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I'm reading over my computational theory book before quarter starts and it's giving me the following definitions.

alphabet: any nonempty finite set

symbols: members of the alphabet

I don't understand the relevance of why this term even needs to be defined. It seems unnecessary. What exactly differentiates this from any regular set that has a cardinality greater than 0?

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    $\begingroup$ Nothing but the context in which the term is used. The terminology turns out to be useful, and it’s the terminology that’s being defined here, not the concepts. $\endgroup$ – Brian M. Scott Dec 20 '13 at 0:51
  • $\begingroup$ Oh, I see. Contextually useful. Simple enough. Thanks! $\endgroup$ – Daniel Cazares Dec 20 '13 at 0:52
  • $\begingroup$ You’re welcome! $\endgroup$ – Brian M. Scott Dec 20 '13 at 0:53
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    $\begingroup$ Just as a heads up, there are sets of cardinality strictly greater than zero that are not finite. $\endgroup$ – parsiad Dec 20 '13 at 0:54
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    $\begingroup$ Yes, I had the same issue with forcing. When I started learning about that the term "forcing notion" is just a partial order. Why not use "partial order"? Now I know that when you say "forcing notion" you effectively set the context for using the partial order and it allows you to skip some assumptions on that order (e.g. it has a maximum, the terms "stronger" and "extending", and so on). $\endgroup$ – Asaf Karagila Dec 20 '13 at 1:14
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Answered in the comments by Brian M. Scott:

Nothing but the context in which the term is used. The terminology turns out to be useful, and it’s the terminology that’s being defined here, not the concepts.

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