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When something (like a limit) is said to "exist" is this perfectly equivalent to "is well-defined"? And, is "well-defined" more-or-less equivalent to, "computers could use this definition and there would (theoretically) be no miscommunications"?

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No, they are very different. For one thing, generally "well-defined" is a property of a definition, while "exists" is a property of a particular example. – Alex Becker Apr 29 '14 at 2:21
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Well-defined has slightly different meanings in different mathematical context, but the idea is generally that it's defined independent from a particular representation.

For example, the following function $f\,:\, \mathbb{R} \to \{0,1\}$ $$ f(x) = \begin{cases} 1 &\text{if the first fractional digit is 9} \\ 0 &\text{otherwise} \end{cases} $$ is not well-defined. Since we may write $1$ as either $1.000\ldots$ or $0.999\ldots$, the value the function takes depends on which particular representation of $1$ we pick.

There is a connection between existence and well-definedness, but the two aren't quite the same thing. For one thing, well-definedness is a stronger property. Saying "Let $x$ be the real number with $x^2 = 1$" doesn't well-define $x$, because it doesn tell us whether $x=1$ or $x=-1$. Existence is, however, necessary for well-definedness. If we attempt to define a function by placing impossible requirements on it - e.g. "Let $g$ be a real function such that $(g(x))^2 = x$ for all $x$", then $g$ is certainly not well-defined.

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The two ideas are orthogonal.

There exists something in my pocket. I can definitely hear things in my pocket jingling around, so something certainly exists. However "something in my pocket" is not well-defined because there are many things in my pocket and the phrase "something in my pocket" does not unambiguously pick out one and only one of those things. None of those objects is well-defined by that phrase, although they all exist.

An electron having a temperature of absolute zero is a well-defined object. It does not, however, exist.

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