God's Omniscience and Cantor: Is There a Set of All Truths?

Yes, God is omniscient, and God's omniscience means that He knows every truth and believes nothing false. Patrick Grim argues from Cantor's theorem that there can be no set of all truths, and so no omniscient being. The mathematics in his argument holds; the premise that omniscience is a set does not, and the classical doctrine of God's knowledge does not hold it. Alvin Plantinga and William Alston have answered it in different ways, and the one premise worth contesting belongs to the way finite minds know.

Key takeaways

• God's omniscience means knowing every truth and believing no falsehood. Classical theology adds that God knows by one simple act, seeing all things together and not one after another.
• Patrick Grim's Cantorian argument is sound mathematics. Cantor's theorem rules out a set of all truths, and Grim concludes that no omniscient being can exist.
• The argument needs one more premise: that an omniscient being's knowledge would form a set of all truths. Alvin Plantinga denied it in 1993.
• Frame-Indexed Classical Theism (FICT) grants Grim's mathematics and rejects his conclusion that omniscience is impossible. It agrees with Plantinga's reply that knowing every truth requires no set, and it explains why the set picture was the finite mind's model of knowing from the start.

What does omniscient mean?

"Omniscient" means all-knowing. The word joins the Latin omnis, "all," to scientia, "knowledge," and said of God it means that He knows every truth there is.

Philosophers make the definition exact. The Stanford Encyclopedia of Philosophy, in its entry on omniscience revised in 2025, gives the standard form: a being S is omniscient if, for every proposition p, if p is true then S knows p. A second version adds that S believes no falsehoods, and the entry treats the two as equivalent. Neither version says how many truths there are, or whether they can be gathered into one collection.

Dictionaries stop at what God knows. Classical theology also asks how He knows it, and its answer is short. Thomas Aquinas wrote that "God sees all things together, and not successively" (Summa Theologiae I, q. 14, a. 7). A finite mind learns one truth, then another, and builds up a store over time. On the classical doctrine God does not know that way, and the answer to Grim rests on that difference in the mode of knowing.

The same method is worked through on a different question in Volume One, free.

Does God know everything, according to Scripture?

Yes. Scripture states God's omniscience without qualification, and at both ends of the scale, the vast and the trivial.

The psalmist sets the two side by side:

"He determines the number of the stars and calls them each by name. Great is our Lord and mighty in power; his understanding has no limit." (Psalm 147:4–5, NIV)

The stars are Scripture's standing image of a multitude past counting, and God numbers them and names each one. Jesus carries the same knowledge down to the smallest case: "And even the very hairs of your head are all numbered" (Matthew 10:30, NIV). John states it in the widest terms: "If our hearts condemn us, we know that God is greater than our hearts, and he knows everything" (1 John 3:20, NIV).

The Hebrew behind "has no limit" in Psalm 147:5 is ein mispar, literally "without number." In these texts the omniscience of God is a perfection of the Lord Himself, confessed in the same breath as His power.

Whether God knows the future, and what that means for human freedom, has its own treatment: does God know the future.

The series takes the text of Scripture as its authority throughout, and the first volume is free.

What is Grim's Cantorian argument against omniscience?

Patrick Grim argues that Cantor's theorem rules out a set of all truths, and that the knowledge of an omniscient God would have to form such a set. So, he concludes, there can be no omniscient being.

Take any set and gather up its subsets, including the empty set and the whole set itself. That collection is the power set. Cantor's theorem says the power set is strictly larger than the set it came from. A set with three members, {a, b, c}, has eight subsets: the empty set, three single-member sets, three pairs, and the whole. The theorem holds for infinite sets too, so an infinite set is still smaller than its own power set.

Grim's 1993 statement of the argument can be laid out in eight steps:

1. Suppose there is a set T of all truths.
2. Take any truth, and call it t.
3. For each subset of T there is a truth saying whether t belongs to that subset.
4. So there are at least as many truths as there are subsets of T.
5. By Cantor's theorem, there are more subsets of T than members of T.
6. So some truth is left out of T, and there is no set of all truths.
7. An omniscient being's knowledge would constitute a set of all truths.
8. So there is no omniscient being.

In his own words: "Were there an omniscient being, what that being would know would constitute a set of all truths. But there can be no set of all truths, and so can be no omniscient being."

