Re: Dropping sets from V


"Marc Alcobé García" <malcobe@xxxxxxxxx> wrote in message news:32282dc4-f49f-46a2-b280-07e554638f20@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
After reading Ignasi Jané's "The iterative conception of sets from a
cantorian perspective" and "¿De qué trata la teoría de conjuntos?" (I
ignore if there is an english translation of the latter), and about
the notion of maximal extensionality (i. e. that a domain has this
property whenever nothing new can be added without loss of
extensionality within that domain), I thought that maybe the opposite
view would be worth analysing: that of 'what can be dropped without
seriously dammaging the domain?'

The universe of sets seems to have an amazing recovering capacity.

For example, let us imagine we drop the empty set from V. This must be
done recursively. Of course, if we drop 0 and every set that contains
0 and every set that contains a set that contains 0 etc. we are left
with nothing at all, this is not the kind of recursive dropping we
wish to have. So we start from 0 and throw it to the clear bin.

----------------------------------------------------------
So, if we consider the class of all ordinals (by way of example), and notationally pretend that such a S is a set,

S = 0 U
{0} U {1} U {2} U ... U
{0,1} U {0,2} ... U {1,2} U {1,3} ... U
{0,1,2} U {0,1,3} U ....

Where the first row is rank 0, the second row is rank 1, etc?
----------------------------------------------------------


Now we
go to the only set of rank 1 and drop 0 from it, this way we recover
0. Then we go to the sets of rank 2, and drop 0 and replace any set of
lower rank living in them by its new version with 0 dropped (this way
both sets of rank 2 become 1), Then we go to sets of rank 3, etc. If
we ignore repetitions, this is going to leave us with the very same
sets we had before we started. As a sort of 'symmetry operation' over
V. The sets 'lost' at first, are soon 'recovered' due to (not too)
higher rank sets being transformed.

---------------------------------
I need an example.
----------------------------------

Now, what happens if we drop this way an arbitrary set x from V?
Again, given any set affected by the dropping we have a higher rank
set that lets us recover what we had lost.

So, can we drop this way all the sets of a given rank and still keep
on having the whole V? It seems so, for the same reason.

What about dropping all the sets that form the backbone of the
cumulative hierarchy, i. e. the ordinals? It seems nothing is lost,
either.

There is certainly a way of destroying V: drop all sets from a given
rank above.

My question is: is there any way of formalizing this 'property' of
sets within set theory? Is there a possibility that we can learn
something from it?

------------------------------
I love the idea, its just that I don't understand it.
------------------------------


.

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