On Aug 21, 9:23 am, aatu.koskensi...@xortec.fi wrote: > MoeBlee wrote: > > Would you say something about the distinction between finitistically > > meaningful and finitistically justified? > > A statement is finitistically meaningful if we can formulate it in > terms of particular finite structures, and parameters naming arbitrary > finite structures, without involving quantification over an infinite > totality. Formally this corresponds to the class of Pi-1 sentences in > the language of arithmetic, or, more perspicuously, to quantifier-free > formulas in the language of primitive recursive arithmetic. Some such > claims are finitistically justified in the sense that on basis of our > understanding of finitary inductively defined classes of objects, such > as the naturals, we can offer compelling arguments for them, using > nothing but principles that are immediately evident on basis of the > finitary inductive definitions involved. Some on the other hand are > meaningful, in the sense that we can make finitistic sense of them, > but no finitistic compelling argument exists for them. An example of > such a finitistically meaningful but finitistically unjustified > statement would be "ZFC + 'there is a proper class of Woodin > cardinals' is consistent".