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Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? How are you to prove that any number "exists"? Why can't we define $\sqrt2$ as a number that fits under some arbitrary definition of existence, while asserting that its most concise expression is with a functional root?

I'm sorry if these questions seem a bit sophomoric; in some ways it resembles an 8 year old repeatedly asking "why". But given that his prose is very concise and technical, his usage of "exist" was out of the ordinary.

(I used two tags representing the book's field of study; and one representing the actual relevant tag.)

edit

Oh, I'm sorry. I misquoted. My question still stands, though; how has he defined existence such that $\sqrt2$ might possible not be within it.

Direct quote: "We have not proved that any such number exists..." in reference to $\sqrt2$.

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    Where in the book, exactly?2012-06-10
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    Page 26 in the Prologue.2012-06-10
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    I don't think Spivak says anything like that. What I read in his wonderful book is that "*at present a proof [of its existence] is impossible for us*...unless you can give an exact quote (with the page number, say) where he says otherwise.2012-06-10
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    Oh, I'm sorry. I misquoted. My question still stands, though; how has he defined existence such that $\sqrt2$ might possible not be within it. Let me edit the original post.2012-06-10
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    Exactly: in page 26 Spivaks does ***not*** say that, but only that at that moment the square root of 2 can't be proved to exist (in some sense).2012-06-10
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    It annoys me that the phrases "least upper bound", "greatest lower bound", lub, and glb do not appear in this post, so I am adding them now 2012-06-10
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    Regarding your question "How do you prove that a number 'exists'", you may enjoy this Numberphile video on that very topic: http://youtu.be/1EGDCh75SpQ2012-06-10

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