How Train Cars and Set Theory Prove the Existence of God

Perhaps the strongest argument against atheism is the argument from contingency. In its barest form, it goes something like this:

A. All of reality, without exception, breaks down into one of two possible categories:
(1) that which is contingent, and (2) that which is non-contingent.

Let me explain what I mean by both the terms, “thing” and “contingent.” I mean “thing” in its broadest
possible sense: a physical object, a scientific law, whatever. If a thing requires something else to exist, it’s “contingent.”

Family Tree

You and I are radically contingent. Without our parents (and their parents, and their parents, all the way back), we wouldn’t exist. Even the slightest change: your great-great-great-great-great grandfather dies before meeting your great-great-great-great-great grandmother (or moves, or isn’t attracted to her, or marries someone else instead, etc., etc., etc.), and you don’t exist. But more than that, innumerous other factors had to line up just so: we had to live on a planet with all sorts of conditions making human life (and procreation possible), in just the right sort of universe, and so forth.

I’m not interested in debating here whether the coalescence of these factors was probable or improbable, mere chance or Grand Design.

My point is simpler: in order for you to exist, some other conditions much logically precede your existence. Your existence is logically dependent upon these other conditions. Therefore, if I know that a particular universe has you in it, I can deduce all sorts of other things about that universe: that it’s life-sustaining, contains (or contained) your parents, etc.

This is what we mean by a contingent reality. Under particular conditions, you come into existence. Had those conditions not been there, you wouldn’t.

All created things are in Category 1: they’re all contingent. They don’t arise necessarily. Sometimes this is stated as “everything that begins to exist has a cause.”

So all created things are contingent. But the opposite isn’t true: a thing could be infinitely old and still logically-contingent. So Category 1, as you might suspect, is enormous. It includes the entire universe.

B) You cannot simply have an infinite series of contingent causes;
an uncaused cause is logically necessary.

Contingent causes require something else to exist: that chain can’t simply go back forever. If A requires the existence of B, B requires the existence of C, and C requires the existence of A, you can’t posit A, B, or C as the cause of the existence of the set of A, B, and C.  A philosophy professor of mine gave me two helpful illustrations of this principle: one involving set theory, and one involving trains.

Let’s talk about set theory first. All causes can be divided into [Caused Causes] and [Uncaused Causes]. No matter how large the set of [Caused Causes] is, even infinitely large, it can’t cause itself. It needs an Uncaused Cause to begin. We tend to think of this temporally (that the uncaused cause exists prior to the caused cause), but this would be true even if the uncaused cause and caused cause(s) are both infinitely old.

That illustration is clean and simple, but it’s not for everyone. For that reason, he gave the illustration of trains. The picture below shows two cars – the car on the right is a locomotive, which has the engine; while the car on the left is an ordinary train car, which doesn’t:

If you’re at a railroad crossing and see an ordinary train car go by, you may be sure that it’s being pulled by a locomotive, even if you can’t see the front of the train. The car may be pulled by the locomotive directly, or by another train car or series of train cars which are themselves pulled along by a locomotive.

A train that was made up only of train cars, with no locomotive, could never move. It doesn’t matter if that locomotiveless-train was one car long, or a thousand, or a million, or infinite. Without something moving it, it won’t move an inch.

So it is with contingent realities: they are ultimately set in motion, so to speak, by what is necessary. It doesn’t matter if you have a single contingent reality, or a thousand, or an infinite number: they require the existence of a non-contingent reality.

By the way, this also explains why the atheist argument, “Who created God?” is philosophical nonsense. It’s like asking, “Well, if all train cars must be moved by a locomotive, who moves the locomotive?”

C) This necessary reality is what we call “God.”

At this point, we can say with certainty that Someone or Something necessary created the entire universe. We can say with absolute certainty that the entire universe – from every person to every particle to every scientific law – traces back to a common origin, some kind of Creator. This gives us a starting place to explore what sort of Creator we’re talking about. Here ends the proof.

