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On page 6 of Munkres' book Topology (Second Edition) he says "the empty set is only a convention" in a few different spots. I am wondering why he says this...

My understanding is that the axioms of ZFC set theory (the axiomatic system which basically all mathematicians use) the existence of the empty set is either taken as an axiom or can be deduced from other axioms; see this post for details. With this said, it appears that the empty set is more than a "convention." Rather, it is a mathematical object that exists. So, assuming the ZFC axioms, statements involving the empty set seem to be completely rigorous statements and not just notational conventions as Munkres suggests. For example, on pages 12-13 he says not all mathematicians follow the "convention" that the empty intersection is the entire space, in contrast to the answer provided in this post.

Furthermore, regarding the empty set, he says that "mathematics could very well get along with it." I don't fully disagree with this, but in the definition of a topology (along with some other mathematical spaces) the empty set is assumed to be contained in the topology. Can we somehow get away without the empty set object in the context of topology?

Any insight on this matter would be appreciated. Is Munkres' wrong? Or should his words be interpreted differently?

Asaf Karagila
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Satana
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    I guess the point is that every time you use $\emptyset$, you could just as well use words. For example, we could say that the intersection of two open sets must either be another open set or contain no elements. Eventually, it becomes expeditious to replace "contain no elements" with "is the empty set," but it isn't required per se. – Charles Hudgins Jan 22 '23 at 17:10
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    The mathematician RL Moore, who was influential in his time (1920s or so), didn't accept the empty set. Perhaps he thought the idea of a collection of objects which doesn't contain any objects doesn't make sense. It's possible to do math without the empty set, but the phrasing is more awkward. – littleO Jan 22 '23 at 17:51
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    In much older literature (well before Munkres, before 1910 or so, maybe even up to the mid 1920s or later) it was fairly standard to say "doesn't exist". For example, one would say something like the following: "The intersection of two open sets, if it exists, is an open set." – Dave L. Renfro Jan 22 '23 at 17:52
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    @littleO But assuming the axioms of Zermelo-Fraenkel set theory, the empty set does exist. Therefore, unless Munkres held the same belief as RL Moore, it should be more than a "convention," correct? – Satana Jan 22 '23 at 19:39
  • @DaveL.Renfro This makes sense. So you are saying that you could get by perfectly well without the empty set in topology? In particular, you could refuse to accept the existence of the empty set and instead use the words "if it exists" to mean that the set(s) under consideration must have elements in them? – Satana Jan 22 '23 at 19:43
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    I think so. Historically I think the empty set came into general use in mathematics well after its initial appearances in logic and foundations and set theory. Keep in mind that the vast, vast majority of mathematical work in the late 1880s to 1920s was NOT involved with these set-theoretical foundational issues (but see Kanamori's 2003 historical survey paper for that). For some examples, look at the snippet views from this google-books search. – Dave L. Renfro Jan 22 '23 at 21:11

1 Answers1

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Your example is about the empty intersection, not the empty set per se. In general, we define (and this is specifically allowed by one of the ZFC axioms) $B=\bigcup_{i\in I}A_i$ as the set given by $$ x\in B\iff \exists i\in I\colon x\in A_i.$$ This poses no problem (thank you, Union Axiom!). We’d love to define analogously $C=\bigcap_{i\in I}A_i$ as the set given by $$x\in C \iff \forall i\in I\colon x\in I.$$ However, such a set $C$ does not exist when $I=\emptyset$! Indeed, the right hand side is vacuously true for all $x$, that is, $C$ would have to be the “set of all sets”, which is absurd.

Therefore, by convention one often defines the intersection in case $I=\emptyset$ as some convenient universal set. For example, while considering a specific topological space (i.e., whenever $I\ne \emptyset$ in the current context, then $A_i\subseteq X$ for all $i\in I$), that convenient universal set might be $X$.

Hagen von Eitzen
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  • Thanks! Maybe I should clarify some more because I was a bit imprecise. Of course the empty intersection does not exist because it would only make sense for it to be the set of all sets (which is a set that doesn't exist). However, if we fix some "universal set" $X$ and define the intersection of any collection of subsets ${A_i}_{i\in\mathcal{I}}$ of $X$ to be the largest subset of $X$ that is contained in $A_i$ for all $i\in\mathcal{I}$, then it is (vacuously) true (as opposed to a "convention") that the intersection is equal to $X$ when $\mathcal{I}=\emptyset$. – Satana Jan 22 '23 at 19:32
  • I should also add that I was under the assumption that we are operating in some universal set $X$ throughout Munkres' book, as opposed to the universe of all sets. In this case, the intersection of the empty set should be a well-defined object (namely the whole set), as opposed to a "convention." Correct? – Satana Jan 22 '23 at 19:34