# How can a class not be a set? [duplicate]

A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class.

So apparently, there are classes that are not sets. However, in the definition of class on the same page it says:

a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

If we take collection to be a synonym of set, then these two statements contradict eachother. I don't see the difference between "collection" and "set".

So my question is: How can a class not be a set, if it is defined to be a "collection" (i.e. set) of objects based on a well defined property?

## marked as duplicate by Eric Wofsey, Rohan, Asaf Karagila♦ elementary-set-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 26 '17 at 11:46

• At least in this context, it is very important that we don't take "collection" to be a synonym of "set". – Zev Chonoles Dec 26 '17 at 8:57
• Not everything you don't see isn't there. Collections aren't sets, neither in mathematics nor even in Java, period. Mathematics is about well-defined properties, not about arbitrary "synonyms". – Professor Vector Dec 26 '17 at 9:01
• @professorvector, well thank you for that very informative comment... but obviously my question then is, what is the difference between a set and a collection. – user56834 Dec 26 '17 at 9:07
• Every set is a collection, but not every collection is a set. Look at the definition, in mathematics (in set theory, the axioms define the properties of objects, usually). – Professor Vector Dec 26 '17 at 9:10
• In usual formalizations of the class/set difference, a set is a class that belongs to another class : so theres a difference here – Max Dec 26 '17 at 9:12

• " we may as well say a set is anything that is an element of something." So does this mean that the number 5 is a set, since it is an element of $\{5\}$ and of $\mathbb N$? That would change my intuition significantly as to what the word set means. So basically, not only is not every collection a set, it is not even the case that every set is a collection? – user56834 Dec 26 '17 at 9:30
• @Programmer2134 Each set is a collection (we say class), though if your theory has urelements you have to say classes are of sets and/or urelements. For the other half of your question, the usual approach is to agree a definition according to which $5$ is a set, e.g. the non-negative integers are the elements of $\omega$ (and you can Google popular constructions for integers, rationals, reals, complex numbers and so on). Then our theory doesn't need urelements. As I mentioned, if you do have urelements then a set is a non-urelement that belongs to a class. – J.G. Dec 26 '17 at 9:57