On 25/02/2013 8:54 PM, Nam Nguyen wrote: > On 24/02/2013 7:14 AM, Jim Burns wrote: >> On 2/23/2013 4:48 PM, Nam Nguyen wrote: >>> On 23/02/2013 2:38 PM, Virgil wrote: >>>> In article >> >>>> In mathematics [...] proofs of existence do >>>> not always require that one find an example of the thing claimed to >>>> exist. >>> >>> So, how would one prove the existence of the infinite set of >>> counter examples of Goldbach Conjecture, given that it does not >>> "not [...] require that one find an example" of such existences? >> >> Your use of the subjunctive mood ("how would one prove") requires >> anyone wishing to answer your question (challenge?) to assume >> the existence of an infinite set of counter-examples to the >> Goldbach conjecture, whether it truly exists or not. >> http://en.wikipedia.org/wiki/Subjunctive_mood >> >> I strongly suspect that you do not intend your challenge to be >> read this way. However, on the slight chance that I am wrong on >> this point, I will answer your challenge *as you asked it* : >> Let A stand for "There exists an infinite set of counter-examples >> to the Goldbach conjecture". The proof your are looking for is >> "From A, we conclude A". >> >> If the subjunctive reading gives a less-than-trivial question, >> then what did you mean instead? >> >> Did you mean to assert, as part of your question, that there exists >> an infinite set of counter-examples to the Goldbach conjecture? >> How that is proven would be helpful information to anyone >> attempting to answer you, if you had that information. Do you? > > Half of my asking Virgil is intended for what's related to that > infinite set although I was quite clear on the such "related-ness".
I meant "I wasn't quite clear". (Sorry).
> > In the naturals - as a language structure - there can exist > only 1 of the 3 _mutually exclusive_ set-existences: > > - S0 = _The_ empty set of counter-examples to the Goldbach conjecture. > - S1 = _The_ finite set of counter-examples to the Goldbach conjecture. > - S2 = _The_ infinite set of counter-examples to the Goldbach > conjecture. > > It wouldn't matter what, say, Virgil (or anyone) would pick amongst > S0, S1, S2, how could such one existence be possibly proven, without > "an example" of the underlying set, where the example-set is nothing > but the underlying set itself? > >> >> If your intended question was intended to challenge someone's >> assertion that a proof of existence NEVER needs an example, >> then that would have made some sense. However, let me remind you >> that Virgil did not assert that. (See above.) >> >> All that is needed to support what Virgil ACTUALLY wrote is >> a single example of a proof of existence that does not >> require one to find an example of the thing claimed to >> exist. One example, and so "not always". >> >> Examples come to mind such as the Banach-Tarski paradox, >> in which something can be shown to exist by using the >> Axiom of Choice, which asserts the existence of a choice function >> axiomatically but does not provide the choice function. > > Right. _IF_ we're talking about "Axiom of Choice" as a > _language formula_ then yes, the formula does _mean_ > that assertion semantically. > >> Do you see a problem with this? > > Unfortunately Yes: what about when we construct a model (which is > a language structure) for, say, the formal system ZFC? > > If we don't provide a _needed_ choice function then we'd would have > an _incomplete_ "structure" (violating Tarski's truth evaluation that, > e.g.: "There exists a choice function" _iff_ there _actually exists_ > a choice function. > > And, when such an incomplete "structure" existence occurs, relativity in > model theoretical reasoning would be a consequence! > >
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.