[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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