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Cool.

It's impossible using an unmarked straight-edge and pair of compasses to trisect a general angle.

It's impossible using an unmarked straight-edge and pair of compasses to compute the cube root of 2.

It's impossible using an unmarked straight-edge and pair of compasses to construct two lines whose lengths are in the ratio 1:pi.

It's impossible to show that given Euclid's first four postulates, the fifth must be true.

It's impossible to show that there is no infinity between aleph_0 and 2^aleph_0.

It's impossible to create a Turing machine that can predict if a general Turing machine will halt in finite time.

Do you want some more?

1 comments

I don't think I said that "nothing is impossible" just that I, personally, don't like the word "impossible".

Sure, some things are indeed impossible, but it doesn't mean we shouldn't try to prove them otherwise.

Then I guess I don't really understand your point. You said:

    One of my least favorite words is “impossible”:
    it simply has no real value and just goes against
    everything I believe in,

I'm trying to demonstrate that it does have real value. There are some things that genuinely are impossible, and knowing that can save you a great deal of wasted time.

    Now, notice how I don’t say that I hate the word.
    Why not? Simply because it is an awesome motivator!

I know people who, once told that something is impossible, immediately go to work trying to make it happen. In some cases they are just wasting time, and not only theirs. Sometimes they pester others, endlessly, with work that is provably wrong, but they are unwilling to take the time to understand that.

    What should you do when someone tells you something's
    impossible?  I say "keep calm and carry on"

I disagree so much with this. Don't dismiss someone's advice or opinion because they claim something is impossible. Instead, try to understand their point of view, and if you are inclined, try to find an error in their argument, a chink in their armour. If you can't, perhaps you aren't really cut out to work in that area and you are better advised to work on something else.

They might be right.