Date: Feb 25, 2013 10:54 PM
Subject: Re: Matheology § 222 Back to the root<br> s
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
>> 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.
> 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".
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
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.