Grim developed the argument in Noûs in 1988 ("Logic and Limits of Knowledge and Truth"), pressed it in The Incomplete Universe (MIT Press, 1991), a book arguing against totalities of truth and knowledge, and debated it with Plantinga in Philosophical Studies in 1993. He calls it "a short and sweet Cantorian argument against omniscience."

Steps 1 through 6 are good mathematics, and they are not in dispute here. Six steps concern sets; step 7 is the one step about God's omniscience, and it says that to know every truth is to possess a set of them. The Stanford Encyclopedia puts the pivot in the same place: "Grim thinks that this is a problem for omniscience because he thinks that a being could know all truths only if there were a set of all truths."

The books state their own arguments in numbered steps like these; Volume One costs nothing to check.

Is there a set of all truths?

No. Grim is right about that, and a defender of God's omniscience need not resist the point.

Mathematics has met this result before. The same pattern turns up wherever mathematics tries to gather an absolute totality into a single object. Russell's paradox shows there is no set of all sets. The Burali-Forti paradox shows there is no set of all ordinal numbers. Volume Five treats these results as one family: totalities of this kind cannot be gathered as sets, and the failure is proved within mathematics itself, before any theologian arrives. Grim's argument against a set of all truths is a member of the same family, and it is as sound as its relatives.

A theist who tried to rescue a set of all truths would be fighting a theorem, and theorems do not lose to theology. The more useful move is to ask what the theorem has touched. It has shown that one kind of object cannot exist. It has not yet shown that God's knowledge was ever that kind of object.

If omniscience were a set of known truths, Grim would win outright. The live question is whether omniscience is a set at all.

Volume Five follows the mathematics of the infinite chapter by chapter; see the Volume Five page.

How have philosophers answered Grim?

The replies divide at step 7. Grim keeps it and gives up God's omniscience; Plantinga rejects it and keeps the omniscience of God without a set; a third line, from William Alston, changes what divine knowledge is.

Grim's own conclusion. Grim treats his result as a verdict. Asked whether omniscience is impossible, he answers: "Within any logic we have, I think, the answer is 'yes'." His strongest move concerns the word "all." To say that God knows every truth is to quantify over truths, and on Grim's view our only account of what quantifying means is given in terms of sets. If that is right, the phrase "every truth" does not escape the set. It carries the set back in under a different name.

Plantinga's reply. Alvin Plantinga answered in the same 1993 exchange with a question: "why buy the dogma that quantification essentially involves sets?" The Stanford Encyclopedia gives his central observation: "Plantinga notes that a parallel argument shows that there is no set of all propositions, yet it is intelligible to say, for example, that every proposition is either true or false." If "every proposition" makes sense without a set of propositions, "every truth" can make sense without a set of truths. So Plantinga keeps the definition and drops step 7. An omniscient being like God "knows every true proposition and believes no false ones. We must then concede that there is no set of all the propositions God knows." The distributive reading of omniscience, for every truth God knows it, is Plantinga's reply, and the credit is his.

Grim's rejoinder. Grim held his ground: "the only semantics we have for quantification is in terms of sets." The exchange turns on whether "all" needs a set-theoretic semantics, and the literature has not closed that question. Simmons (1993) has given a more technical reply in terms of levels of sets.

Alston's non-propositional route. Alston's 1986 paper offers a different picture of divine knowing altogether. In the Stanford Encyclopedia's summary: "According to Alston, God has an intuitive, immediate awareness of all truth, which gives him knowledge without belief." If God's knowledge is not a store of beliefs, there is no store to gather into a set. The route carries a price. Critics, Hasker (1988) among them, ask whether knowledge without belief is still knowledge of truths.

One method runs through the whole series, and its first volume is free to read.

How does Frame-Indexed Classical Theism answer?

It grants Grim every theorem and, with Plantinga, denies step 7. God's omniscience is not a set, because His knowledge is not an inventory, and FICT explains why the set picture looked compulsory to begin with.

Frame-Indexed Classical Theism (FICT) is a research program in analytic theology that resolves classical theological dilemmas by diagnosing them as artifacts of a single formal error, and it is the position argued in the Many Beings Framework book series. Set beside Grim's conclusion that omniscience is impossible, Plantinga's reply that "every truth" needs no set, and Alston's knowledge without belief, it keeps what Plantinga keeps and traces the set premise to the finite frame's way of knowing.

