This question is about the Turing computability of the $\epsilon-N$ definition of a limit of an infinite sequence $S$. First, a proposition:

There cannot exist a Turing Machine $M$ which, given a program $P_S$ whose output is the sequence $S$, has output $$ M(P_S) = \left\{ \begin{array}{rl} \operatorname{true} & \text{if}\,\lim(S)\, \text{exists} \\ \operatorname{false} & \text{otherwise.} \end{array}\right. $$

In other words, a Turing Machine cannot decide if $\lim(S)$ exists. A short argument as to why this is the case is below.

My questions are:

  1. is there another (perhaps weaker) definition of limit which is decidable?

  2. a definition is "a statement of the exact meaning". Does an undecidable concept have an exact meaning? Perhaps undecidable statements are less suited to be definitions than decidable statements.

Argument: A Turing Machine that decides the existence of $\lim(S)$ could be used to solve the Halting Problem in the following way.

Denote $P_S\oplus0$ as the program that runs $P_S$ and upon observing the termination symbol appends an infinite number of zeros. Similarly $P_S\oplus 01$ is the program that appends an infinite alternating sequence $01010101\ldots$.

Then $P_S$ halts iff ($M(P_S\oplus 0)$ and not $M(P_S\oplus 01)$).

Hence the existence of limits, for sequences given by algorithms, cannot be decided by a Turing Machine.

closed as off-topic by Jeremy Rickard, Andrés E. Caicedo, Chris Godsil, Andy Putman, Alex Degtyarev Aug 5 '15 at 8:29

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    I'm voting to close this as off-topic because MathOverflow is not a discussion forum and this post is seeking to start a discussion rather than asking a question. – Jeremy Rickard Aug 5 '15 at 1:48
  • Now there is a question, but I don't understand what it means. What would you consider a satisfactory answer? – Noah Schweber Aug 5 '15 at 4:03
  • Actually, when you look closely at elementary analysis, lots of things are undecidable. For example, you might consider Richardson's Theorem which says that, for a very natural class of expressions, it's impossible to tell if they are equal to $0$. – Robert Israel Aug 5 '15 at 4:38
  • Thanks for helping me to grow my question. I've split this into two parts: first I'd like to know if there is a decidable version of $\lim$. For the second question I'm interested in people's opinion on what constitutes a definition. – Daniel Mansfield Aug 5 '15 at 5:37
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    Formally, mathematics is founded on axiomatic systems, rather than on computational models; the Church-Turing thesis does not apply to (say) structures of ZFC. In an axiomatic system, a term is well defined so long as it provably has a unique interpretation for all admissible choices of parameters. Computability of this interpretation would be a desirable bonus if available, but is not necessary in order to usefully take advantage of a mathematical definition. – Terry Tao Aug 5 '15 at 5:52

You're right, and moreover this is sharp, i.e., you cannot do more than compute the Halting Problem using limits of computable sets.

This is called the Limit Lemma in computability theory.

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    To save future readers the trouble of comparing times, I point out that this answer was for the original version of the question, which essentially just asked whether the OP's argument was correct. – Andreas Blass Aug 5 '15 at 16:25

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