The Axiom of Separation (Axiom Schema) Axiomatic Set Theory

A description of the axiom schema of separation, an infinite list of axioms relating to the properties of a set.

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)

[okotray]

A little clear from this channel. Thanks

[movaxh]

It is to note that axiom of separation (or rather AXIOMS of separations) has infinite number of axioms. Not just because of its definitions of usage of dot dot dot to denote axioms with different number of classes at the star and number of "variables" for substituion in rho. But also because it also says (informally) "FOR ALL rho", aka for every possible logical formula rho. So even the first "axiom", with just one class, is already infinite number of axioms.

[defenderofwisdom]

How do you feel about this way of phrasing it?

The collection of elements x in A satisfying property P forms the set B. B equals the set of all x in A, such that P of x holds true.

[beenaalavudheen]

Will you talk about Vladimir Voevodsky?

[tyler-iyjk]

I get what it means conceptually but one thing that is frustrating is the notation/terms. Wikipedia uses different notation and so does Britannica(just says S(x) as a quantifier). So a class is a collection of sets, A is a class, so it's saying for any number of classes, and A is just all classes considered. Then it says all well formed formulas(not sure what that means) with a free x(don't really know what that means either). I also don't understand how it gets rid of Russell's paradox.