What is the Null Set? (Empty Set, Void set, ∅, {}) Axiomatic Set Theory

A description of the Null set (class) (also known as the Empty set, or the Void Set) and the symbols for it such as ∅ or {}.

This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.

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Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)

[AltisiaK]

Can you please explain how the seperation axiom and the extentionality axiom necessarily assert that there is only one null class/set? Could you not have internal classes/sets of non-equal size such as 0={} and 0^{n} = {} where n is a member of the real number set such that n > 0? Does the many non-identical n's get ignored because it eventually works out that the set is equivalent to 0 and therefore the empty set?

[cliffordhodge]

To demonstrate the existence of the null class, ending at 2:30, instead of appealing to something as odd as non-self-identity, could you not simply say, "There exists a class B such that for any object x (be it a set or otherwise),   x is not a member of B." ?   Also, regarding the fact only one null set exists, (ending at 3.30) it seems odd, and wrong to assert that the sets have the same members, since the noteworthy feature the null set has is the lack of members.  I suppose nothingness is self-identical too, but it seems it would be more natural to have an axiom for the identity of indiscernibles and rely on that to show there can only be one null set.  I know you want to be able to say two references referring (as it turns out) to the same set - the null set - with different names or defining descriptions ("the set of all animals with kidneys but no heart" or whatever) are actually the same set because they have the same members.  Still, this seems unnatural, and all you really need is to show that each is actually a memberless set, hence indiscernible from one another, hence the (one and only) null set.