Tuesday, March 06, 2012

Terminology Tuesday: Infinity

Infinity: The characteristic of having no bounds or limits. In classical theism, many of God's attributes, such as knowledge, power and love, are viewed as infinite. In set theory, infinite is often defined as the property of a set that has a subset whose members can be placed in a one-to-one correspondence with the original set, as is the case for the natural numbers and the even integers. With respect to a series, there is debate among philosophers as to whether an actual infinite is possible, as opposed to a procedure that can in principle be repeated endlessly. Those who deny an actual infinite must also deny that the universe is infinitely old.1

1. C.Stephen Evans, Pocket Dictionary of Apologetics & Philosophy of Religion (Downers Grove, IL: InterVarsity Press, 2002), p. 61.


Ex N1hilo said...

Those who deny an actual infinite must also deny that God has attributes that are infinite.

Brian said...

When we speak of the attributes of God, however, we are not speaking of a quantitative infinity. Instead, it is more of a qualitative infinite.

God does not have a certain quantity of knowledge; rather, more of an infinite quality of knowledge.

Ex N1hilo said...


God is simple. That is, He is not composed of parts. And yet infinite. Yes, this is hard for us creatures to grasp.

But wouldn't you say God has exhaustive knowledge of all future events?

I posed this question on another blog in the comments section of a post in which the author explained and promoted the Kalam argument.

The response from the blog owner was, as best I can recollect, that the set of future events consists of a potential, but not an actual, infinite. Therefore, God's knowledge of those events does not constitute an actual infinite.

Would you concur with his assessment or answer differently?


Brian said...

My personal view would tend towards something like this, I think:

If there exists an actual infinite, I don't think it could exist in a finite construct such as creation.

If an actual infinite does exist, then it could exist in an infinite mind: which would be the mind of God.

But as for a sequence of events that are taking place and stretching into the future, because they have not happened yet, then they are not an actual infinite, just a potential infinite. That seems to make sense to me. And I don't think that does any damage to our view of God in any way as far as his attributes being infinite (whatever the correct understanding of that may be).

Ex N1hilo said...

Thanks for responding and clarifying your views. I don't think we are far apart. I don't believe there are--present tense--actual infinites in creation. But it kind of irks me to hear Christians say "There are NO actual infinites." Without qualification, this could be taken to imply that God's knowledge is finite, or, at best "potentially infinite."

And this points out what I think is a serious weakness in the Kalam argument. I haven't heard or read a solid demonstration of the claim that here are no actual infinites. The discussion invariably reduces to "I cannot conceive in my mind of what an actual infinite would look like. Therefore, none exist."

Kief said...

Do you guys think that those who deny that an actual infinite exists or could exist have to deny that space is infinite in extent?

Ex N1hilo said...


They would have to, to be consistent, wouldn't they?

emmzee said...

Whenever I've heard William Lane Craig lecture on the Kalam argument, I've always thought that he was arguing as you've stated: "I don't believe there are--present tense--actual infinites in creation." For example, during his discussion of Kalam here (login is unfortunately req'd):

Craig states: "a key feature of the kalam cosmological argument for a Personal Creator of the universe based on the finitude of the past is the claim that an actually infinite number of things cannot exist."

It may be the case that some argue that "There are NO actual infinites" but a more precise philosopher like Craig does not do so. (Or if he does say such a thing he means within the material realm.) Even in my own teaching notes & handouts for my grade 11 Sunday school class I wrote "Actual infinite sets of things do not (cannot) exist." (I had to go back and check after reading this post & its comments to make sure I was accurate.)

Anonymous said...

Hey, Brian,

This is off topic, but you're kind of my go to source for things like this. Can you offer me any resources on the Documentary Hypothesis? I need something that actually list, explains and defends (or critiques) the support for the theory as well as the criteria used in dividing the sources.

Brian said...

When arguing against actual infinites, Craig cites numerous examples given by mathematician David Hilbert. The point of the examples, such as Hilbert's Hotel is that when you try to give actual examples of infinites in the real world, it leads to what appears to be logical absurdities.

