Talk:Empty set

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Notation

Currently the article contains (in two places) the sentence "Common notations for the empty set include "{}", " $\emptyset$ ", and "∅"." Recently, Mamtaattrinain edited this to add "𝜙" to the list, and I reverted on the grounds that using the Greek letter phi is a typesetting cludge, to avoid having to create specialized symbols, and that it's not really a different notation. However, it seems to me that $\emptyset$ and ∅ are in the same boat: they are two different ways of writing the same thing (the letter Oh with a slash through it from top-right to bottom-left). I am sort of inclined to remove one of them, or to replace them with a description in words. Does anyone else have thoughts about this? --JBL (talk) 19:37, 11 June 2019 (UTC)[reply]

I agree, the use of phi (or varphi) is a kludge used primarily in older texts to keep printing costs down and doesn't really reflect a different notation. On the other hand, serious readers will come across these variations from time to time and maybe we should say something about it in the article. I'm inclined to not write out all the variations, as this gives them the appearance of equal status, but rather separate the common symbols and the degenerate ones and label them as such. --Bill Cherowitzo (talk) 18:31, 21 June 2019 (UTC)[reply]

What is the complement of the empty set?

The current 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.

So, if the universal set doesn't exist in a set theory, what does that theory do about the complement of the empty set? - leave it undefined?

Tashiro~enwiki (talk) 05:21, 22 September 2019 (UTC)[reply]

A set complement is always a relative notion: you start with A contained in B, and the complement is B - A. No set has a complement if you don't specify complement with respect to what. --JBL (talk) 12:14, 22 September 2019 (UTC)[reply]

According to that, the notation "

A^{C}

" is ambiguous. In particular "

\emptyset ^{C}

" doesn't denote a particular set in a set theory that does not allow the existence of a universal set. I'm not disturbed by such an outcome. I'm only curious about how the technicalities are handled in a rigorous presentation of set theory. Do we say that the notation

A^{C}

only denotes a particular set if the writer has chosen a particular set

B

to use for defining

A^{c}=A\setminus B

? Tashiro~enwiki (talk) 18:05, 22 September 2019 (UTC)[reply]

I am not a set theorist, but in my practice as a mathematician the meaning of "

A^{C}

" is always context-dependent, and the answer to your last question is "yes, of course". I'm sorry that this disturbs you but I think that says more about your expectations than it does about any problem with set theory. I also see that you have spread this question across several different article talk pages; that's not what article talk pages are for, maybe you would like the math reference desk instead. --JBL (talk) 18:08, 22 September 2019 (UTC)[reply]

The answer depends very much on the set theory that you are using. In a set theory with classes, such as von Neumann–Bernays–Gödel set theory, the complement of the null set is the universal class, which is not a set. In Zermelo–Fraenkel set theory , there is no complement of the null set. In New Foundations, the complement of the null set is the universal set, which is a set. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:01, 15 February 2024 (UTC)[reply]

It's a joke, son!

Wow! Wikipedia has really outdone itself this time. What you call the “popular syllogism” of eternal happiness/ham sandwich is, in philosophical circles (and I know, I was a philosophy major at MIT of all places) considered a joke which is used to illustrate an invalid form of reasoning, that is, if A is B, and B is C, then A is C is invalid if the first B and the second B actually mean two different things. And that’s what’s happening in this case: The “nothing” that is better than eternal happiness means “of all things, none of them” and is itself not a thing but merely a figure of speech. The “nothing” that a ham sandwich is better than means an absence or lack of something, and that absence or lack is itself a thing. Oops! 74.104.189.176 (talk)

That's explained very clearly in the article. So your point is ...? --JBL (talk) 16:33, 16 May 2021 (UTC)[reply]

My point is this has nothing to do with the empty set, but rather the mistake of using an ambiguous word in a logical argument.74.104.189.176 (talk) 17:05, 16 May 2021 (UTC)[reply]

Alternate names

I was going to add void set to the Notations section, when I noticed that the last paragraph^[a] in the lead might better be moved to Notations, followed by the term void set. Thoughts? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:08, 1 April 2024 (UTC)[reply]

First thought: We should never cite MathWorld on terminology. The term "null set" is indeed sometimes used to mean "empty set", but we can find a better ref than that.

Second thought: Do you have attestations for "void set" in this sense? I imagine someone somewhere has used it but I don't think it's common. --Trovatore (talk) 19:43, 1 April 2024 (UTC)[reply]

I agree that the reference in the quoted text is bad, and we should replace it regardless of whether we move the text out of the lead.

Do you have suggested better sources for empty set, null set^[2] and void set?^[3]

Notes

^ In some textbooks and popularizations, the empty set is referred to as the "null set".^[1] However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty).

References

^ Weisstein, Eric W. "Empty Set". mathworld.wolfram.com. Retrieved 2020-08-11.
^ James Dugundji (1966). Topology (PDF). Allyn and Bacon Series in Advanced Mathematics. Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. Retrieved April 2, 2024.
^ John L. Kelley (1955). General Topology (PDF). University Series in Higher Mathematics. D. van Nostrand Company, Inc. Retrieved April 2, 2024.

[2] In some textbooks and popularizations, the empty set is referred to as the "null set".^[1] However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty).

[:1-1] Weisstein, Eric W. "Empty Set". mathworld.wolfram.com. Retrieved 2020-08-11.

[3] James Dugundji (1966). Topology (PDF). Allyn and Bacon Series in Advanced Mathematics. Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. Retrieved April 2, 2024.

[4] John L. Kelley (1955). General Topology (PDF). University Series in Higher Mathematics. D. van Nostrand Company, Inc. Retrieved April 2, 2024.

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