 Example: Goldbach conjectured in 1742 that every even number bigger than 2 could be written as the sum of two primes. Until today this conjecture remains unproved. Consider the term “The first number that violates Goldbach’s conjecture”. It does not give us all information about the number, since the number might not exist. The prefix “the first” ensures the description, if it exists, is unique, and it gives us an algorithm to find the number. It is a partial uniquely identifying description. This algorithm is only effective if the number really exists, otherwise it will run forever. If we find the number this will be great news, but from the perspective of descriptive complexity the number itself will be totally uninteresting, since we already know the relevant properties to find it. Observe that, even if we have a number n that is a counter example to Goldbach’s conjecture, it might be difficult to verify this: we might have to check almost all primes \( \leq n\). This can be done effectively (we will always get a result) but not, as far as we know, efficiently (it might take “close” to n different computations) . 