Does the empty set have a complement?

[Stephen Tashi]

TL;DR Summary

The concept of a "universal set" is problematical in rigorous versions of set theory. Do such versions leave $\emptyset ^ C$ undefined?

In an elementary school version of set theory, we can take the complement of the empty set to obtain $\emptyset ^ C = \mathbb{U}$ However, in a sophisticated version of set theory, the concept of a "universal set" $\mathbb{U}$ is problematical. ( So says the current Wikipedia article on "unversal set" https://en.wikipedia.org/wiki/Universal_set ).

How do sophisticated versions of set theory treat the concept of a complement to the empty set. Do they leave it undefined?

 

[fresh_42]

Where do we get with the infinity axiom and the power set axiom? Isn't the power set of the empty set the set we are looking for?

 

To answer your question, what is your defintion of complement of a set? I think the existence of a complement inherently assumes the existence of a set $U$ which acts as a universe. More concretely, given a subset $X \subseteq U$, we can define

$$X^c = \{u \in U \mid u \notin X\}$$

I see no way to define a complement without specifying a "universe".

Disclaimer: I did not study formal set theory yet.

 

[mathman]

In practice, there is not a universal empty set, but rather each empty set A is in relation to a given set B, so the complement of A is B.

 

[Stephen Tashi]

In practice, there is not a universal empty set, but rather each empty set A is in relation to a given set B, so the complement of A is B.

I agree with "in practice", meaning in applications. But what happens in the theory? Is the complement of the empty set undefined?

I thought as you do - that there should be different types of empty sets, but in https://www.physicsforums.com/threa...oms-specify-that-the-empty-set-is-open.773047 , post#11 @Fredrik says that the empty set is unique set in ZFC theory.

 

[MisterX]

Couldn't you look at as the complement operator actually taking two arguments, the set to complement and the set that is the "universe"? The empty set would be unique that way. It just doesn't make sense to complement anything without defining (perhaps implicitly) what is the entire set. So you might say "the complement of A in G" where A is the subset to complement and G is the complement of the empty set.

This seems to be the way I have been thinking about it for some time and it seems okay to me for most situations.

 

[Stephen Tashi]

Couldn't you look at as the complement operator actually taking two arguments, the set to complement and the set that is the "universe"? The empty set would be unique that way. It just doesn't make sense to complement anything without defining (perhaps implicitly) what is the entire set. So you might say "the complement of A in G" where A is the subset to complement and G is the complement of the empty set.

That is the approach of the current Wikipedia article on "Complement( set theory)". https://en.wikipedia.org/wiki/Complement_(set_theory) That article speaks of a "universe" U as opposed to a "universal set". (The Wikipedia article on "Universe(mathematics)" is different than its article on "Universal set".) However the article on Complement treats "U" as if it denotes a set. For example, it assumes "$x \in U$" has a defined meaning.

The Wikipedia article on Universal set says

Reasons for nonexistence
Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, do not allow for the existence of a universal set. It is directly contradicted by the axiom of regularity, and its existence would cause paradoxes which would make the theory inconsistent.

This seems to be the way I have been thinking about it for some time and it seems okay to me for most situations.

Likewise, I have no trouble thinking (intuitively) of a universal set. However, my question concerns a rigorous formulation of set theory. This is a question for mathematicians who study "foundations". Most mathematicans don't worry about such things.

 

[mfb]

Without such a universe no complement can be defined in a meaningful way. What is the complement to {4}? If you can find that, add 4 as element and you have the complement to the empty set.

 

[fresh_42]

Without such a universe no complement can be defined in a meaningful way. What is the complement to {4}? If you can find that, add 4 as element and you have the complement to the empty set.

We have the existence of $\{\,\,\}$ and by the infinity axiom all sets $\{\,\,\}\cup\{\,\{\,\,\}\,\}$. For this set we can build the powerset per axiom. These results will give us the smallest possible sets we can build a complement to.

 

[Stephen Tashi]

Without such a universe no complement can be defined in a meaningful way.

Complements are defined in a context dependent manner.

My current understanding is this: In mainstream versions of set theory, no universal set exists. What does exist are context dependent definitions for the notation "$A^C$". To use that notation unambiguously, we must establish that we have selected a set $S$ that is one of the (non-universal) sets allowed by set theory and define "$A^C$" to denote the set difference $S \setminus A$.

What is the complement to {4}? If you can find that, add 4 as element and you have the complement to the empty set.

The complement to $\{4\}$ is only defined if we have explicitly or implicitly defined a set $S$ and agreed that $\{4\}^C$ denotes $S \setminus \{4\}$. For example, if $S$ is the set of integers, then $\{4\}^C$ does not contain the number 2 + 3i or the person Bobbie Watson the commercial traveler.

 

[SSequence]

Complements are defined in a context dependent manner.

My current understanding is this: In mainstream versions of set theory, no universal set exists. What does exist are context dependent definitions for the notation "$A^C$". To use that notation unambiguously, we must establish that we have selected a set $S$ that is one of the (non-universal) sets allowed by set theory and define "$A^C$" to denote the set difference $S \setminus A$.

Yes, you are right. One option is to fix a set as you mentioned.

Paraphrasing from the view of a set theorist***, what you mentioned seems correct to me (informally) [my knowledge is limited/incomplete in this domain so please bear any mistakes].
----For example, one way to observe is that if we have an $\alpha \in Ord$. And now we defined a "collection" $S$ which contains all those ordinals which aren't in $\alpha$ then $S$ is a class (and not a set).

----As far as empty set is concerned, if you try to take its complement you have essentially what is informally called $V$ (the hierarchy formed by iteration of power set --- through ordinals). Once again, $V$ is a class and not a set. If you further adopt "constructibility" then the complement of the empty set will be $L$ (once again a class and not a set).


Edit: With little search this page came up (link may be of some use for further ref. or keywords etc.):
https://math.boisestate.edu/~holmes/holmes/setbiblio.html


*** I expressed my concerns in other places, so I don't need the feel to repeat them at every place. The concerns are both qualitative/philosophy-based and logical [the latter of which can be definitively addressed if it is the case that the logical part has some unwarranted assumptions or other issues].