So all we need is an algorithm—given any real number or formula representing a real number, determine whether that number is an integer. But there is no such algorithm. It isn’t so much a difficulty in simply encoding or defining the word “integer”, it is a difficulty in producing an operational or algorithmic definition of “integer”.
Yes, this was approximately what I meant by “encoding the word integer”. Somehow it didn’t occur to me that it may be difficult, or even impossible, to test real numbers for integerness. Still, I am not sure if I understand. A concrete example of a uniquely defined real number whose integerness is unknown would help.
I think you misunderstood. I said there exists no algorithm such that for all formulas it can decide whether the formula represents an integer. Which is equivalent to saying that forall algorithms there exists a formula which can’t be decided.
In other words, the way this game is played, first you specify the algorithm and then I supply the formula that breaks it.
I understood that the algorithm valid for all formulae had to be produced in advance, but still I found it strange that it doesn’t exist. Further thinking about it it doesn’t seem so strange after all.
Yes, this was approximately what I meant by “encoding the word integer”. Somehow it didn’t occur to me that it may be difficult, or even impossible, to test real numbers for integerness. Still, I am not sure if I understand. A concrete example of a uniquely defined real number whose integerness is unknown would help.
I think you misunderstood. I said there exists no algorithm such that for all formulas it can decide whether the formula represents an integer. Which is equivalent to saying that forall algorithms there exists a formula which can’t be decided.
In other words, the way this game is played, first you specify the algorithm and then I supply the formula that breaks it.
I understood that the algorithm valid for all formulae had to be produced in advance, but still I found it strange that it doesn’t exist. Further thinking about it it doesn’t seem so strange after all.