No. There are many heavens because the difference is more nominal than real.
Basilius autem, et Damascenus sequens eum, dicunt plures esse caelos. Sed haec diversitas magis est in voce quam in re.
In order, then, to understand the distinction of heavens, it must be borne in mind that Scripture speaks of heaven in a threefold sense. Sometimes it uses the word in its proper and natural meaning, when it denotes that body on high which is luminous actually or potentially, and incorruptible by nature. In this body there are three heavens; the first is the empyrean, which is wholly luminous; the second is the aqueous or crystalline, wholly transparent; and the third is called the starry heaven, in part transparent, and in part actually luminous, and divided into eight spheres. One of these is the sphere of the fixed stars; the other seven, which may be called the seven heavens, are the spheres of the planets.
Ad distinctionem ergo caelorum sciendam, considerandum est quod caelum tripliciter dicitur in Scripturis. Quandoque enim dicitur proprie et naturaliter. Et sic dicitur caelum corpus aliquod sublime, et luminosum actu vel potentia, et incorruptibile per naturam. Et secundum hoc, ponuntur tres caeli. Primum totaliter lucidum, quod vocant Empyreum. Secundum totaliter diaphanum, quod vocant caelum aqueum vel crystallinum. Tertium partim diaphanum et partim lucidum actu, quod vocant caelum sidereum, et dividitur in octo sphaeras, scilicet in sphaeram stellarum fixarum, et septem sphaeras planetarum; quae possunt dici octo caeli.
In the second place, the name heaven is applied to a body that participates in any property of the heavenly body, as sublimity and luminosity, actual or potential.
Secundo dicitur caelum per participationem alicuius proprietatis caelestis corporis, scilicet sublimitatis et luminositatis actu vel potentia.
Thirdly, there are metaphorical uses of the word heaven.
Tertio dicitur caelum metaphorice.