Bunches
> On 11/27/05, John E Clifford
> <clifford-j@sbcglobal.net> wrote:
> > --- Jorge LlambÃas <jjllambias@gmail.com>
> wrote:
> > > I was thinking of special abstract things
> like
> > > the number seven.
> > > Any kind of seven is seven, at least from
> some
> > > point
> > > of view.
> >
> > I don't think I follow this. The number seven
> is
> > a kind in that there is a natural seven and a
> > rational one and a real one and so on,
>
> That's why I hedged "at least from some point
> of view".
> If natural sevens, rational sevens and real
> sevens are
> different kinds of sevens, then obviously
> sevens are not
> an ultimate kind.
But natural sevens, real sevens, and rational
sevens might be.
> > but that
> > seems to be breaking down in ultimate kinds,
> > pretty much any way you extend the list.
>
> I'm not sure that if sevens can differ like
> that then
> natural sevens won't in turn be able to differ
> in some other
> way, but it doesn't really matter.
>
> > And of
> > course anything is some subkind is also a
> thing
> > of the kind itself. So I have missed the
> point
> > here.
>
> The point was that the restriction "every bunch
> breaks down
> completely into individuals" for the system of
> bunches is an
> independent restriction of all the other thesis
> listed, it doesn't
> follow from them but must be imposed. There are
> systems
> where it need not hold.
Ahah! Yes, I think that that is true, though I
don't think your remarks prove it. And ordinary
system of intensional kinds might work for this
however, either a poset or some sort of nexus
(where a memger could fall under two or more
members which are not directly related) and then
the possibility of infinitely subdividing would
keep generating new sets. But that might also
mean that the set of individuals was infinite,
not empty. It doesn't quite work yet.
>
> > In any case, the
> > fact that it can be divide indefinitely only
> > complicates the claim that we always get to
> the
> > bottom; it does not deny it.
>
> You can impose it as an additional condition,
> but it does
> not follow from the rest of the thesis.
Well, of course, that is something one wants of
one's axioms (which I suppose this must be then):
that they are independent of other axioms. I
hope that turns out to be true for all of them
(if I ever get around to sorting out the theses
into various types.
> > At the bottom level, the next lowest kind has
> > only individuals as subkinds, each "one of a
> > kind."
>
> That is, if you assume there is a bottom level.
Well, it turns out I was taking extensional kinds
as my model and that does make a difference. The
bottom level is more obviously the case here --
though I think it always is for anything that
really might be called a kinds. I await a
contrary case, but won't be either surprised or
dismayed if one turns up, since my interest is
not in kinds (about whivh I haven't given much
thought) but about bunches, which I think I am
close to defining.
> > But some might object to taking
> > individuals as kinds. In that case the lowest
> > kinds would have no subkinds and so are the
> > sought lowest level into which the kind
> divides.
> > I gather I am still missing your point.
>
> My point is that you need not assume a bottom
> level.
>
> > > The natural numbers greater than one follow
> the
> > > same rules
> > > as bunches, with "+" being the product and
> "in"
> > > being "is a divisor of".
> > > Then the primes are the individuals and
> "every
> > > bunch breaks down
> > > completely into individuals".
Well, it turns out they aren't. The model
disconfirms a+a=a at least.
>>> Kinds are
> more
> > > like the real numbers.
> >
> > I do hope this is true, relative consistency
> > proofs using set theory are always a bit
> chancey.
But alas this does not help in that matter.
> > But I still don't get the kinds claim: it
> would
> > seem that every number is then "in" every
> other
> > and so there are no individuals.
>
> Well, I was keeping the "greater than one"
> restriction,
What restriction is that? We are looking only at
real numbers (strictly?) greater than 1. But
that does in no way take away from the fact that
every one divides every one. I suppose you mean
that the quotient in such a division has to be
greater than one (or, apparently, one or greater)
This will get an ordering and, of course, since
there are no least real greater than 1 (I think
this needs "strictly greater") there are no
individuals, but the result is not very plausibly
called a system of kinds, I don't think.
>so
> every number is "in" every number greater than
> or equal to
> itself. And indeed this is an example where
> there are no
> individuals, there is no bottom level.
But also no idempotence (is that the word I want?
a+a=a) and probably not asymmetry: a in b & b in
a => a=b, as this will dramatically reduce the
number of kinds (and, indeed, give individuals).
So it is still not the proof of independence that
is wanted.