[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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