It doesn’t immediately prove the truth of Catholicism (that’s known through a combination of reason and Divine self-revelation), but it does disprove the possibility of atheism. There’s simply no way to start with no uncaused causes, and end up with anything, be it physical laws or material realities or anything else.

P.S. What sort of Creator?

The proof might suggest that the universe has a Cause, but a wholly impersonal one. It turns out, using this proof, we can actually determine a whole lot about our Creator. You see, nothing can exist in an effect that doesn’t exist (at least in potency) in the sum total of the causes. You can’t put two and two together and get five. That whole notion violates basic causality.

And since all of the other causes derive from the same necessary cause, we can say that nothing can exist in the universe that doesn’t come from the Creator. So, for example, we exist in a universe that contains (and permits) personality and sentience. But the effects of personality and sentience can’t exist without the ultimate Cause possessing personality and sentience. So we know right away that we’re dealing with a personal God


  1. Thank you.
    Now for some proofreading.
    Probable errors…

    Something [Sometimes] this is stated as “everything that begins to exist has a cause.”

    We tend to think of this temporarily [temporally]

  2. To play Devil’s Advocate to your Post Script for a moment…
    “You see, nothing can exist in an effect that doesn’t exist (at least in potency) in the sum total of the causes.” is used in conjunction with the existence of personality and sentience to argue that “the ultimate Cause [possesses] personality and sentience.”

    I’m not completely sure that follows, though it does sound good. Couldn’t I theoretically get personality and sentience by chance in a universe where an impersonal, perhaps nonsentience Deist-style God creates gravity (My potentially-erroneous understanding of M-theory indicates that the Law of Gravitation + nothing -> Big Bang) or just kicks off the Big Bang directly. Billions of years pass, stars form and explode, producing steadily heavier elements. Somewhere along the line the Earth coalesces, life evolves from extremely simple forms to progressively more ‘aware’ creatures. Eventually, humans are walking about, reading blog posts on the internet and having debates and being personal and sentient.

    It seems at this point I’d have contigent personality and sentience from an impersonal and potentially nonsentient, but non-contigent God. This gravity-creating It is the sum total of the first causes of me, but I don’t see how that would mean It would necessarily be personal or sentient.

    Is there some error in my counterexample, or do I not understand potency? I agree with the necessity from this argument for the First Cause, or Unmoved Mover. I’m not confident the from-reason-alone proof of God as personal/sentient as presented here holds.


    1. Hi Kevin,

      You asked:
      I’m not completely sure that follows, though it does sound good. Couldn’t I theoretically get personality and sentience by chance ….

      No. I ignored the rest of your message because this is your premise. And if your premise is false, that which follows remains false.

      So, why can’t it happen by chance.

      There is no such thing as pure “chance”. Another and better word for “chance” is “probability”. Both words imply a delimiting Actor. When we speak of chance, we mean that one action is always related to the next in some way. If the actions are not somehow related or delimited, there is “no chance”. Therefore it can’t happen.

      Let me give an example. A die (singular for dice). It has six sides. When a die is cast, the actions, the casts are related to each other in the sense that each time it is cast, it is a die which is cast. The sides are all on the same item. But each side of the item has an equal chance of coming up.

      However, if we have “pure” chance (aka no chance), there is an unlimited number of possibilities and none of them related to one another since there is no sentient being to delimit the possibilities. If there is no sentient being to delimit the possibilities, there is also no sentient being to cause the possibilities. Therefore, the answer to your question is, “no’.


      De Maria

    2. Kevin,

      That was my first objection, also. But see what Mark Duch says below: the effect cannot contain anything not found (at least in potency) in the totality of all of the causes. You can’t claim that “2 + 2 = 2 + 2 + something else,” because it introduces something to the effects not found in the causes.

