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I want to express the following two sentences in mathematical syntax (if possible):

The number of solutions to the problem should be finite and each solution should be of polynomial length

More difficult (maybe impossible to express in math syntax):

Given a Polynomial Length solution, we should be able to verify if an answer is true of false

Please Note: I just want to express these sentences as elegantly as possible, so please do not downvote me if this question does not meet the mathematical high standarts of yourself or this web site. If anything is unclear, please ask and I will edit my question to meet the high standarts of this community.

Asaf Karagila
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    Plain prose is the best route. – lhf Jun 02 '14 at 15:55
  • Yes maybe.. But at least "must be finite" should be expressable in math shortags if I may call them so, no? –  Jun 02 '14 at 16:00
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    In my opinion you just did express yourself in mathematical syntax. What do you need more? – dtldarek Jun 02 '14 at 16:00
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    The point made by lhf and dtldarek is a good point. Mathematical syntax is not just formal strings and formulas. It is the language written in English. And you have written the sentences in a mathematical syntax which is readable. Always choose readability and clarity over "cool symbols math formula". – Asaf Karagila Jun 02 '14 at 16:06
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    If I really wanted to use symbols, I would say that $n < \infty$, where $n$ is the number of solutions. – Gamma Function Jun 02 '14 at 16:07
  • Great Answers, thanks to all! –  Jun 02 '14 at 16:19
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    Try $n\in \mathbb N$ – Mark Bennet Jun 02 '14 at 16:20
  • @Mark Bennet n ∈ N, because it is countable infinte? –  Jun 02 '14 at 17:29
  • Any particular natural number is finite. So to say that a number is a natural number is to imply that it is finite. You can do $r\in \mathbb R$ for the reals, too. – Mark Bennet Jun 02 '14 at 17:41
  • Interesting so although N's cardinality is countably infinite because there is no finite number to accurately describe it, we use n ∈ N to describe a finite number. –  Jun 02 '14 at 19:51

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