[FOM] Axioms of reducibility and infinity
Aatu Koskensilta
Aatu.Koskensilta at uta.fi
Tue Aug 9 12:24:41 EDT 2011
Lainaus steve newberry <stevnewb at att.net>:
> My understanding of the Axiom of Reducibility is that it was
> intended to state that:
>
> To every proposition of higher-order, there is an equivalent
> proposition of First-order, or more precisely, to every entity
> definable in Higher-order logic there is an equivalent such entity
> definable in First-0rder logic.
Well, not quite. The axiom of reducibility says that for any
propositional function of type t there is an extensionally equivalent
predicative propositional function of type t. We can express this in
terms of orders -- roughly: any propositional function of order n is
extensionally equivalent to a propositional function of order 0 -- but
here we must recall that the relevant notion of order has to do with
the ramification of types with a hierarchy of orders at each type, not
order in the sense of first-order predicates, second-order predicates,
and so on, which correspond to different types in the system of
Principia. The axiom has the effect of collapsing the ramified
hierarchy in one dimension, giving us impredicative propositional
function comprehension for extensional predicates.
--
Aatu Koskensilta (aatu.koskensilta at uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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