Is there a 'Third Heaven'? 2 Corinthians 12:2-4 (New International Version)


  2 I know a man in Christ who fourteen years ago was caught up to the
  third heaven. Whether it was in the body or out of the body I do not
  know—God knows. 3 And I know that this man—whether in the body or
  apart from the body I do not know, but God knows— 4 was caught up to
  paradise and heard inexpressible things, things that no one is
  permitted to tell.


Here Paul is relating a story of an experience that (presumably) he had, where he mentions going to 'the third heaven' also called 'paradise' in the same passage. 


What is he referring to? Is he talking about a distinct place other than heaven, the abode of God'? 
Is this phrase, 'the third heaven' have some meaning in Greek that is lost in English? 
How would a first century Corinthian have understood this?
From the FaithLife Study Bible, this (three heavens) represents the understanding of the universe held by the Greeks:


  According to ancient cosmology, there are seven levels: the sky, the clouds, the sky above the clouds, the firmament, the waters above the firmament, the heavens, and the heaven of heavens, where God dwells. The three heavens view understands the first heaven to be the visible sky or the “firmament” (see Gen 1:8 and note), the second heaven to either be the “heavens” or the division between the “heavens” (the “waters above the firmament”; Psa 148:4), and the third to be the “heavens of heavens” or “highest heavens” (1 Kgs 8:27; Psa 148:4). All views agree on the places, but they label them differently.


In a nutshell then:


the first "heaven" is the atmosphere, the sky - the "heavens" that you can imagine a Victorian talking about.
the second heaven is the border between the sky and what we think of as "heaven," or, what I like to think of as "Space" 
the third heaven is that bit with the pearly gates, the gold streets, and free harps. 


In identifying "the third heaven," Paul is saying he was in the presence of God directly.