[FOM] Replacement
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Sun Aug 19 23:40:53 EDT 2007
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
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