# Paradox of the extend of space [closed]

We are in a house, the house is in the earth, the earth is in space, so what is space in? In something bigger? Then what is it in? It can go on like this right....? What is the truth?

## closed as unclear what you're asking by user19563, WillO, Nick R, Swami Vishwananda, John AmJul 11 '16 at 18:00

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You did not specify a formal meaning for your terminology, so we have to do some guesswork. Believe it or not, the precise meaning of the word "in" matters here. When one starts exploring the largest (or smallest) limits of a way of thinking, as you are, the funniest words become surprisingly important.

One valid meaning is to refer to space in either a Newtonian sense, or a relativistic sense such as spacetime. In these cases, space is not actually a "thing." Rather it is the domain upon which the equations of physics are applied. If one did not desire to calculate the trajectory of objects, one would not need a spacetime to make sense of the universe.

Another valid meaning is mathematical. For me, your question immediately conjures up visions of set theory. I can treat all of your propositions as set theoretic phrases. "We are in a house" could be `we ∈ house`, or "'we' is an element of 'house'". These can continue forth with `house ∈ earth`* and `house ∈ space`, but what is space an element of? Can we write `space ∈ ???` if we fill in the question marks with the right thing?

Potentially, or potentially not. We can always construct a new set which contains space, and everything in it. We could even give it a name, like "superspace" or "megaspace" or "user17325's thing bigger than space itself" However, this almost certainly not the direction your question was going. You want to know what is bigger than "everything."

Well, that gets to be complicated. In set theory, there is the concept of U, the "universe of all sets." However, some complications arise here. U is not actually a set, itself. It is a category. There is nothing that "contains" U because U cannot be an element in any set. Sets can only contain other sets (in most set theories, anyways). Since U is a category, it cannot be contained in a set.

This sort of distinction becomes important because there's all sorts of paradoxes that arise if one tries to presume U is a set. The most famous is Russel's Paradox. Russel's paradox arises when one seeks to use a naive assumption that all definable collections are sets. If this were true, U would be a set, because it is defined to be "a collection of everything." However, this definition creates a paradox:

Let's define two types of sets: tail chasing sets and normal sets. A tail chasing as a set which contains itself, like a dog chasing its tail. Clearly U would have to be such a tail chasing set, if it is indeed a set itself. Normal sets are those which do not contain themselves. These are the well behaved sets we're used to, such as {1, 2, 3} {{12}, {3, 4}}.

Since we defined these two types, we can define two sets by them: the set of tail chasing sets, and the set of normal sets. But which set is the set of normal sets part of? Is it tail chasing, or normal? If it is normal, then it must contain itself, thus it is tail chasing. If it is tail chasing, then it does not contain itself, thus it must be normal. This is a contradiction, and it is this contradiction that shows that our assumptions are invalid. As it is, the "collection of all normal sets," which is clearly either U or a subset of U, is not a set at all. They coined the word "class" to describe these concepts. U is a class, not a set.

So this is where your question can lead, using modern science and math. One can take other approaches as well, if desired. The Chinese have the concept of the Dao. Everything is part of the dao. However, if one asks what the dao is a part of, the question falls apart. It becomes hard to construct phrases related to your question. The most common response I have seen is "the dao is," and it is left at that.

It also could be turtles all the way down.

* I'm handwaving away some technicalities in set theory. In the strictest sense, if `a ∈ b` and `b ∈ c` does not entail `a ∈ c` in all cases. However it is possible to construct my sets for "we" "house" "earth" and "space" such that this relationship does in fact hold. For the exact construction, see von Neuman ordinals as a well respected formal construction that uses this approach. This distinction will only be important if you continue exploring set theory, and is not really required to understand the rest of the answer.

Actually, this is a question that Aristotle asks in his Metaphysics; he asks "what is the place of place"; where before he defined place as something like our notion of space - it is like a container which can contain a body.

He admits this as being as being an obvious question to ask, but struggles to answer it himself, admitting it as a difficult queston.