how can I prove that there are real numbers that cannot be defined by English phrases?? I guess I have to use counterexample ???
Well what do you mean by 'phrase'? Are 'one', 'two', 'three', 'four point six', 'pi squared', phrases?
I think you can cover them all, but would be happy to be proved wrong.
I think the proof that there aren't any has been given by tonio.
OP, don't listen to them; you are being trolled.
First, what proves that you defined a unique number? There may be many numbers that can't be defined in English.Let x be a real number that cannot be defined by an english phrase....but I just defined it by an english phrase!
Second, this is Richard's paradox (Richard was French, so presumably, "d" in his name is not pronounced and the stress is on the last syllable.) Unlike Tonio's suggestion, this paradox's statement constructs a unique real number that cannot be defined by any English phrase. But in this case, the statement actually defined it! Berry paradox is similar.
I am not a specialist on paradoxes, but the logic flaw here is that it is not well-defined which numbers are defined by which English phrases.
Yes, these are phrases.Well what do you mean by 'phrase'? Are 'one', 'two', 'three', 'four point six', 'pi squared', phrases?
I think the OP's instructor expects an answer based on cardinality. There can be at most countably many reals defined by English phrases provided each qualified phrase defines exactly one number, since there is at most countable many phrases.I think you can cover them all, but would be happy to be proved wrong.
Let x be a unique number that cannot be defined by an english phrase then.
Sorry to be glib, but there is no mention of cardinality or countability in the OP's question.
Maybe Godel could help.
I presumed this was going to be about computable numbers; surely it is related though!
Turing proved that not all numbers were computable by a Turing Machine. Now, if a number if definable by an English phrase, it can't be that hard to encode it into a Turing Machine (i.e. make a computer program to work it out).
I assume the original problem refers to phrases that identify a unique real number. That is, the problem says: "Consider all English phrases that define a unique real number. Prove that not all real numbers are defined by these phrases." If we admit phrases that define (uncountable) sets of numbers, then all real numbers can be covered.
So, why is there a unique number that is not defined by qualified English phrases?
Problem statements often give just propositions to prove leaving the choice of proof method to the student.Sorry to be glib, but there is no mention of cardinality or countability in the OP's question.
"A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds in the standard model of set theory" (Wikipedia). Not all definable numbers are computable. For example, let be a computable enumeration of all programs with no argument. Consider a real number x whose binary expansion is where if halts and otherwise. Then x is definable but not computable. One must note, though, that one can construct successive rational approximations to x that get arbitrarily close. The reason x is not computable is that in general one does not know whether the found approximation to is correct if this approximation is 0.Originally Posted by Swlabr
Sorry-I think I was trying to go the other way; that if it is computable then it is definable. If you can write it as a program, then take the program as your definition. But that doesn't answer the question...hmm...I think, actually, I just wasn't thinking straight....
Tonio's example, which I am sure he intended as a joke, is not valid because it does NOT define a specific real number- it defines a class of real numbers. In any phrase in English (or any other language) there must be an finite number of words, each consisting of a finite number of letters, of which there are 26 different possible letters. If we allow arbitrarily large words- though still a finite number of letters, or an arbitrarily large, though finite, number of words in a phrase, we will still have a countable number of possible English phrases. And the set of all real numbers is uncountable.
Well, I must say I didn't intend my answer to be a joke...completely. I tried to make the OP think of the (a) paradox as was later
mentioned somewhere else, and as somebody else also pointed out, even if we get into cardinalities stuff and, logically, deduce
there must be real numbers which cannot be defined by an english phrase, pinpointing any of these numbers must be as impossible
as pinpointing the smallest positive real number...
Tonio