Brian said...


I remember some lectures by Phil Fernandes that deal with this.
See these ones:
Advanced Apologetics 21 (on OT reliability)

Historical Apologetics 3 (shorter treatment)

Intro to Apologetics 5 (gives history of documentary hypothesis and then refutes it, arguing for Mosaic authorship)

Here is a summary of Intro to Apologetics 5 covering the documentary hypothesis:

Documentary hypothesis: a theory that the Pentateuch was a compilation of different written documents composed by different authors at different places and times long after Moses. The documentary hypothesis denies the Bible as God's Word.

Form criticism: seeks to find the supposed oral traditions that lay behind the written documents . . . extremely subjective.
Common liberal bias is atheistic evolution. This automatically rules out divine revelation before even looking at the evidence. It assumes that religion started with polytheism.

Documentary hypothesis: brief history
This assumes a different author with the use of each different name of God.
1. Scholars today do not hold to this hypothesis.
2. They use circular reasoning.
3. They explain away opposing evidence with a hypothetical editor
4. They teach that only Hebrews couldn't use only one name for God.
5. They assume that secular history is always right when it conflicts with the biblical account.
6. They assume the Hebrew religion evolved into monotheism.
7. They take passages out of context to prove contradictions in OT, yet they accept no solutions.
8. They reject evidence for much Semitic repetition in literature by one author.
9. They assume they can scientifically reconstruct the text 3000 years after the fact.

Some of the evidences for the Mosaic authorship of the Pentateuch:
1445 BC
1. The unity of the first 5 books
2. The Pentateuch itself, the rest of OT and NT all refer to Moses as the author
3. Many eyewitness details in the Pentateuch
4. The author is acquainted with Egypt, but unfamiliar with the land of Canaan
5. Desert atmosphere and point of view
6. The customs are 2nd millennium BC customs
7. Contain a greater percentage of Egyptian words than the rest of the OT
8. Moses' qualifications show he was the author (education, training, knowledge, etc)
9. The different divine names are used for different contexts
10. The variation in diction and style can be explained by several reasons
- Different types of literature
- The author varied the text to prevent monotony
- The parallel accounts are poetic style
11. The biblical evidence shows the Jewish faith was originally monotheistic and that the Jews later became idolatrous
12. The study of ancient religions shows that primitive peoples had technical sacrificial language.
13. Evidence shows that writing existed before Moses' time
14. Archeological finds confirm the Pent.
17. Numbers is similar to another text
18. Deuteronomy is the same format as a treaty during that time
19. The Jews accepted the law as Mosaic

Anonymous said...

Thanks a lot, Brian. It's okay if I blog that, right?

Anonymous said...

1. Numbers exist.
2. There are an infinite amount of numbers.
3. Therefore, an actual infinite exists.

Brian said...


Mathematicians accept that the concept of the infinite is useful in doing maths, but it's a different thing to say that an infinite set of things exist in the material world. In math, infinity is just a concept in your mind.

Brian said...


Feel free to reference those audios—although I would suggest listening to them to ensure the summary points I have listed there are accurate.

Marc Belcastro said...


Suppose we say that God’s knowledge is finite if He knows a finite amount of propositions. It seems to me that claiming that God’s knowledge is finite is only problematic if it’s possible for an actually infinite to exist, which is the issue in question. But if it’s impossible for an actual infinite to exist, then it seems perfectly proper (and therefore harmless) to say that God knows a finite amount of propositions. So, to put it differently, our approach to divine omniscience could be conditional in nature. If an actual infinite can exist in reality, and there’s a distinct proposition corresponding to the actual infinite, then God knows an actually infinite number of propositions. But if it’s impossible for an actual infinite to exist, then God’s knowledge is finite.