Because a knower's mode of existence fixes its mode of knowing, natures that differ in kind entail perceptual frames that differ in kind. God knows reality in the Absolute Metaphysical Frame, which is atemporal, simple and a se. We know it in the Human Finite Frame, which is sequential, composite and contingent. Predicates whose content involves time, modality, agency or value carry their frame with them and are said of God and of us analogically. Treating such a term as though it meant the same thing on both sides of that line is the error named the Many Beings Fallacy. This is not relativism: one reality is known in both frames, and God's frame is the ground of ours.

A finite mind knows by holding truths one at a time and adding to the store, so for us "knowing everything" could only mean a finished collection. Step 7 writes that picture into a premise, and Volume Five's verdict is that the premise is "the finite frame describing its own mode and calling it God's."

On the classical doctrine, God's knowledge "is not an aggregation of items but one simple act," His knowing of Himself and of all He wills and sustains. He does not know infinite things "as if He enumerated part after part; since He knows all things simultaneously, and not successively" (Summa Theologiae I, q. 14, a. 12). Propositions are how we articulate what He knows, packaging into countable units a reality that is not itself parceled. In God's frame, Volume Five adds, truth is identity, and He does not learn facts. He sustains them, and He knows the free acts among them as the free acts they are.

With Plantinga, FICT relies on "every truth" needing no set. What it adds is an account of why: God's act of knowing is not itself a gathering of items, so the set picture was not a picture of His knowing to begin with. With Alston, it denies that God's knowing is a store of beliefs formed one after another, without denying that what God knows is every truth.

"His understanding has no limit" (Psalm 147:5) praises an understanding that belongs to the Lord Himself. "He knows everything" (1 John 3:20) confesses its scope and does not speak of a collection.

The position has a real cost. It gives up the sentence "God knows the set of all truths." It says instead, and only, that for every truth God knows it, and that there is no completed inventory.

The position's full statement is in the books, and you can begin with Volume One, free.

Doesn't "God knows every truth" bring the set back?

No. "For every truth, God knows it" is a claim about each truth taken one by one, and it posits no gathered object that holds them all.

The difference is between a distributive claim and a collective one. Every student in a school has a birthday. That is true of each student, and it stays true whether or not anyone has compiled a list of birthdays, or could. The claim ranges over the students. It does not require a register. Volume Five draws the same line for divine knowledge: "for every truth, God knows it; what never existed is the collective object, the completed inventory, and omniscience never needed one."

Here is the objection at full strength. The books' own formal notation states God's knowledge one truth at a time: of any given truth, God knows it if and only if it is true. If the formalism itemizes, then the prose's "one simple act" seems contradicted by the books' own logic, and the inventory has come back through the notation.

The answer is that the notation is distributive, which is what the prose affirms. It says, of each truth, that God knows it. It itemizes our statement about God's knowledge, and our statements come in items because we are finite speakers who say one thing at a time. It does not itemize God's act of knowing, which the notation describes from the outside and does not reproduce.

You can test the argument against the first volume at no cost: get your copy.

Is God's knowledge propositional or not?

An omniscient God knows the truth of every proposition, and He knows it by one simple act and not by holding a store of propositional beliefs. The two claims fit together, because one concerns what He knows and the other how He knows it.

The objection runs like this. Volume Three, in its chapter on tensed facts, argues that God knows every indexical truth exhaustively. An indexical truth is one whose statement depends on who or when is speaking, such as what time it is "now" for a creature. Volume Three calls that knowledge propositional. Volume Five, meanwhile, calls God's knowledge one simple act and treats propositions as the creature's packaging. Alston's distinction, as the Stanford Encyclopedia presents it, is where the tension bites: one volume seems to put God's knowledge on the propositional side, and the other seems to take it off.

The contrast in Volume Three is drawn somewhere else. It sets knowing the truth value of an indexical statement against being the finite subject located inside the timeline, which is propositional knowledge against situated experience, and not propositional knowledge against one simple act. God knows every such truth value exhaustively without occupying the standpoint from which the statement is voiced. His knowing of those truths is not a queue of beliefs formed one after another. The content is propositional as we articulate it; the mode is simple as God knows it.

