Paul use both terms and there aren't any crossnotes.
Romans 8:1 - 9:33
ESV - 1 There is therefore now no condemnation for those who are in Christ Jesus. 2 For the law of the Spirit of life has set you free in Christ Jesus from the law of sin and death.
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I don't sense any great theological difference between the terms “Spirit of God” and “Holy Spirit,” (but see the technicalities below). Many times a writer will resort to a simple stylistic difference of words in order to highlight a section of his letter or body of thought or such, or to simply avoid redundancy when using the same concept over and over again. What is more, sometimes differences in words are meant to convey what we call "poetic parallelism," in which different words or phrases compliment one another as they attempt to express the beauty of the exact same thing using various original words, yet in poetic fashion (like in Ps 119). Ultimately, God is unfathomable in his existence! The Spirit of God is God’s Spirit—viz, the Spirit of Holiness—viz, the Holy Spirit. God is a Spirit, yet his Holy Spirit is a separate member of the "trinity," yet there is only ONE God. Ponder that one for a while... The Greek word for Spirit is πνεῦμα “pneuma,” which can be translated variously as “breath,” or “wind,” or “ghost.” The Hebrew word for Spirit is רוּחַ “ruach,” which can also be translated variously as “breath,” or “wind,” or “ghost.” When Messianic Jews such as myself refer to the Holy Spirit, quite often we use the term “Ruach HaKodesh.” Technically since the Hebrew word “kodesh” is a noun, a verse like Ps 51:11 where the phrase “Ruach HaKodesh” is found literally conveys the sense of “the Spirit of Holiness.” But “Holy Spirit” (with “holy” functioning as an adjective) works just fine as well.
If you study the literal language of the original manuscripts (using various literal interlinear translation tools available for free on the Internet) you'll see the original language was... Spirit of Sanctification. The words "Holy Spirit" are not a wrong translation, but it's a question of accuracy. So wherever you see "Holy Spirit" or "Holy Ghost" in the KJV or NIV the more correct set of words is "Spirit of Sanctification". It's important to dig a lot deeper than modern translations (and I'm calling the KJV a relatively modern translation here). I recommend you download a tool from http://scripture4all.org and study the original words for yourself. It's very interesting to say the least. You'll find a few surprises.
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