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

Spivak says that $\,\sqrt{2}\,$ exists means that there's a real number $\,x\,\,s.t.\,\,x^2=2\,$ , and this cannot be proved with the knowledge assumed in page 26 of his book.
