Is there any specific reference to Heaven or Hell in the Old Testament or did this notion emerge solely as a result of the Persians' Zoroastrian influence on the Jews?
The OT contains numerous references to heaven. Many of these refer to the physical heavens (Gen. 1:1, Psalm 19:1, etc.). Nevertheless, there do also seem to be a number of references to heaven as the dwelling place of God (1 Kings 8:30, Psalm 11:4, etc.).
As for the term "hell," it depends on which English translation you consult. The KJV, for instance, translates the Hebrew term "Sheol" as "hell." The NASB, on the other hand, simply renders this term "Sheol." The NIV translates this term in a variety of ways: the grave, death, the depths, etc., depending on the context. Strictly speaking, sheol (the Hebrew term) does not refer to hell in my judgment. It might refer to Hades (i.e., a temporary place of punishment for the unrighteous dead between death and resurrection) in some contexts. But hell, as I understand it, is properly understood as the second death, the Lake of Fire, the place of eternal punishment. And this is not true of either Sheol or Hades (see Revelation 20:13-15). Thus, the Hebrew term Sheol can, in certain contexts, be used in a manner similar to the NT term Hades (e.g. Job 26:6; etc.), but I personally don't think it refers to hell (strictly speaking).
I do not think it's necessary to suppose that Zoroastrianism was solely responsible for the NT doctrines of heaven and hell. In the first place, the OT does refer to heaven as the dwelling place of God, distinct from the physical universe. For another, the OT concept of Sheol is often used to refer to the place of the dead (i.e., the place of the dead between death and resurrection). This actually parallels the NT doctrines of Abraham's Bosom or Paradise and Hades (see Luke 16:19-31). In the OT, Sheol was apparently a place for both the righteous and unrighteous dead. It may have been a place of rest for the righteous and a place of torment for the unrighteous. However, in the course of progressive revelation, we have been given a clearer vision of the afterlife (including the eternal state) in the NT. Thus, I think this can be easily explained in terms of progressive revelation, rather than as borrowing from Zoroastrianism.
In case you're interested, I have written a previous reply about Zoroastrianism. Although this reply is attempting to answer some questions other than what you've asked about, it may nonetheless be of benefit to you.
I hope this helps.
Sincerely in Christ,
About the Author
Michael Gleghorn is a research associate with Probe Ministries. He earned a B.A. in psychology from Baylor University and a Th.M. in systematic theology from Dallas Theological Seminary. He is currently pursuing a Ph.D. in theology from Dallas Theological Seminary. Before coming on staff with Probe, Michael taught history and theology at Christway Academy in Duncanville, Texas. Michael and his wife Hannah have two children. His personal website is michaelgleghorn.com.
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