On May 5, 7:50 am, fom <fomJ...@nyms.net> wrote: > On 5/5/2013 9:27 AM, Julio Di Egidio wrote: > > > > > > > > > > > "Ross A. Finlayson" <ross.finlay...@gmail.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.
What is the set of natural integers? Sets are defined by their elements, so it is a set that satisfies the predicate contains(n) for each n that is a natural integer. Then, in a pure set theory there are only sets: what set is a natural integer? Generally the notion is that the finite ordinals are defined, with a constant zero and a rule to generate for each a successor, then that the set of finite ordinals is an inductive set, where the set has the property that in induction, the ordinal elements alpha can be used to build inductive cases that for each ordinal alpha there exists a unique and distinct ordinal alpha+1 that is not the successor of any previous number. But, that is not so clearly all that a natural number is, an ordinal, though the ordinals are given labels matching those of the natural integers. Natural integers have arithmetic defined, to define the PA or PA (Presburger (+) or then Peano (+,*) Arithmetic), while the labels uniquely identify elements of the domain and range of those operations as functions, addends and sums, multiplicans and products, the establishment of the values of sums is a consequent raft of theorems that: go into the definition of natural integers, thus, into any pure set they are in as to what elements they are.
Then the notion as above is that to have elements of the naturals and inductive set identify distinct elements, and then to have a copy of an inductive set indicate elements of those elements, basically building the space of rows and columns instead of just induction, has then built a sequence and not just course-of-passage, and as the complement exists an omega-sequence, with the compactification of N (point at infinity) thus resulting as a theorem instead of as a definition as it is in ZFC, with the only constant besides 0, the ordinals, being omega, the next limit ordinal and itself defined as the collection of finite ordinals.
Basically then is a consideration that the operation of a set that is an ordinal alpha to successor is then to the closure of a set w that is of finite ordinals to correlation with sets of finite ordinals, that builds systems with the completed infinity from the potential: back into the language of the collection of elements. Then, well- foundedness aka regularity isn't necessarily a true axiom, not that it is anyways from pure naive set theory, but that N e N isn't paradoxical because it is only from building up the systems to that, variously:
the naturals are implicitly compact there are infinite elements in the naturals
These features of the numbers, in their entire collection, aren't so directly relevant to most one-off computations of finite arithmetic or their unbounded associations. Then where that is ignored, having the natural numbers as sets as ordinals, is a reasonable abstraction to keep the overall machinery out of the way of simple computation.
In pure set theory, sets are defined by their elements and contain only sets. In number theory, numbers are indviduals generally built from the natural integers, and all the operations on them and relations among them, and as to equality of the values of the numbers in extended number systems toward the continuum of real numbers, with 1 = 1.0, and so on. Then, as pure sets, numbers are all those things.
Elements of an inductive set, ordinals that contain their predecessor, and naturals, aren't interchangeable. Ordinals and naturals can serve as representations of elements of an inductive set, and in various cases as representations, or labels, or names, of each other.
Then as above where the sequence is the representation, in the 1-ary representation of a value, succession was used to build the sequences, then to go over all of them and build a different sequence, puts the omega-sequence into the language, of the finite sequences.
Then, Cantor's result is interpreted not as making the collection of finite sequences incomplete, instead, introducing the compactification to then: that of the language of the representations, the resulting element of that representation, is a point at infinity. Basically then this is the notion of having a construction of omega, as a set and not necessarily and indeed not a well-founded set, instead of defining it as a constant in the language which would then be inconsistent with the resulting construction.
Then number theorists might find it easy to work up using sets for containing numbers, in provisional set theory where the notion of sets is just for quantifiers over elements satisfying predicates and as to induction: that the natural integers can have a point at infinity without breaking ZFC.