Cardinality of [N]^ω: What Does ω Stand For?

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The cardinality of set of [N]^\omega . what does omega stands for?
 
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\omega is usually the http://en.wikipedia.org/wiki/Ordinal_number_(mathematics)" .
 
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Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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