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I think this question is not asked here. I apologize in advance if I am wrong.

I have the following two definitions (Joaquín Olivert, Estructuras de álgebra multilineal, 1996):

Class.- A class is an abstract object $C$ which permits to decide if their elements belong to it or not.

Set.- A class $C$ is said to be a set if there exists a class $D$ such that $C\in D$. A class which is not a set is called a proper class.

With this in mind, I'm trying to understand if $A=\{\{\emptyset\}\}$ is a set or not and why. It seems to me the answer should be not but with these definitions, I think $\{ A \}$ is a class and then $A$ is a set. On the other hand, with this reasoning, every class would be a set and that is false (e.g. Russell's set).

Any help please?

Thanks

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  • $\begingroup$ Well, you have the abstract object $\;\{\emptyset\}\;$ which determines that only objects of the kind "set containing only the empty set" belong to it...so yes, $\;\{\{\emptyset\}\}\;$ is a set. $\endgroup$ – DonAntonio Jan 9 '18 at 22:08
  • $\begingroup$ Ugh... given this 'definition' alone you really can't answer this question. My best guess is that this is meant as an informal view of sets -- not an effort to characterise them. $\endgroup$ – Stefan Mesken Jan 9 '18 at 22:08
  • $\begingroup$ What has been asked here already is this question. $\endgroup$ – Dietrich Burde Jan 9 '18 at 22:10
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    $\begingroup$ @StefanMesken It has been asked, as you can see. I did not say that it was much related. $\endgroup$ – Dietrich Burde Jan 9 '18 at 22:12
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    $\begingroup$ NBG is likely what you want. (It's basically ZFC -- the 'standard' formalization of set theory -- expanded to deal with classes). $\endgroup$ – Stefan Mesken Jan 9 '18 at 22:37
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In ZF(C) set theory, the Axiom of Pairing states that if $x$ and $y$ are sets, then $\{x,y\}$ is a set too. Applying this with $x=y$ gives that whenever $x$ is a set, $\{x\}$ is a set too (because it has the same elements as $\{x,x\}$ and is therefore equal to it).

Thus,

  • $\varnothing$ is a set because of the Axiom of the Empty Set. (Sometimes this is not considered an axiom, in which case $\varnothing$ is still a set because the Axiom of Infinity states that there exists some set, and then the Axiom of Separation can be used to remove all of its elements).
  • $\{\varnothing\}$ is a set because of Pairing applied to $\varnothing$.
  • $\{\{\varnothing\}\}$ is a set because of Pairing applied to $\{\varnothing\}$.

Your definition of "set" sounds like it doesn't belong to pure ZF(C), but more likely to NBG or MK set theory. However, these theories have similar axioms of Empty Set and Pairing, so the argument (at least at the non-formal level I'm writing at) is the same.

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  • $\begingroup$ Really interesting your final remark about my definitions. Thanks a lot. $\endgroup$ – Dog_69 Jan 9 '18 at 22:37
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$A=\{\{\varnothing\}\}$ is a set, not a proper class, because we have $A=\{\{\varnothing\}\}\in\{A\}=\{\{\{\varnothing\}\}\}.$ And $\{A\}$ is not a class either, because we have $\{A\}\in\{\{A\}\}.$

Axiomatically, one usually assumes that $\varnothing$ is a set, and for any set, the singleton containing that set is a sett (for example via the asxiom of pairing), so $\{\varnothing\},$ $\{\{\varnothing\}\}$, etc are all sets, and not proper classes.

In the usual universes of axiomatic set theory, for something to be a proper class, and not a set, it must be very big. For example the class of all sets, the class of all cardinals, or the class of all ordinals. These classes are not contained in any other classes.

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  • $\begingroup$ Wait... who says that, in this framework, $\{\{\{ \emptyset \}\}\}$ is a class? $\endgroup$ – Stefan Mesken Jan 9 '18 at 22:07
  • $\begingroup$ "one usually assumes" [emphasize is mine] My point exactly. There isn't sufficient information to answer OP's question. Who are we to judge what OP or the author considers to be a set/class if he doesn't tell us? Is $\{\{\{\emptyset\}\}\}\}$ an abstract object that allows us to decide if their members belong to it? I don't know... sounds like utter nonsense to me. $\endgroup$ – Stefan Mesken Jan 9 '18 at 22:13
  • $\begingroup$ @StefanMesken If you need to make some assumptions in order to make the question answerable, I think it is allowable to do so, as long as you state your assumptions. Hopefully my answer has done that. If not, please feel free to let me know how it may be improved. $\endgroup$ – ziggurism Jan 9 '18 at 22:18
  • $\begingroup$ @StefanMesken I don't' know if the following helps you but the author says: ''Esentially, the axiomatic we are going to follow is ZF together with notable contributions of Neumann-Bernays-Gödel (...) and some details given by J. L. Kelly (...).". $\endgroup$ – Dog_69 Jan 9 '18 at 22:23
  • $\begingroup$ It's not really clear what you have assumed. For example you say "For something to be a proper class, and not a set, it must be very big." That's not true at all. Classes are 'very big' in most common set theories but they certainly don't need to be. Singletons can be classes in the right framework. $\endgroup$ – Stefan Mesken Jan 9 '18 at 22:23

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