      When Darwinians claim that X trait naturally evolved in a species, they’re implicitly saying that the pre-evolved species contained the potential to evolve X trait (or that the pre-evolved species, coupled with the right external stimuli, contained this potential). But if that’s the case, then the effect (X trait) is located in the sum of the causes (the pre-evolved species + external stimuli).

      So the pure (materialist) Darwinian would have to say that sentience exists potentially in primordial matter: that the first particles from the Big Bang already contained the potency for sentience, and that it just took a long time for that potency to be actualized. If that claim sounds absurd, that’s a problem with the pure Darwinian understanding, not with the principle that the effect can’t have anything not found within the sum of the causes.

      So if the sole cause of the universe is a necessary Cause [and that’s the only explanation that makes sense, as A-C show], then everything contained in the universe must be contained (at least in potency) within the Cause.



    3. De Maria

      A mildly expanded or colloquial understanding of the word chance might have done you well. Chance is also commonly used to describe unplanned events. A chance meeting, I happened upon it by chance, etc. Lack of planning for a given event or process is meant.

      My argument relies on a few points
      There is no scientific evidence for a purpose (telos) in nature, or a guiding hand. This indicates that there is at least the possibility that the development of Creation was unguided.
      Therefore, there is a possibility that the Creator did not have a plan, didn’t care, or wasn’t capable of caring. Perhaps It just created gravity because. Or perhaps It wanted gravity, but didn’t care whether or not we existed or were persons or what a person was.
      Therefore, personality and sentience could have arisen without a plan for them, and without necessarily being a reflection from our Creator.

      I’m afraid your discussion of probability is irrelevant to the argument at hand. I’m objecting that the domain of possible self-contingent Creators is not shown to contain only the personal and sentient by this argument. Though it is worth noting that some theoretical physics models indicate that our set of physical laws is one realization or sample from a probability distribution of all possible physical laws, an ensemble ‘integrated out’ over multiple universes. Certainly still theory, and probably lifetimes from being tested, but the math shows that “infinitely many unrelated possibilities” is certainly possible.


    4. Mr. Heschmeyer,

      Thank you for the reply.

      I don’t object to the “in potency”, I don’t think. Sure, the particles of the Big Bang would contain personality and sentience in potency, but that’s kind of meaningless. Fundamental particles are not sentient or personal, but contain that in potency. Likewise, a theoretical God could have created the universe, and thus would in potency contain personality and sentience. But we don’t talk about dirt or fundamental particles being personal because they contain that in potency. We talk about persons as personal who have attained that. How does showing that God is in-potency personal make Him *actually* personal?


    5. @Joe: I used to believe that very thing, that sentience exists in potentia in everything that makes up the universe and therefore in the universe itself. I was congratulating myself for having concluded “God” then. Even now I haven’t crucified my pride completely, but at least I recognize my flaws a little better now and want to try.

    1. Ha! I’ve increasingly noticed that the ordinary understanding people have of quantum physics is that it’s magic, and things like causality and logic don’t apply, because magic.

      I’ve had the Bootstrap Paradox put forward as a “solution” to this problem twice already: that the material universe was created by something like a backwards-moving particle. (That is, a particle from our universe went back in time and created the universe). Magic!



  3. Kevin, Joe has skipped a lot of steps in his post script, and I don’t like the way he’s phrased it, but I think he’s assumed some Thomist principles (not that such an assumption is invalid per se, but that it was not made evident for the purposes of discussion, which would have been helpful for such a bold claim). See if this summation of those principles helps:

    1. The characteristics of a thing–say, a triangle, have their raison d’etre in the essence (nature) of that thing (which is to have three sides, etc.).
    2. The existence of an effect has its raison d’etre in the cause which both (a) produces AND (b) preserves that existence–that is to say, in the cause which is the reason not only of the “becoming,” but also of the continued being of that effect.
    3. Thus, that which “is being” by participation has its raison d’etre in that which “is being” by essence. E.g., A triangle is a triangle and continues to be a triangle only to the extent that it participates in triangularity.

