What is a Swelled Set? (Axiomatic Set Theory)

A explanation of a swelled class or set including examples and a formal definition.

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.

Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!

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)

[stapleman]

I was completely lost on the example until the G explanation, then it all makes sense. Set Theory is crazy.

[sinecurve]

But what is a Swole set?

Jokes aside enjoying the series so far. 😁

[levipoon]

The null set is (vacuously) a swelled set but it doesn't contain the null set as a member. At 1:18 you should have said that every non-empty swelled set contains the null set.

[movaxh]

So, the necessary condition (but not suffient) is to have a nullset as a member.

The non intuitive part is that not all sets have nullset as member, but all sets do have nullset as a subset. Including nullset, having no elements, but having a subset that is nullset.

So take away is, you can have subsets, even if there are no elements in a set. Aka, no matter what kind of set you have, you will have at least one subset.

[movaxh]

Is there some way to define swelled set check as a graph, and properties of the graph?

[manueldelrio]

So the Swelled set is like the Union Set of the Powersets of its elements?

[Sci]

[for all a who are subsets of b and b is a element of x implies that a is a element of x]
is the set of all sets that follow the thing in square brackets the same as the set of a swollen sets?

[ferdinandkraft]

1:50 by the axiom of Regularity (every non-empty set contains an element that is disjoint from itself), you cannot have circular definitions such as A = {B, C, D, 0} and C = {A}. To see why, consider J = {A, C} and apply the axiom to J. Since J has non-empty intersection with both A and C, you have a contradiction.

[MK]

I don’t get how A is in C, even though c is in A....my brain is breaking....please explain