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God Himself first used both of these titles. (In other words, they were designations that God used to refer to Himself, and were not names that were applied to Him by any human.) God Almighty (El Shaddai in Hebrew) was the title that God applied to Himself when speaking to Abraham in Genesis 17:1. It emphasizes God's omnipotence and control over all things. The Tetragrammaton (represented, as the word implies, by the four letters YHWH, and commonly rendered as "Yahweh", or as LORD (in all capital letters)) was the name that God applied to Himself in Exodus 3:13-15 when He appeared to Moses, and Moses asked Him what His name was. It means "I Am Who I Am", or "I Will Be What I Will Be", and emphasizes God as the eternal, supreme form of existence and source of being. (As God explained in Exodus 6:3, He had also previously appeared to Abraham, Isaac, and Jacob, but He had not fully made Himself known by this name to them.) This new name was a more personal, intimate designation than El Shaddai, and indicated an intention on God's part for a closer relationship with humanity, which would culminate in God taking on human form (while remaining fully God) in the person of Jesus.
There are thousands of years of theology behind the usual answers to this question which is rather male centric and not overly biblical, but there is an alternate view based on law/prophecy (biblically these are the same thing) which is beginning to gain traction and makes much more sense in view of various laws and prophecies that should be considered the most basic Bible interpretation tools. The traditional view is that the root word for shadday is shadad, which is translated as power, but has a connotation that simply does not fit what we claim to know about God. The use of power given by this word involves violent destruction, devastation and despoilment. This simply does not fit with a God that portrays himself as a loving creator, parent and husband, portrayals that are widely used throughout scripture, not just in the NT as many theologians teach. A more likely choice for the root of shadday is shad, which is the breast. The theology being built around this interpretation says that shadday means the breasted one and speaks of God as a nurturer, more specifically as a mother. This makes a certain sense if we take note that all religion started upon the eviction from Eden and variations in the religions of men come from their misunderstandings of the rites and symbols given. Most Pagan religions give greater reverence to a mother figure god over a father figure god and even portray this goddess as protecting her children from a father god who is motivated by hatred and vengeance rather than love and caring for his children. Sound familiar? This same doctrine exists in Christianity except that the roles are assigned to the Father and Jesus, or the Father/Jesus and Mary or the OT God and the NT God/Jesus. El Shaddai speaks of God as a creator and nurturer, having given birth to creation and raising it up as a loving parent (mother) until it is mature enough for the child to begin learning his father's business (Ex 4:22, Lu 2:49). When creation was ready to learn its Father's business God introduced himself to Moses as Yahveh. This word is usually translated as "I Am" but because Hebrew has only 2 tenses, not 3, and the present and future are combined into one this name can also mean, "I Will Become" with the added connotation of "anything I please." It is at this same time that we begin seeing a specific prophecy in scripture that states, Yahveh will become my Yeshua (salvation) (Ex 15:2, Ps 118:14, Is 12:2, 2Pe 3:15, along with over 50 other verses that in some way paraphrase this prophecy.) Yeshua is the Hebrew form of the Greek Iesous transliterated into English as Jesus. Literally, Yahveh became Jesus. Literally God is our salvation. All three names are names, not simply titles, and all three summarize a function God is performing in his creation at that point in time and shadad simply does not fit. While some Christian, and much non-Christian theology portrays God as a destroyer (Abaddon/Appolyon -- Hebrew and Greek words for the Persian god The Destroyer) Rev 19:11), this is not a biblical portrayal of God. El Shaddai is the nurturing God, not the destroying god, and speaks of our God as creator and as parent raising his creation to maturity.
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