The question as to why God is necessarily “all-powerful” is a formidable question. In order to answer it properly, one must also reflect upon what the term “all-powerful” is, as even that term is not always well-understood. I can think of a couple of reasons why God must be “all-powerful.” The first (and weaker) is related to the ontological argument for God's existence, and the second (and stronger) to the cosmological arguments.
The ontological argument is that God is the greatest Being who can be thought to exist: no greater Being can be imagined. Human imagination is nearly limitless, which means that God must Himself be limitless; since people can imagine a Being of great power, and since a powerful being is greater than a powerless one, God's power must be greater than that imaginable by humans. But this is only possible if humans are limited in their ability to imagine, or if God is unlimited in His power; the two pose a conceptual difference, but do they pose a significant difference?
I would say not essentially, for this reason: it is possible to imagine in the abstract a thing which God cannot do. CS Lewis treats such ideas in his The Problem of Pain: he rightly calls them “intrinsic impossibilities.” A commonly used example is that a God who can do everything still cannot create an object which is too heavy for Him to lift; if He could create such an object, then he cannot lift it, and thus can not do everything. Hence, the use of the words “all-powerful” cannot denote that God can do everything, but rather that He can do everything which is not intrinsically impossible. Thus, our ability to imagine things which God cannot do implies that we can imagine ultimately everything which may be done—and thus a Being who can do all of these things. Hence, God must be all-powerful, because He is the greatest Being whom we can imagine.
The cosmological argument actually has two branches, reflecting two different version of the argument: the first attributed to Kalam, the second to St. Thomas Aquinas. Kalam’s argument essentially states that everything which began to exist must have been caused to exist, and that the First Cause was God, who never “began” to exist, but rather has existed eternally and thus was not caused. As applied to the question as to why God must then be all-powerful, the answer is that since He caused everything else in existence to begin to exist, he must have the power to do so. What’s more, “everything” in the universe must be more than the mere matter, but also the abstracts, including the laws that govern the universe. As God created these things, He must also have the power to change them.
That being said, one definition of omnipotence (being “all-powerful”) is that He is the Being who can change anything which He wants changed. This includes not only the universe itself, but also the laws which govern nature. Furthermore, He and He alone is capable of changing all of these things in such a way that everything else is affected by the changes; God himself would not necessarily be affected by such “universal” changes, but everything else must be. Hence, God is essentially and uniquely all-powerful.
St. Thomas Aquinas formulated his cosmological argument in a different manner. He essentially said that anything which is contingent on something else for existence cannot exist without that something else’s also existing. But the only thing which exists and which is not contingent on any other thing is God; hence, everything else which exists must be contingent on something else and therefore ultimately on God. But since everything else which exists depends upon God for its continuing existence, God must have the power to allow these existences to continue. He thus also has the power to cause any or everything to cease to exist.
Such a power must be at least as infinite as the universe itself, including everything within the universe and everything underlying it, for all time. The universe itself is essentially infinite, as it is bounded in every direction by that object which exists farthest from its center (most likely light), and is thus essentially unbounded, meaning infinite. Hence, the Being who must sustain the universe must also have the power required to sustain the universe, meaning and infinite amount of power. Thus, God must have infinite power, which means that He must be all-powerful.
If you enjoyed this post, here are some related ones:
Unstoppable Objects, Immovable Walls, and Omnipotence (Nicene Guys)
Does Hell Matter?
The Problem of Pain: A Brief Theology of Theodicy
Who Made Evil?
Of Infants and Salvation (Nicene Guys)
Cosmological Evil: Some Thoughts
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