Talk:Singleton (mathematics)

Clarification please: Definition by indicator functions

Regarding section "Definition by indicator functions":

First off, if I'm reading this correctly, this section pertains to singleton *classes*, not the singleton sets which were described in the article intro. Since a class and a set are not the same thing, might this call for a clarifying remark?

Next, the actual narrative "Let S be a class [...] for some y in X" is inordinately obtuse without any explanation. I *think* this specifies S = {y}, and if so it would greatly help to say so, and tell why one would want to use such a verbose way of saying so. But I'm not entirely sure, because of the "for some y in X" phrase. Since for each x, "some y" could be set to x, that could be interpreted to mean that x=y for all x, and hence S = X.

Finally, the definition of natural number 1. Is there an expectation that the reader will understand this, without any definition of the constituent variables? What is its relevance to this article? Gwideman (talk) 11:05, 27 August 2012 (UTC)

I've separated the PM definitions to a separate section since there appears to be no connection between the treatment in PM and indicator functions. I've also added some clarification. I agree with foregoing comments in respect of what remains. The last line seems to require one of 0 or 1 to be a proposition. It may assume an assignment of 0 or 1 to propositions according to the truth value as in some programming languages, but if so it should specify.Martin Rattigan (talk) 05:17, 19 April 2018 (UTC)

Distinction from the contained element

What happens if the contained element is the same as the singleton? Does this break the axiom of regularity? 178.138.35.124 (talk) 23:58, 12 September 2023 (UTC)

Terminology

I think the redundant phrasing "singleton set" is more established than "singleton" alone, but I'm not sure. It could be worth reviewing some textbooks to check this. —Kodiologist (t) 15:02, 29 November 2023 (UTC)

Axiomatic definition in set/class theories

@Georgydunaev, wouldn't it be more appropriate to integrate this subsection in the "Properties" subsection? Currently the revision comes across as a bit idiosyncratic; lacking context, wiki hyperlinks and references. Roffaduft (talk) 06:38, 28 May 2024 (UTC)

the idea is that it is the specificaton.Better to keep, since it is valid. Georgydunaev (talk) 11:41, 28 May 2024 (UTC)

It being valid is not sufficiënt. It lacks context and references. In its current state it should be removed from the article. Roffaduft (talk) 13:25, 28 May 2024 (UTC)

notational flaw

Article says: ”...a singleton is necessarily distinct from the element it contains, thus 1 and {1} are not the same thing...”

This is a puzzling assertion, since 1 is itself a set, just as the integers or the reals constitute a set, regardless of whether or not one employs the curly bracket notation. Thus, when you refer to 1, you are referring to both the number and the set, since they are the same thing. This seems to be a logical flaw in the notational use of curly brackets, which ought not undermine the fundamental concept of a set, although it clearly does. 96.237.184.133 (talk) 13:52, 21 October 2024 (UTC)

This is not correct. 1 and {1} are different things. Maybe someone else will explain it in terms of formal set theory. See Russell's paradox for why we are suspicious of sets that contain themselves. McKay (talk) 04:56, 24 October 2024 (UTC)

Are you saying 1 is not a set? How so? And the trouble is that 1 is a set that cannot be expressed with curly brackets, since as you say that results in a set that is a member of itself. 96.237.184.133 (talk) 15:54, 25 October 2024 (UTC)