    Dense and rough, but it is a combox, so there you go…

    To the main discussion, Islamic philosophy has a similar argument from contingency to that of Aquinas et al. Avicenna’s argument, as summarized by Dr. Edward Feser, is as follows:

    1. Something exists.
    2. Whatever exists is either possible or necessary.
    3. If that something which exists is necessary, then there is a necessary existent.
    4. Whatever is possible has a cause.
    5. So if that something which exists is possible, then it has a cause.
    6. The totality of possible things is either necessary in itself or possible in itself.
    7. The totality cannot be necessary in itself since it exists only through the existence of its members.
    8. So the totality of possible things is possible in itself.
    9. So the totality of possible things has a cause.
    10. This cause is either internal to the totality or external to it.
    11. If it is internal to the totality, then it is either necessary or possible.
    12. But it cannot in that case be necessary, because the totality is comprised of possible things.
    13. And it also cannot in that case be possible, since as the cause of all possible things it would in that case be its own cause, which would make it necessary and not possible after all, which is a contradiction.
    14. So the cause of the totality of possible things is not internal to that totality, but external to it.
    15. But if it is outside the totality of possible things, then it is necessary.
    16. So there is a necessary existent.

    1. Because if it was non-contingent, it would be necessary. The idea that quantum vacuums are necessary and non-contingent strikes me as self-refuting. After all, quantum vacuums arise as a result of particular conditions, and are in perpetual states of flux. They also still contain matter (albeit a very small amount). They’re not true voids (true voids don’t exist).

      A necessary thing can’t be unstable, because it’s not dependent upon anything else: it’s not altered by external stimuli, etc. (that would be a form of contingency). So a quantum state can’t possibly be non-contingent.



    2. You say that quantum vacuums arise as a result of particular conditions; that may not be the case. They may be the default state.
      Yes, they are in perpetual flux, and yes they generate and destroy small amounts of matter, and yes, they’re not true voids. I agree with all of those, but I still don’t see why a QV could not be non-contingent.

      “A necessary thing can’t be unstable, because it’s not dependent upon anything else”
      But if the instability is part of the nature of the necessary thing, it can be unstable and still not depend upon anything else. It’s not altered by external stimuli, it’s altered by its own nature.

    3. First, the quantum state you’re describing still contains matter and requires time, so it could hardly be held to be the cause of time or matter.

      Second, I’m not sure what it means to claim that a thing is altered by its own nature.

      If it isn’t impacted by external stimuli, then it’s a system containing internal stimuli. But if that’s the case, you’ve just kicked the problem back further. One of the internal components of the quantum state would have to be necessary and non-continent.

      Third, you’ve also introduced a problem of traversing an actual infinite. For the sake of clarity, imagine what you’re describing: a quantum vacuum in a perpetual state of flux. At one moment, it’s in hat we can call Quantum Position [QP] 1, then QP 2, QP 3, etc.

      And in your account, it’s done this forever. This would mean that by the time it arrived at a life-producing QP, it had already been in flux for an infinite amount of time. So the live-producing position is QP Infinity (which can’t exist, because it would be an actual infinite, etc.).



    4. Appreciate the reply. 🙂

      The quantum state that I’m describing generates matter. We know how that works, physically. I don’t see the problem there.
      If the QS that I’m describing requires time, as you say, that does seem like a problem, but I’m not sure that you’re right that it does. Can you expand on that?

      As for the “altered by its own nature” bit. Imagine for the sake of argument that the necessary thing at the root of the universe is a quantum vacuum operating in accordance with known scientific laws. Such a vacuum generates matter (and maybe time – like I said above, I don’t know) naturally. Without an external stimulus; it just does it ‘cuz it does it.

