On 25/02/2013 5:49 PM, Virgil wrote: > 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, > > I am not aware of how one would prove the existence of even one > counterexample to Goldbach without finding one. > > Nevertheless, in standard non-WM mathematics, proofs of existence do > not always require that one find an example of the thing claimed to > exist.
As I've explained to Jim Burns, that depends on the context the word "proof" is in. If you talk about a _formula_ expressing the existence, then your original statement would make sense: no need to find an "example" for the semantic, the meaning, of the formula.
But if the context is a structure, then your statement would not be true.
For example, let T = {Ex[Red(x)]}. How would you construct a model of T without an (example) individual being Red, given that the universe U of this model must be non-empty by definition of model?
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.
