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?


closed as not a real question by Igor Rivin, Qiaochu Yuan, Steve Huntsman, user5810, user9072 Oct 3 '11 at 19:13

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question.

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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 '11 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 '11 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 '11 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 '11 at 23:14
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    $\begingroup$ on META: tea.mathoverflow.net/discussion/1158/… $\endgroup$ – Will Jagy Oct 4 '11 at 0:36