# Nothingness: What philosophical concept relates to how the empty set is a subset of every set?

Harry Potter and the Deathly Hallows:

“Where do vanished objects go?"

"Into nonbeing, which is to say, everything," replied Professor McGonagall.

"Nicely phrased," replied the eagle door knocker, and the door swung open.”

I wasn't able to find refences that JK Rowling is referring to the mathematical fact that the Empty Set is Subset of All Sets, i.e. of every set.

I have a feeling the empty set is perhaps constructed in such a way to match or at least is coincidentally consistent with some philosophical idea that nothing (as in nothingness / emptiness) is contained in anything / everything and then HP7 is referring to that philosophical idea.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Apr 12 at 9:03

In mathematics the empty set is defined as the set which has no element. A set A is a subset of a set B, if each element of A is also an element of B. If A is the empty set and B arbitrary, then there are no elements to check. Hence the check is OK, because no element of A contradicts.

Now one can further discuss whether the introduction of the empty set is a useful concept. Yes it is. E.g. the empty set is necessary to provide the intersection of two disjoint sets.

The empty set is useful for set theory like the number zero is useful for elementary number theory.

Whether Rowling is right with her literary interpretation of the empty set, that's a different kettle of fish :-)

• +1 for the main answer; but re the last para, it’s not fair to ask “Is Rowling right in her literary interpretation of the empty set?” without first asking “Did Rowling really intend this passage as a reference to the empty set?”, to which I strongly suspect the answer is “No, not at all.” Other philosophical/mystical/folkloric ideas about nothingness, very possibly, but the more mathematised view of the empty set isn’t her style at all. Commented Apr 8 at 15:15

Nothingness — a void, the non-existing — is the prerequisite for anything coming into existence. Something that has been created (the world by God, a poem by a poet, a pot by a potter) fills a place where nothing was before. This difference is what signifies the creation. Without voids, without places of nothingness, no creation could happen.

The void is filled, but it does not cease to exist; it continues to co-exist with what is. Nothingness is everywhere and nowhere, because it does not make the difference; the difference is in what is, not in what is not. Take what is away and the emptiness is once more revealed. Like the empty set whose contents is part of any set but does not make a difference. Take the other elements away, and the empty set reappears.

Reading up on the very long chapter on Nothingness in the SEP, one does not find any particular stream of philosophers which expounded on the concept of the empty set as a trivial subset of every set as related to the philosophical concept of Nothingness. Philosophers seem to like to make things more complicated than necessary, often, but seemingly not in this particular case.

The quote from the book you gave is just that, a quote from a fantasy book. I do not see any particular "philosophy" in it - it sounds cool but is nonsensical as far as I can tell. Obviously, if something disappears it is not then part of "everything".

In our real, physical universe, the laws of conservation make it so that we have no indication that a physical object could disappear - sure, matter can convert to energy and vice versa, but then the new form exists. Sure, stuff can fall into a black hole, but then the black hole has more mass. Sure, black holes can evaporate, but then the stuff that did the evaporation exists. If our current scientific understanding of reality is correct (or - until we have a better version) nothing can ever disappear in the sense that would apply to the quote.

Also be sure to notice that a lot of the historic discussions about Nothingness is very much sophistry or rhetorics, mostly. Often this kind of issue arises when some logic building (i.e., the old, intuitive understanding of what a set is) is too simple, and thus leads to internal contradictions related to Nothingness. This mostly means that the logic building needs to be enhanced to allow for the obvious cases where it makes sense to talk about Nothingness (i.e., the intersection of disjunct sets).

Also, a lot of these arguments very much come from a time where people where trying to come to grips with God vs Science, and also tried to come up with arguments that somehow prove God's existence by trying to find some conflicts when talking about concepts related to Nothingness. All of this makes perfect sense if you already believe in God, or live in an era where ostensibly not believing would bring you into mortal danger. But if you do not believe in God, then all of this is just words with no particular meaning.

Nothingness: What philosophical concept relates to how the empty set is a subset of every set?

First, you are confusing the notion of nothingness with the idea that there is nothing, which is always that there is nothing somewhere. In the concept of nothingness, on the other hand, there isn't the idea of a place where nothingness would somehow be.

Second, the notion of empty set is a purely technical solution to a purely theoretic problem. In other words, the concept of empty set is irrelevant to the intuitive notion of set. In other words, in real life, nobody ever thinks of sets as containing as a subset what mathematicians call the empty set.

I wasn't able to find refences that JK Rowling is referring to the mathematical fact that the Empty Set is Subset of All Sets, i.e. of every set.

Mathematical fact? Words are really cheap.

I have a feeling the empty set is perhaps constructed in such a way to match or at least is coincidentally consistent with some philosophical idea that nothing (as in nothingness / emptiness) is contained in anything / everything and then HP7 is referring to that philosophical idea.

Sorry to disappoint you but the only justification for the notion of empty set is purely theoretic. Mathematicians don't know how to make their totally implausible theory of logic works without it.

Gottlob Frege criticised the idea of empty set with an analogy we can all understand: A wood with no trees is not a wood.

A set with no element is not a set.

consistent with some philosophical idea that nothing (as in nothingness / emptiness) is contained in anything / everything

The idea that nothing is in everything is inconsistent . . .

And how could an idea be consistent with an inconsistent idea?

Still, I would agree with you that words are really cheap.

• If you accept the Peano Axioms, you accept the null set. Commented Apr 8 at 18:02
• Empty set is fairly well represented in the real world by considering an empty folder on a computer. If Frege lived to see computers, I imagine the notion of empty set would match his intuitions better. A folder with no files is still a folder. Thinking of sets as containers is fairly standard- and most consider an empty box to still be a box. Commented Apr 9 at 2:09
• @MichaelCarey - For Frege there is a difference between empty concept: he used the concept x≠x to define the number zero, and set (Menge) that he read as a "collection", to which he applied the consideration above: "no object, no collection"; see Gottlob Frege, Philosophical Writings (1952) page 89 (origianl paper: review (1895) of Schroeder's Vorlesungen) Commented Apr 9 at 8:58
• @MichaelCarey "it does mean they behave like sets in some ways- and does provide some notion of credibility to the set theoretic constructions, relevence to the real world." This is as fallacious as claiming that when the conclusion of a false theory is true, then the theory is true. In the real world, there is no set if there is no members. A set is a collection of things, not a bag with things in it. The idea of empty set is an abdication. These people couldn't understand how it worked, so they adopted a zany theory that seemed to works in at least some trivial cases. Commented Apr 10 at 9:29
• @MichaelCarey "that basic operations on sets, are applicable to databases" Sure, and we don't need the notion of empty set because a set is not a folder. We knew of folders well before we could build our first computers and our first databases, and I'm sure no one missed the notion of empty set then. Commented Apr 10 at 9:36