x = [x] !!!!

Julio wrote :

Porky Pig Jr wrote:

On May 1, 7:55 am, "Mark" <u...@xxxxxxxx> wrote:
Hi, most definitions of element and set I have
come
across, say something
like,

An element is any object of our perception or of
our thought.
A set is a collection of unique elements.


OK, that was discussed many times, but let's
repeat
it anyway. A set
is undefined concept and can only be characterized
by
what's known as
"set membership". We say that s in S and call s
"an
element" of a set.
However, at least in ZF set axioms we do not
distinguish elements of
set from the set. An element of set is a set on
its
own, so s is a
set.

thus x = [x]

2 is an element of a set , the set [2]

thus x = (x) ( () is same as [] )

just as i always said : x = [x]

the foundational axiom of tommy set theory.

with or without ZFC.

without russel-like paradoxes.




For instance, we can think of a set of all
natural numbers N,
and say that 1 in N, but 1 is a set on its own
(check
Von Neumann
numerals).


ZF is more basic then Von Neumann.


That sounds interesting, although I am going to
contradict it. Struggling a formalization: x = [x] a
bit like: _element_ IS _set_ (or whichever
terminology you might wish there).

as you say x = [x] , yes.

which i said long ago here on sci.math ( and earlier elsewhere )



Keeping in mind that elements in a set are supposed
to be "unique", an immediate consequence is that
either an element is in the set or is it not. The
_contra-diction_ is in the fact that non-belonging is
recursive and belonging is not... from outside!

what ??

2 is in [2] , in fact its the same.

2 is not in [3] ... what recurvisive ? ... what from outside ?



Now, I'll take it the other way round: elements in a
set _do_ share a "specific" property, that being that
they belong to *that* set.

"specific" but not unique :

2 and 3 are in [2,3,5] and not in [5,7]

but 2 is also in [2,7] and 3 also in [3]

That property, I have
(ultimately) called _exclusiveness_ (which, BTW,
sounds good enough to me)

exclusive sounds like unique to me ... which i countered above.

, and it seems enough to: on
a side, found elements and sets, whatever they are;
on the other, provide a (formal) definition of the
complement of a set, from within the set itself!

That (I'd say) makes our objects self-contained
universes, at least as far as mathematics goes...

-LV



from the comments in between i reach another conclusion.

x = [x]



So whats a collection?

There is no formal definition of collection.
Informally we say that
set is a collection of unique elements, OK, but
"collection" is not
defined just as "set".

In practice, however, we often have the following
situation. We work
with a set, say S, and even its members s1, s2,
...
are also sets, we
would like to pretend that they are atomic (if
this
assumption simply
our logic but doesn't screw anything). Now suppose
we
need to create
the 'higher-level' set, A, consisting of S1, S2,
...;
often to avoid
confusion we would like to call that second level
aggregation by some
name other than set; a "family" or "collection"
are
often used, but
once again, that's just a matter of convenience;
formally A is a set
consisting of S1, S2, ..., in turn each Sn
consists
of something else,
and that something else is what we treat as
'atomic
element' simply
because we don't care about its internal
structure.
E.g., when I work
with a set N, normally I don't care about set
representation of 1,
2, ... (but if we do need to prove the laws
governing
the natural
numbers, we have to look at the internal structure
of
the elements of
N, and this is where Von Neumann representation
comes
to play).

So you may run into something like "consider the
family of all compact
sets on something". Here the "family" is still a
set,
we just call it
a family for convenience. You'll also see the
'first
level set' are
designated by uppercase, the second level - by
fancy
script letter in
the beginning of alphabet, and the third level by
the
fancy script
letter closer to the end of alphabet. If I
remember
correctly, Halmos
in Naive Set Theory discusses those conventions.
(Normally we wouldn't
go higher than three levels - at least I hope so.)



So whats a collection?
Wolfram says it's a multiset.
Wiki says it's a multiset.


No, they don't. Neither Wolfram nor Wiki use the
word
'collection'.
Try to understand what they say and not to read
"between the lines".

This basically gives the following definitions.

A multiset is a collection of elements
A set is a multiset of unique elements.


No, both Wolfram and Wiki give exactly the same
definition: multiset
is generalization of set; if multiset we allow
multiplicity of
elements; a set is multiset with multiplicity of
1.
So multiset is defined as generalization of set,
this
is as much as we
can say. Again, seems like you just can't
comprehend
neither Wolfram
nor Wiki definitions. That's *your* problem.

So whats a collection?

An informal synonym for a set.

Would this be a good definition of colletion,

Just as a set, it's undefined.

A collection is any elements which have
something
in common.


All elements of any set have something in common
by
the virtue of the
fact that they belong to that set. A good example
would be collection
of crackpots posting on sci.math.

Or could someone give a better definition?

You can't define what's fundamentally is
undefined.
Any definition in
this situation would boil down to creating yet
another undefined
synonym of something being undefined. A set is a
collection is a
family is an aggregation is a set.


regards

tommy1729
.

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