Volume Three works through the indexical case at length, and the series starts with a free first volume.

If God knows infinitely many truths, how can He be simple?

God's omniscience and His simplicity fit together because the many-ness belongs to what creatures are and receive, not to God. The divine ideas are many only in relation to creatures, and in God there is one simple act.

Divine simplicity holds that God has no parts and no composition, so that His knowing is not really distinct from His being. Knowing infinitely many distinct truths seems to install an infinite store of distinctions in His mind, an infinite quantity of information inside a being with no inner plurality. If the truths are distinct and God knows each, His knowledge looks at least as divided as its objects.

Aquinas saw the difficulty and answered it: "it is not repugnant to the simplicity of the divine mind that it understand many things; though it would be repugnant to its simplicity were His understanding to be formed by a plurality of images" (Summa Theologiae I, q. 15, a. 2). God knows many things through one essence, which He knows as imitable by creatures in many ways.

Volume Five applies the same discipline. The divine ideas are many "only in relation to creatures; in God there is one simple act, one essence knowable as imitable in many ways." Mathematics and the order of creation preexist in God "virtually, as white light contains the colors, not as a drawer contains its contents." The infinity of distinctions lies on the creature's side, in the many ways God's one essence can be imitated. It is not furniture inside God.

Simplicity raises a sister question, whether a God without distinctions could have willed otherwise, and that is the problem of modal collapse.

Volume Four carries the chapters that argue divine simplicity itself.

How do the answers compare?

Three of the four positions accept the mathematics outright; all four part ways over step 7 and over what each gives up.

Frequently asked questions

Is God omniscient?

Yes. God's omniscience means He knows every truth and believes nothing false. Classical theology adds that He knows by one simple act, seeing all things together and not one after another, as Aquinas put it. Scripture states the scope without qualification: "God is greater than our hearts, and he knows everything" (1 John 3:20). The Cantorian argument against an omniscient God shows only that there is no set of all truths, and the classical doctrine does not claim that God's knowledge is one.

Does God know everything, including the future?

Yes. On the classical doctrine, God's knowledge takes in the future, since He knows all things together and not in sequence. How that knowledge fits with human freedom is its own question, taken up under does God know the future. Open theism is the view that limits what there is to know about the future: God knows every truth, but the free future is not yet there to be known.

What is the paradox of omniscience?

The phrase "paradox of omniscience" is used loosely for arguments claiming that complete knowledge is incoherent. The omniscience paradox treated here is Patrick Grim's Cantorian argument: by Cantor's theorem there is no set of all truths, an omniscient being's knowledge would have to be such a set, and so there is no omniscient being. Plantinga's reply grants the theorem and denies that omniscience is a set, and Frame-Indexed Classical Theism takes the same line while explaining where the set premise comes from.

Does Cantor's theorem disprove God?

No. Cantor's theorem proves that any set is smaller than its power set, and Grim uses it to show that there is no set of all truths. That result stands. What it disproves is a set of all truths, which the classical doctrine does not assert. God's knowledge was held to be one simple act, not an inventory of items. The theorem reaches God's omniscience only if His knowing is a collection. Volume One shows how the method handles questions of this kind, and it is free.

Can God know infinitely many things?

Yes. Aquinas held that God knows infinite things, but not "as if He enumerated part after part; since He knows all things simultaneously, and not successively" (Summa Theologiae I, q. 14, a. 12). His knowing does not proceed item by item. He knows through one simple act, His knowing of His own essence and of all He wills and sustains, and the many truths are many on the side of what He knows, while His knowing of them is one.

Where does this leave God's omniscience?

The omniscience of God stands as the classical doctrine stated it, and what falls is the inventory. Grim's argument is best read as a result about what the finite frame can gather, and on that it is correct: truth cannot be collected into a set. It becomes an argument about God only by assuming that God knows the way we do. Grim's conclusion that omniscience is impossible needs that assumption, Plantinga's reply showed that the definition of omniscience does not, and Frame-Indexed Classical Theism names the assumption as the finite mind's own mode read into God. The doctrine beneath this answer is divine simplicity.

Volume One begins the argument from its foundations, and it is free to read.