- Question 0: What is the answer to question 0?
In actual fact, Question 0 can be any question you like. Not all questions have answers, though, so we need to ask the question:
- Question 1: Is it possible to answer question 0?
What if it is impossible to answer question 1? To make sure, we need to ask:
- Question 2: Is it possible to answer question 1?
We can make up an entire infinite series of questions along these lines:
- Question 0: What is the answer to question 0?
- Question 1: Is it possible to answer question 0?
- Question 2: Is it possible to answer question 1?
- Question 3: Is it possible to answer question 2?
- ...
- Question n: Is it possible to answer question n-1?
- ...
We will call this series of questions series 0. Obviously, higher numbered questions in series 0 get easier, so surely (even if Question 0 cannot be answered) one of the questions in the series can be answered. The answer to all successive questions will then be "Yes".
- Question x: Which is the first question in series 0 which can be answered?
Perhaps it is impossible to discover the answer to question x.
- Question x+1: Is it possible to answer question x?
But then we must ask:
- Question x+2: Is it possible to answer question x+1?
And we get another infinite series (series 1):
- Question x: Which is the first question in series 0 which can be answered?
- Question x+1: Is it possible to answer question x?
- Question x+2: Is it possible to answer question x+1?
- Question x+3: Is it possible to answer question x+2?
- ...
- Question x+n: Is it possible to answer question x+n-1?
- ...
It must be possible to answer one of the questions in series 1, but which? In asking this, we get another series of questions, series 2:
- Question 2x: Which is the first question in series 1 which can be answered?
- Question 2x+1: Is it possible to answer question 2x?
- Question 2x+2: Is it possible to answer question 2x+1?
- Question 2x+3: Is it possible to answer question 2x+2?
- ...
- Question 2x+n: Is it possible to answer question 2x+n-1?
- ...
And then series 3:
- Question 3x: Which is the first question in series 2 which can be answered?
- Question 3x+1: Is it possible to answer question 3x?
- Question 3x+2: Is it possible to answer question 3x+1?
- Question 3x+3: Is it possible to answer question 3x+2?
- ...
- Question 3x+n: Is it possible to answer question 3x+n-1?
- ...
We can sum up all these series in another question series:
- Question 0: What is the answer to question 0?
- Question x: Which of the questions in series 0 can be answered?
- Question 2x: Which of the questions in series 1 can be answered?
- Question 3x: Which of the questions in series 2 can be answered?
- ...
- Question nx: Which of the questions in series n-1 can be answered?
- ...
This is a completely different type of series to the other series we have been looking at, so we will call the series y (for reasons which may become obvious later). You guessed, next we ask:
- Question x^2: Which is the first question in series y which can be answered?
- Question x^2+1: Is it possible to answer question x^2?
- Question x^2+2: Is it possible to answer question x^2+1?
- Question x^2+3: Is it possible to answer question x^2+2?
- ...
- Question x^2+n: Is it possible to answer question x^2+n-1?
- ...
This is series x. Then there is a question series for which of the questions in the series x, x+1, x+2, etc. can be asked. The number of questions which need to be answered in order to answer question 0 is uncountably infinite.
In general, question number f(x) is "Is it possible to answer question f(x)-1?" unless f(x)=0 for all x, in which case question number f(x) is the original question. Question f(x,x)x is "Which is the first question in series f(x,y) which can be answered?". Series g(x,y) is the series of questions with numbers such that if you remove the constant parts of the question identifiers and divide by x, the series of identifiers you get is g(x,0), g(x,1), g(x,2), ... g(x,n), ...
Series y^2 isn't very interesting - it's just a subseries of series y.
We might wish to think about the series:
- Question 0: What is the answer to question 0?
- Question x^2: Which is the first question in series y which can be answered?
- Question 2x^2: Which is the first question in series 2y which can be answered?
- Question 3x^2: Which is the first question in series 3y which can be answered?
- ...
- Question nx^2: Which is the first question in series ny which can be answered?
- ...
According to our numbering scheme, this series must be series xy. What is the identifier for the question "Which is the first question in series xy which can be answered?"? According to our formula, the answer to this question is x^3.
Using these rules we can easily find the question corresponding to any polynomial f(x) with positive integer coefficients. Interestingly, the question is of the "is it the case that..." (i.e. a question expecting a yes or no answer) if f(0) is not equal to 0, otherwise it is "which" type of question, expecting an answer from the set of natural numbers, or question 0 itself, which can have any form [1].
Are there questions which do not correspond to polynomials in x? Let us consider the series:
- Question x: Which is the first question in series 0 which can be answered?
- Question x^2: Which is the first question in series y which can be
answered? - Question x^3: Which is the first question in series xy which can be
answered? - ...
- Question x^n: Which is the first question in series x^(n-2)y which can be
answered? - ...
According to our numbering scheme, this series must be x^y. Question x^(x+1) ought to be "Which is the first question in series x^y which can be answered?". This is not a polynomial in x.
We can construct as many different levels of series as we like, as many different levels of levels as we like and so on, and as many different levels of levels of ... of levels of levels (where there are n occurrances of the word "levels" in that clause). Quickly we have to use complicated constructs and curious meta-language just in order to enumerate the possibilities.
For every sort of infinity that we can invent, there is another, larger one. We can enumerate our infinities as the aleph series - aleph_0 is the "smallest" infinity, aleph_1 is the next one up, aleph_2 is the next one and so on. But then we can start thinking about some more series:
- aleph_(aleph_0), aleph_(aleph_1), aleph_(aleph_2), ...
- epsilon_0 = aleph_0, epsilon_1 = aleph_(aleph_0), epsilon_2 = aleph_(aleph_(aleph_0)), ...
- epsilon_(aleph_0), epsilon_(aleph_1), epsilon_(aleph_2), ...
- epsilon_(epsilon_0), epsilon_(epsilon_1), epsilon_(epsilon_2), ...
- gamma_0 = epsilon_0, gamma_1 = epsilon_(epsilon_0), gamma_2 = epsilon_(epsilon_(epsilon_0)), gamma_3 = epsilon_(epsilon_(epsilon_(epsilon_0)))
- ...
We can take the series aleph_1, epsilon_1, gamma_1 and continue it - call it a(0), a(1), a(2) - then there's nothing to stop us thinking about a(aleph_0). We can continue doing this sort of thing for as long as we like and we will never have succeeded in even writing down how many questions you need to ask in order to answer question 0 (since we can do exactly the same sort of things with the questions, series of questions, levels of series and so on).
I hope I have succeeded in boggling your mind. I have certainly succeeded in boggling my own. Boggling the mind a little from time to time is good for it. Too much of any good thing is bad, though - thinking about this sort of stuff drove Cantor (the inventor of the aleph notation) to insanity. Be careful next time you ask a question!
For some similarly boggling (but probably more mathematically correct) stuff see this.
[1] On a more philosophical note, there only 4 types of question:
- "What": Used for asking the nature of something. Occasionally also used as a synonym for "which".
- "How": Used for asking for details of a process.
- "Why": Used for asking the reasoning behind something.
- "Which": Used for asking to select from a list, as in "Which of the following is true". "Is it the case that..." type questions could be considered to be selecting from a list of two: True and False. Three other common questions are made from Which:
- "Who" is "Which person"
- "Where" is "Which place"
- "When" is "Which time"