On 5/5/2013 9:27 AM, Julio Di Egidio wrote: > "Ross A. Finlayson" <email@example.com> wrote in message > news:firstname.lastname@example.org... > >> Almost seems like: making indices N x N, has that for the one image, >> each element is finite yet unbounded, while for the other it is >> infinite. This basically gets into considering a copy or instance of >> N, then another, has in some manner the sputnik of quantification or >> here correlation among the two, that for the same specification, one >> has finite and unbounded elements, the other finite and unbounded and >> as well: infinite elements. > > Indeed an infinite set is a set, i.e. an actual (i.e. completed, in the > math sense) infinity, while N is the potentially infinite. My hunch is > that we should be using N* (the compactification of N, to begin with) as > the counting set outside the finite realms: then arithmetic and set > theory could indeed be equivalent. >
Isn't that what modern logic forces upon us?
To interpret the universal quantifier for the Peano axioms under the received paradigm, the natural numbers have to be a totality.
In Markov, the potentially infinite is described relative to a specific set of definitions for constructed syntax and a specific recharacterization of quantifiers with respect to those constructs. In view of such careful reasoning, simply attributing potential infinity to |N without consideration of what is involved with the interpretation of quantifiers seems as if it is missing something.