While watching the below Vsauce video on YouTube, I had an interesting thought: Could the Banach-Tarski Paradox be a way of explaining the Holy Trinity? Or at a minimum, a way of illustrating the relationship between God the Father and Christ?

Jesus is the begotten son of God. He isn’t made by, but made of.

Per my understanding of the video, to make a second of an infinite body, you remove one piece. In doing so, you create an entirely distinct new body, made from the original, that is the same as the original.
(Now for the part that I am struggling with the most to fit my concept)

The Holy Spirit is said to be the breath of love between the father and the son. Could it be said that the complete giving of one to the other through love is that breath? If so, you would then have ∞+∞=∞, which could be said to be a 3rd infinite body from the original and the second equally.

I will admit that #4 is kind of shaky, as it felt like I was reaching logically. Maybe additional thought will yield something better.

My lunch break is over. Let me know what you think.

My understanding of the paradox is limited, but from what I can tell it involves using the Axiom of Choice to take sets that can be meaningfully assigned length and then cutting and rearranging them so that the whole notion of length no longer applies. The resultant sets are no longer measurable. Then the pieces are reassembled in the desired form in which they are once again measurable. No actual contradiction occurs because the lengths never actually increased. Basically the original object was destroyed beyond geometric recognition and the debris was used to build a different object.

Anyway, it’s very technical and doesn’t correspond to reality in any obvious way. I don’t think the paradox is indicative of anything besides the fact that the concept of length is surprisingly tricky or that the Axiom of Choice is surprisingly flexible.

I thought it was that a pea can be chopped up and reassembled into the sun, which is done by applying the axiom of choice an uncountable number of times.

Yes. The “chopping” consists only of translations and rotations, which under normal circumstances would not change the length of segments in geometry. But the use of the Axiom of Choice lets you basically disfigure the segment to the extent that it wouldn’t make sense to say it has a length, then the pieces are rearranged so that you get something longer than before. No contradiction arises because you essentially temporarily broke geometry. As Wikipedia puts it:

[quote=Wikipedia]Unlike most theorems in geometry, the proof of this result depends in a critical way on the choice of axioms for set theory.** It can be proven using the axiom of choice, which allows for the construction of nonmeasurable sets**, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.
[/quote]

Regarding reassembling the pieces:

[quote=Wikipedia]The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here, because in this case it is impossible to define the volumes of the considered subsets, as they are chosen with such a large porosity. Reassembling them reproduces a volume, which happens to be different from the volume at the start.
[/quote]

Okay, I think I understand what you guys are saying. I am guessing that the axiom of choice comes in when you decide which piece of the sphere you decide to remove. In the video, the guy says that the only way the math works is if you have a truly infinite object, and that we can’t physically prove it because our reality is finite. Even if the paradox doesn’t work in reality, do you think it could apply in a theoretical sense? I mean, if God is infinite, and a sphere, you could theoretically apply the paradox, and it would hold, wouldn’t it?

There are multiple reasons why it can’t physically be shown. Let’s say you used a beach ball as the sphere. As the Wikipedia article says, we want to cut the beach ball up into a “fine scattering of points”. There’s our first issue. Points are geometric objects that do not exactly correspond to anything on the beach ball. Instead, the beach ball consists of particles, which are spherical and can’t be broken down further.

A second issue was brought up by Tomdstone: you must apply the Axiom of Choice uncountably many times. I’m not sure how familiar you are with countable vs. uncountable infinities, but basically the ability to arrange an infinite set of objects into a sequence makes it “countable”. If we pretend that the beach ball really is made up of points so that we can keep cutting it forever, then we might imagine making a machine that will keep cutting the ball for all time, long after humanity is extinct. But the machine is limited because it can only cut countably many times–even if all of eternity is allowed–by making the first cut, then the second, and the third, etc., in a sequence.

So this can’t be done physically because we can’t cut that many times and matter doesn’t break down that far.

Even if the paradox doesn’t work in reality, do you think it could apply in a theoretical sense? I mean, if God is infinite, and a sphere, you could theoretically apply the paradox, and it would hold, wouldn’t it?

What they meant by “infinite” was probably “continuous”. In other words, the sphere or whatever you’re cutting has to be an unbroken surface of points, not an array of particles like matter (such a surface would have infinitely many points, whereas there are only finitely many particles). Continuity is a topological concept, and as such I’m not sure how one would apply it outside that context.

I hope I helped with the math at least. I have no clue about the theology, though.

