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Is there a finite-dimensional vector space whose dimension cannot be found? Assume, we have somehow constructed a vector space whose dimension is finite, but yet unknown. Is there always an algorithm to find it?

Take the example by Noam Elkies from the comments: let $V$ be the vector subspace of $\mathbb{R}$ over $\mathbb{Q}$ spanned by $\lbrace 1,e,\pi \rbrace$. Should we believe that $\dim(V)$ is either $2$ or $3$, even if we don't know which?

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    $\begingroup$ Consider the vector space whose dimension is $1$ if a particular Turing machine halts on a particular input and $0$ otherwise. $\endgroup$
    – Qiaochu Yuan
    Oct 3, 2011 at 18:56
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    $\begingroup$ Let $V$ be the span of $\lbrace 1, e, \pi \rbrace$ as a vector space over ${\bf Q}$. Then $\dim(V)$ is finite, and surely equal to $3$, but nobody knows how to prove it. $\endgroup$
    – Noam D. Elkies
    Oct 3, 2011 at 19:28
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    $\begingroup$ From a constructivist point of view, this is a very 'real' question. Noam Elkies' answer is quite a good one. One can expand on that pattern to give quite substantial answers to this question. Vote to reopen. $\endgroup$
    – Jacques Carette
    Oct 3, 2011 at 19:33
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    $\begingroup$ @Reimundo Heluani: you make it sound as if editing and closing where mutually exclusive. In some sense, the opposite is the case. A main point of closing a vague question is to give time for editing, while (temporarily) holding back answers that then, after a clarifiying edit, might seemm off-topic or besides the point. For a more authorative account (not for this specific question but in general) please see François's contribution, in particular 3. in his numbered list, in the following recent meta thread tea.mathoverflow.net/discussion/1156/… $\endgroup$
    – user9072
    Oct 3, 2011 at 23:14
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    $\begingroup$ on META: tea.mathoverflow.net/discussion/1158/… $\endgroup$
    – Will Jagy
    Oct 4, 2011 at 0:36

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