      It kind of seems to me like you’re trying to invoke Euthyphro here. I’m describing a system (quantum vacuum) operating according to a law; you’re saying that’s incoherent because one of them has to be existentially prior to the other. Right? But it seems to me at least conceivable that objection can be handled in the same way that a monotheist handles it – God and Law (morality in that case, but same difference for our purposes) are in fact one and the same. While it is easier for us to talk about system (quantum vacuum) and laws (observed behavior of quantum vacuum) separately, is there any reason that they can’t be considered to be one and the same? If they can be the same, then your objection doesn’t work.

      I see your point about traversing the actual infinite, and I’ve got nothin’. 🙂 But I seem to recall Aquinas saying that an eternal universe was logically tenable, for reasons that I don’t remember offhand. Would you be OK with tabling this point for now?

      God bless,

  4. Well, hello, my old friend the cosmological argument!

    It was in fact this argument that led me back to theism after I had fallen away from the Church (as written by Mortimer J. Adler, God rest his soul). Now I am a confirmed Catholic and couldn’t be happier about it even with all the flaws from which I haven’t recovered and all the progress I still have left to go through.

    Thank you for posting this! God bless!

  5. Okay, my question is more basic, related to this idea: “Contingent causes require something else to exist: that chain can’t simply go back forever.

    Does everyone (atheists included) agree that contingent causes must, at some point, have had a non-contingent cause? Other than God, do they have a theory for what this is (assuming they agree with this)?

    1. Interesting. I personally think God intervened to create at multiple points throughout history, meaning I do not believe in macroevolution. But, doesn’t the idea of evolution prove that they believe in all things having contingent causes, rather than non-contigent causes? They cannot leave room for the non-contingent, or their theories collapse and they must consider the super-natural. However, they can’t identify the original cause, which you call the “brute fact.” I assume the “brute fact” is not intelligent in any way?

      Finally, that paper is really long, so, can you point me to the most important parts of it (or summarize :))?

    2. “doesn’t the idea of evolution prove that they believe in all things having contingent causes, rather than non-contigent causes?”

      I don’t see the connection. What do you mean?

    3. I’m saying that evolution is built on the idea that all things have a contingent cause, i.e., humankind wasn’t created, rather, evolved from the same source as apes—and back and back until you arrive at the first organic life on earth. It was really a side point, so I didn’t mean to side track the conversation.

  6. Your three examples clarify what you mean by existence, when you say, “If a thing requires something else to exist, it’s ‘contingent.’” The argument for the existence of God is based on the ontological meaning of existence, namely the verity of being. None of your three examples refers to being, itself. The biological existence of a biological entity is fully explained by the immediate parents of those biological entities, which are produced sexually, whether lilies or humans. God is not the ultimate biological cause of a sexual entity’s biological existence. Local motion is fully explained by the immediate physical cause of motion. God is not the ultimate physical cause of motion. Sets are purely logical constructs. God is not a logical construct.

    1. Bob,

      Is your complaint that my analogies are only true analogously? Or are you saying something more substantive?

      Also, I am not sure what your basis is for claiming “The biological existence of a biological entity is fully explained by the immediate parents of those biological entities, which are produced sexually, whether lilies or humans. God is not the ultimate biological cause of a sexual entity’s biological existence. Local motion is fully explained by the immediate physical cause of motion.” Can you defend these claims?



    2. Such popular illustrations concede that each element within the series is sufficiently explained by the preceding element. Otherwise, why discuss a series at all? In that the elements are sufficient explanations within the series, they must be equal in nature. Consequently, the first of the series need not be superior to any element since its function does not differ from any other element within the series. If God is the first mover of such a series or the first cause of such a series, he need not be any different in nature from the other elements in the series.

      To discuss series is a distraction. What must be demonstrated is that there is one immediate cause of the existence of each and everything within our experience.

      The reason the locomotive is different in nature from the railway cars is that a train is not a natural phenomenon, but the product of human engineering. Thus, this common analogy is disallowed as well for being anthropic.

      Aquinas rejected the arguments based on series (

Leave a Reply

Your email address will not be published. Required fields are marked *