I suppose that makes sense. How about the following questions:

What is your definition of a continuous surface? Do you assume there is a difference between an infinite solid surface and a surface made of an infinite number of data points that are infinitesimally close together? I ask this because I am not familiar enough with the math to understand why it matters that the sphere would be broken for the paradox to work.

What if we throw our physical reality out the window, so we are not talking about physical particals? As best we understand, God exists outside the universe. That would mean that God does not have to be made of particles. From the purely theoretical perspective, is could my original post work? Asked another way, can we make the assumption that we are dealing with data points, rather than physical particles?

It may be easier to begin with the notion of a continuous function.

Each function has a set of values that can be used as inputs called a domain and sends those inputs to outputs in the codomain. For example, your height is a function of your age since, for any given time after your birth, the number of seconds you’ve been alive (a value in the domain) can be associated with your height in inches at that time (a value in the codomain).

A function is continuous if the change in its outputs can be made as small as you like by restricting the change in the inputs. So your height is a continuous function of time since, no matter how small the change in height we allow, we can find a small enough time interval so that your height will stay in that range. To oversimplify a bit, a continuous function is basically one that doesn’t “jump” instantly from one value to another.

Apply this idea to surfaces. If we have, say, a sheet of paper that is ripped, we are forced to “jump” over the rip. This is a discontinuity. So continuous surfaces are basically those that don’t have holes. Technically no material is continuous since all matter is made up of particles, but many things are approximately continuous.

Do you assume there is a difference between an infinite solid surface and a surface made of an infinite number of data points that are infinitesimally close together?

I’m not sure what you mean by “infinite surface”. Do you have in mind something like an infinitely large plane? The problem there is that the area of such a surface would be infinite, so you couldn’t meaningfully compare its area to whatever new surface you construct from it.

I ask this because I am not familiar enough with the math to understand why it matters that the sphere would be broken for the paradox to work.

My understanding is that you essentially want to poke so many holes in the surface that not even the pieces that are left over will be continuous. The notion of length (area, volume, etc.) depends on certain topological features such as continuity staying intact.

What if we throw our physical reality out the window, so we are not talking about physical particals? As best we understand, God exists outside the universe. That would mean that God does not have to be made of particles. From the purely theoretical perspective, is could my original post work? Asked another way, can we make the assumption that we are dealing with data points, rather than physical particles?

One could think of continuity analytically–as in the definition involving functions–or topologically–as in the definition involving surfaces. So I would have to know what function or surface you have in mind. I don’t know how you would apply notions of continuity and length to God, however.

Wow! Thanks for such a detailed response. Because of it, I think I found our disconnect. In your response to my first set of questions, you described having to jump over the discontinuity created when a point is removed for the Axiom of Choice. I completely agree that this is a problem, and agree that this is why it is physically impossible to prove.

Now, if you look in the video I linked at about 8:20, Hilbert’s Hotel is applied to a circle. Because the circumstance of a circle is based on pie, applying Hilbert’s Hotel to a circle, or a sphere, causes a point that was removed to not be needed to have a continuous surface. At 12:10 of the video, constraints on movement are also applied, in terms of the distance and directions you can travel. The video then goes on to illustrate application of the paradox through the 19 minute mark.

Another disconnect just occurred to me. The example used by the video doesn’t really remove data points. It only separates them for manipulation. The use of Hilbert’s Hotel and pie are key because you mathematically prove that any missing points are still there.

Assuming we are now on the same page, or at least in the same chapter, I will try to redefine my original post:
Assumptions:

God exists outside the universe, and isn’t compos of particles.

All parts of God are equidistant from the center of his being. (Sphere)

God is infinite.

Taking these assumptions into account, all data points on the surface of God can be named using the method described in the video. This will yield two versions, if you will, of God the Father. How do we know that the second sphere would be Jesus? Per the Nicene Creed, we know that Jesus is begotten from and consubstantial with, God the Father.

How does this revision sound? Hopefully my logical jumps are smaller, and my assumptions are more clearly defined. One place I realize I messed up originally is thinking something was removed, when it is all just rearranged.

Thanks Tombstone. Parts is bad terminology. My intend was to describe the outer bounds/extents/limits. While I know he doesn’t have those either, I am trying to define an assumption that God is a sphere for the purposes of this exercise.

If that satisfies the terminology, do you have thoughts on the premise I have presented?

DISCLAIMER: The views and opinions expressed in these forums do not necessarily reflect those of Catholic Answers. For official apologetics resources please visit www.catholic.com.