[FOM] Re:Clarification on Higher Set Theory
wm@cage.rug.ac.be
wm at cage.rug.ac.be
Mon Feb 17 11:56:39 EST 2003
Aanhalen Thomas Forster <T.Forster at dpmms.cam.ac.uk>:
> >Laver, Steel and Dehornoy have given an example where higher set theory
> (the
> large cardinal axiom I3, or the existence of self simular ranks) reveals
> new
>
>
> Maybe i'm not up to speed on this, but i don't recall any *converses* in
> this area. Elementary embedding give nice results about LD-algebras,
> yes,
> but do we know that that is the only way to get those nice results about
> LD-algebras?
>
> Thomas Forster
>
There exist indeed other ways to obtain these results on LD-systems(using
braidgroups),but the main point is as Dehornoy said:"We argue that both the
braidorder as the Laver tables are and are to remain applications of set
theory,if the latter has not clearly shown the way,it is more than likely that
most of the results would not have discovered yet."
Higher set theory is not necessary here,but it has shown the way.
This is an other role for higher set theory which could motivate
mathematicians to study it.
Let me also remark that there are still results about Laver tables (finite
LD-sysems)which have yet not received a combinatorial proof and still depend on
the large cardinal axiom I3. Dehornoy has proved that such a combinatorial
proof (if it exist) cannot be formalised inside PRA.
Wim Mielants.
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