Bruno Marchal skrev:
>
>
> Le 29-nov.-07, à 17:22, Torgny Tholerus a écrit :
>
> There is a difference between "unlimited" and "infinite". "Unlimited"
> just says that it has no limit, but everything is still finite. If
> you
> add something to a finite set, then the new set will always be
> finite.
> It is not possible to create an infinite set.
>
> Come on! Now you talk like a finitist, who accepts the idea of
> "potential infinity" (like Kronecker, Brouwer and the intuitionnist)
> and who rejects only the so called actual infinities, like ordinal and
> cardinal "numbers" (or sets).
Yes, I am more like a finitist than an ultrafinitist in this respect. I
accept that something can be without limit. But I don't want to use the
word "potential infinity", because "infinity" is a meaningless word for me.
>
> At the ontic level, (or ontological, I mean the minimum we have to bet
> on at the third person pov), comp is mainly finitist. Judson Webb put
> comp (he calls it mechanism) in Finitism. But that is no more
> ultrafinitism. With finitism: every object of the "universe" is
> finite, but the universe itself is infinite (potentially or actually).
> With ultrafinitism, every object is finite AND the universe itself is
> finite too.
Here I am an ultrafinitist. I believe that the universe is strictly
finite. The space and time are discrete. And the space today have a
limit. But the time might be without limit, that I don't know.
>
> Jesse wrote:
>
> My instinct would be to say that a "well-defined" criterion is one
> that, given any mathematical object, will give you a clear answer
> as to whether the object fits the criterion or not. And obviously
> this one doesn't, because it's impossible to decide where R fits
> it or not! But I'm not sure if this is the right answer, since my
> notion of "well-defined criteria" is just supposed to be an
> alternate way of conceptualizing the notion of a set, and I don't
> actually know why "the set of all sets that are not members of
> themselves" is not considered to be a valid set in ZFC set theory.
>
> Frege and Cantor did indeed define or identify sets with their
> defining properties. This leads to the Russell's contradiction. (I
> think Frege has abandoned his work in despair after that).
> One solution (among many other one) to save Cantor's work from that
> paradox consists in formalizing set theory, which means using
> "belongness" as an undefined symbol obeying some axioms. Just two
> examples of an axiom of ZF (or its brother ZFC = ZF + axiom of
> choice): is the extensionality axiom:
> AxAyAz ((x b z <-> y b z) -> x = y) "b" is for "belongs". It says that
> two sets are equal if they have the same elements.
> AxEy(z included-in x -> z b y) with "z included-in x" is a macro for
> Ar(r b z -> r b x). This is the power set axiom, saying that the set
> of all subsets of some set is also a set).
For me "belongness" is not a problem, because everything is finite. For
me the axiom of choice always is true, because you can always do a
chioce in a finite world.
>
> Paradoxes a-la Russell are evacuated by restricting Jesse's
> "well-defined criteria" by
> 1) first order formula (in the set language, that is with "b" as
> unique relational symbols (+ equality) ... like the axioms just above.
> 2) but such first order formula have to be applied only to an already
> defined set.
This 2) rule is a very important restriction, and it is just this that
my "type theory" is about. When you construct new things, those things
can only be constructed from things that are already defined. So when
you construct the set of all sets, then that new set will not be
included in the new set.
> For example, you can defined the set of x such that x is in y and has
> such property P(x). With P defined by a set formula, and y an already
> defined set.
>
> Also, ZFC has the foundation axiom which forbids a set to belong to
> itself.
This is a natural consequence of my type theory. When you construct a
set, that set can never belong to itself, because that set is not
defined before it is constructed.
> In particular the informal collection of all sets which does not
> belongs to themselves is the universe itself, which cannot be a set
> (its power set would be bigger than the universe!).
Yes, the set of all sets which does not belongs to themselves is the
universe itself. But this is not a problem for me, because you can
always extend the universe by creating new objects. So you can create
the power set, and the power set will then be bigger than the universe.
But this power set will not be part of the universe.
--
Torgny
--~--~---------~--~----~------------~-------~--~----~
You received this message because you are subscribed to the Google Groups "Everything List" group.
To post to this group, send email to everything-list.domain.name.hidden
To unsubscribe from this group, send email to everything-list-unsubscribe.domain.name.hidden
For more options, visit this group at http://groups.google.com/group/everything-list?hl=en
-~----------~----~----~----~------~----~------~--~---
Received on Fri Nov 30 2007 - 14:15:02 PST