[FOM] Replacement

[Aatu Koskensilta]

Robert M. Solovay wrote:
>  	1) It seems to me that beleiving that ZFC is inconsistent is 
> stronger than believeing replacement is false. It amounts to holding that 
> the idea of replacement [or that of some other axiom of ZFC] is 
> incoherent].
>   

Well, it's stronger even than that. It might be -- however we should 
interpret that modality here -- that ZFC is consistent but proves e.g. 
"ZFC is inconsistent", in which case the idea of the world of sets as 
described by ZFC is certainly incoherent; incoherence is weaker than 
inconsistency.

>  	2) Boolos argues against the existence of a kappa such that kappa 
> = aleph_kappa. He waffles a bit and doesn't quite say this is false. But 
> he certainly argues that we don't have good grounds to believe in the 
> truth of its existence. I haven't reread his paper carefully. But I think 
> the instance of replacement he questions would assert that if alpha is an 
> ordinal, there is a set consisting of aleph_gamma for those gamma less 
> than alpha.

If one accepts the conception of ordinals as abstract order-types of 
well-ordered sets, as Cantor certainly did, then it seems one should 
accept principles stronger than those found in Zermelo set theory, in 
particular all levels V_alpha of the cumulative hierarchy with alpha 
less than the least fixed-point of the Beth-function. That is, one 
should accept the principle that the "set-of" operation can be iterated 
along any well-ordering, based on the idea that the cumulative hierarchy 
should extend as far up as "possible". This doesn't buy full 
replacement, but to me is more natural a stopping than Zermelo set 
theory, from a conceptual point of view.

-- 
Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus