The first two questions were easy enough, but the third left me puzzled. The golden ratio, assumed for the sake of contradiction to be a ratio of some natural numbers, as a result of some computations gets redefined to a ratio of another pair of smaller natural numbers, a process that seems to be able to go on recursively forever, except the Principle of Well Ordering does not agree with a concept of ever decreasing natural numbers. That's great, but how does that relate to square root of 5?
After long hours of thinking and hoping to catch a clue at the discussion boards, it happened. Somebody mentioned quadratic formula, and it actually worked out. I am not sure about some details, such as was there really enough information to assume that phi was a positive number? At first, it was based on assumption that it is a ratio of natural numbers, so it must be positive, but once it becomes clear that it isn't a ratio of natural numbers, what exactly can be used to argue that it must be positive? Someone mentioned the fact that it is a "ratio" and ratios are always positive, but it does not sound very convincing to me. Are ratios universally accepted to only mean positive rational numbers? Velocity can be thought of as a ratio, and in physics it is normally allowed to be negative. Comments?
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