Well, any multple choice question is going to be unanswerable if none of the choices given are a correct answer.
Since I think most would define a multiple choice question as one with a list of answers from which you pick the correct one, I say this question fails to validate.
Thus not only is this multiple choice question unanswerable, it's not even a multiple choice question. Try wrapping that around your head.
Since A and D give you the same result, the question is a "check all that apply question". Excluding contradictory combinations, there are four answers:
* nothing checked (i.e., none of A,B,C,D)
* A+D
* B
* C
Assume the correct choice is among these four answers (it is, since we also include a "none of these" option); then you have a one-in-four chance to get it right. Hence, A+D is right.
If this is not a "check all that apply" question, then having both A and D is contradictory and the person posing this question deserves a whack on the head.
This question is posed to a large number of people who all attempt to answer is correctly. "What fraction of respondents answered the same as you? a) 25% b) 50% c) 0% d) 25%"
If you choose the answer to this question at random, what is the chance it will be correct?
What if you change it to "What fraction of respondents not including you answered the same as you"?
What is the antecedent for 'it,' the 'answer' or the 'question'?
Does this phrase even make sense? I suspect it is supposed to be some 'recursive cleverness,' or just not make sense?
C) 0% because for something to be correct it follows that it fits a given definition of correctness of which none is given.
now the recursion has started to work based on the above reasoning
So if C)0% is the "correct answer"(that is, there is no correct answer) then choosing an answer at random will be correct 25% of the time, if you shift to viewing 25% as the "correct answer" then you have a 50% chance of randomly selecting a "correct answer"
what is the chance [the question] will be correct?
C)0%
q1=what is the chance [the question] will be correct?
what is the chance [an answer to q1] will be correct?
25%
q2=what is the chance [an answer to q1] will be correct?
what is the chance [an answer to q2] will be correct?
50%
If I choose to answer this question at random (with the constraints of having to choose one of the choices) then the chance of it being correct is going to be 25%, because I am answering at RANDOM, therefore not using any intellect and not applying any sense to the question.
However, the question merely states IF you choose, not that you have to.
Therefore, I am entirely correct by answering either A or D.
Thus it is not unanswerable and is incorrect in stating so.
But then you're implying that you'll get a correct answer 1/2 of the time, which means that the answer is B, but then the answer is correct only 1/4 of the time, and you've got yourself in an infinite loop.
If the correct answer is 50% then there needs to be 2 possible answers of 50%, otherwise picking randomly will only result in that answer being selected 25% of the time, rendering the "correct" answer false.
x^y/y = z
x= chance of a correct answer (1/4)
y= number of random attempts
1/4^2 divided by 2 = 1/32 (The odds of picking the correct answer twice in two attempts) and so on..... the more you try the less chance you have of succeeding so just give up and cheat off the guy next to you. I made this crap up so what's the chance of it being right? a)0% b)0^ c)what he wrote d)no chance
i don't get it... maybe i'm being incredibly dense but it looks like it should be B. this makes the correct answer 25%, so the probability of getting it right at random is 50% because 2 out of 4 answers are 25% - so the answer is B = 50%, and indeed the chance of choosing B is 25%
answering the question at random is not the same as answering the question posed - so the fact that the two answers are not equal shouldn't be a problem. right?
edit: i hate paradoxes, i'm pretty certain i'm wrong, but i can't put my finger on precisely why. :)
Gödel's first incompleteness theorem states that any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250) per http://en.wikipedia.org/wiki/Gödels_incompleteness_theorems
Just because you title something an "Unanswerable question" doesn't mean it actually is, and certainly doesn't mean the first incompleteness theorem is relevant.
No. Godel's incompleteness theorems say nothing about non-arithmetical or non-mathematical statements nor do they apply in contexts where no formal system exists. Some sort of paradox, sure, but Godel doesn't apply.
Truth is not a mathematical concept, and determining the "truth" or "falsehood" of a sentence has nothing to do with Godel's incompleteness theorems.
"Godel's first incompleteness theorem is a Godel sentence"
Anyways, it is my belief that Godel's Incompleteness Theorem is false by definition of Truth. That is, what is true is provable and vice versa. Once you depart from this definition, you get statements that are paradoxical, and by my definition of Truth, the Godel Incompleteness Theorem is certainly paradoxical (thus False).
The thing is, that Godel's theorem is provable. So by your definition, it is true.
The usual definition of false is not "I can't prove it's true" as that is pretty hard to decide.
Suppose I have a statement S that I can't prove. Is it false, or am I just not clever enough to prove that it is true?
The normal definition of S being false is that the negation of S is true.
Part of what the incompleteness theorem says is that in any system of logic that doesn't contradict itself, there will be statements that are neither provably true nor provably false. Thus you can take these statements to be true OR false as an axiom and it won't lead to contradictions.
Well, my definition does not imply that what is False is easy to decide, and that's OK. If N != NP, then the statement "X is factorable" would be hard to decide, even if it's False (or True for that matter).
Godel, in the proof of GIT, chose inconsistency (by concluding that G is True, even though it is also False).
So, by your own conclusions, I choose GIT to be false, and there are no contradictions.
Godel's "proof" states that G is undecidable, and since that is what G states, G must be True, and this G is a True statement that cannot be proven (or disproven). I say that the conclusion that G is True does not follow. Calling G True is no better than calling G False.
Nobody can prove GIT (Godel's Incompleteness Theorem). I tried to disprove it, but I can't do that either. Godel's Incompleteness Theorem itself is a Godel Sentence.
You can add GIT as an axiom in my system, then it would become True. I'm saying that you don't need to do that to have a complete system. You can either have a complete and consistent system, OR you can have GIT.
Godel's Incompleteness Theorems are probably some of--if not the most---misunderstood concepts in all of mathematics (rivaling Cantor's Uncountability Theorem).
While the usual analogy for the first theorem is drawn to the liar's paradox ("This is a false statement."), it's important to remember that it is only an analogy. The first theorem states, in layman's terms, that we can construct a valid mathematical statement which is complete and utter nonsense. (Much like "This is a false statement." is neither true nor false, but nonsensical.)
The second incompleteness theorem simply states (again, in layman's terms) that the consistency (where we say something is consistent if it contains no contradictions) of certain systems cannot be shown from the rules of the system itself. (Or alternatively and more correctly, if you're able to show the consistency of these systems using their own rules, then they're inconsistent.)
That said, please remember that these are mathematical theorems and as such their "applications" to other areas such as metaphysics, even by their creator, are not rigorous or necessarily meaningful. They live where they belong: in the depths of mathematical logic, in the heart of mathematics.
Your rubric doesn't denote whether I can answer more than one item for the question therefore based on "you can only choose one" then 25% is the chance of being right, as the correct answer is still one of A,B,C,D.
Answering randomly may give you a result that coincidentally matches the truth, but you have provided no logical or epistemological support for your choice, nor any chain of reasoning that leads you from the available evidence to a conclusion that there even is a correct answer.
You have a 25% chance of randomly choosing the "right" answer, (C) 0%, but no paradox is created because you answered without establishing knowledge of the answer, and therefore your answer cannot rightly be called correct.
Since I think most would define a multiple choice question as one with a list of answers from which you pick the correct one, I say this question fails to validate.
Thus not only is this multiple choice question unanswerable, it's not even a multiple choice question. Try wrapping that around your head.