You asked Brian if he thinks that God has exhaustive foreknowledge of all future events. I wonder if the term “all” is conceptually misleading when it pertains to quantity and an unending future. In numerous contexts, we use the term to refer to a static, determinate number: e.g., all of the people at the basketball game, all of the crayons in the desk, all of the humans to have ever lived, etc. But with respect to future events, Christian theism teaches that the future will have no end. This means that when we refer to the future by speaking of all future events, we’re not talking about a static, determinate number. For any event we select, there’s always another successor. So although there’s no static, determinate number of future events, I don’t think we need to commit ourselves to the existence of an actually infinite number of future events when considering God’s exhaustive foreknowledge of all future events.

Anonymous said...

It may not exist in the material world, but it seems that numbers do exist (abstractly), since they have properties, and if that is true, then there is an actual infinite. For example, the set of all natural numbers actually has all natural numbers in it.

LittleGoose said...

@ Marc

"For any event we select, there’s always another successor."

If God knows every future event and knows it's successor and there is a never ending chain of future events, well, how is this not an actual infinite?

This philosophy stuff is confusing but oh so interesting.

Marc Belcastro said...


Sorry about the delay in responding. I unfortunately didn’t see your comment until recently.

>>“If God knows every future event and knows it's successor and there is a never ending chain of future events, well, how is this not an actual infinite.”<<

I don’t think we can identify or determine the specific number of future events, but I do think it’s plausible to say that the quantity will be finite. For any event E we might choose, there’ll always be another event – indeed, two, three, four, etc, succeeding events. And, for any E, the number of prior events will be finite, and the number of future events will be indeterminate: the same which holds for E will also hold for any other future event E*.

Similarly, if you started counting the natural numbers today—1, 2, 3, etc.—and you never stopped, I don’t think we could identify or determine the number of natural numbers which you’ll count. But I do think it’s plausible to say that the quantity will be finite. For any number we might select, there’ll always be another number – and, like before, two, three, four, etc. succeeding numbers. Given any number, the number of prior numbers will be finite, and the number of subsequent numbers will be indeterminate.

With respect to future events, no matter how far into the future we proceed, the quantity of transpired events will be finite. With respect to natural numbers, no matter how far into the natural number system you count, the quantity of counted numbers will be finite. At this point, someone may suggest that when we ask how many events there will be, or how many numbers you will count, the only sensible answer is an actually infinite number. But I don’t believe this is a meaningful answer. If the number of future events never will reach an actual infinite, since it’ll be finite at every point, it doesn’t seem meaningful to say that there will be an actually infinite number of future events. What could the term “will” refer to? Similarly, if you’ll never reach infinity during your counting process, since it’ll be finite at every point, it doesn’t seem meaningful to say that you will count an actually infinite number of natural numbers. Again, what could the term “will” denote? What does it mean to say that something will happen in the future if there will never be a point in the future when it happens?

Anonymous said...

Could God create something infinite? Like could he make an infinite amount of books? Or is that a logical impossibility? Also does infinity include everything or just possible things?

Marc Belcastro said...


>>"Could God create something infinite? Like could he make an infinite amount of books? Or is that a logical impossibility?"<<

William Lane Craig would argue that, if it's metaphysically impossible for an actual infinite to exist in reality, then it's not even within God's power to create an actually infinite number of things. And since Craig thinks that the existence of actual infinites is metaphysically impossible, and since omnipotence is (basically) the power to do what's metaphysically possible, the creation of an actual infinite doesn't represent a genuine possibility.

>>"Also does infinity include everything or just possible things?"<<

If one accepts that actual infinites could or do exist, then an actually infinite collection of things wouldn't have to include literally everything which exists. Imagine that there are two walls which are parallel to each other, and suppose that each wall is composed of an actually infinite number of bricks. This gives us two collections which both have an actually infinite number of items, yet this leaves open the possibility of other things existing, like a third or a fourth wall of bricks.

Anonymous said...

Could an actual infinity exist in other logically possible worlds, or different dimensions? Or if we say an actual infinity cannot exist, does that mean anywhere, including different dimensions, and worlds?

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