And if you go to the first post in this thread, you will find that that is
the question that I posed.

The
empty set can be a member of other sets, using the common interpretation.
The empty set is a member of {{}} using the common interpretation.

But what you asked is (see subject):
"Is {} an element of all sets",
and using a common interpretation the answer is NO.

How can you answer that question without specifying the definitions and/or
axioms of the system to which the question is applied?

I can conceive of some form of set theory where things are either sets or
objects. And where "primitive element of" means being an object within a
set, and where "element of" means being either a primitive element within
a set, or being a subset of that set.

The word "element" actually derives from the Latin "elementum" which means
irreducible constituent. It is fairly close in meaning to the
Greek "atom". Making that assumption (that elements are indivisible), and
understanding that sets of elements constitute a distinct category
from "things (elements) put into sets", one can reasonable consider subsets
as members of a set and yet they are not elements. The only set which is
irreducible (in the sense of having no internal structure) is the empty
set.

In that case the empty set is not a
primitive element of every set, but is an element of every set (that is,
it is a subset of every set). Changing terminology does not help in
comprehension.

Clarifying terminology does, however.

But to answer your very first question, whether that is common, the answer
is no. It is quite uncommon.

Thank you. That was the question.
--
Nil conscire sibi
.

Quantcast