On Jun 4, 12:00 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Jun 4, 4:45 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > > > "K_h" <KHol...@SX729.com> writes: > > > Curt was suggesting induction fails. > > > You yourself were suggesting that anyone claiming the axiom of > > infinity is false "needs to actually produce a finite counting number m > > such that m > m + 1". What's the justification for this on the face of > > it rather baffling and arbitrary suggestion? > > > -- > > Aatu Koskensilta (aatu.koskensi...@uta.fi) > > > "Wovon man nicht sprechen kann, dar ber muss man schweigen" > > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > > Does induction complete? > > Variously it does. > > Regards, > > Ross Finlayson
Then here the "axiom of infinity" is not "that there exists completion of induction or an inductive set", but "that the structure is of a particular form". So, arguing about infinity, you can have induction without infinity, you can have infinity with or without regular infinity. Then the "axiom of infinity" is basically the same as that it is a "regular" infinity from the axiom of infinity with a(n) e- minimal element the empty set but no e-maximal element.
Then, actually arguing about the axioms of infinity and regularity in ZF set theory, while independent in the theory they have co-completion requirements for satisfaction, with infinity having possible a collection of each constructible set then for regularity though there is no infinite constancy of mode, here in the "anti-foundational" that may actually be the primitive or fundamental "real" infinity of for example a continuum.
There is where that each other structure is recursively enumerable, generally here in ZF with the axiom of regularity or foundation with infinity (and thus the existence of transfinite ordinals), the constructible is recursively enumerable.
