Axioms of set Theory - Lec 02 - Frederic Schuller

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Recommends AlvinL · 2019-01 · Mathematics Stack Exchange · on “Where does $\in$ come from and where is it defined?”

I did catch a lecture of F. Schuller [this video] where he calls it a 'fundamental relation' and he emphasises there is no definition of epsilon. At any rate, the series is very interesting to watch.

Cites The curious amateur · 2018-01 · Mathematics Stack Exchange · on “Proving the schema of separation from replacement”

I have been looking at how separation can be deduced from replacement. I have found already this question: Proving Separation from Replacement [LINK] It was very enlightening, but it caused some thoughts to arise. First of all, I have noticed that people have different versions of the axiom of replacement. Omitting for brevity the requirement on $P$ to be a functional predicate, the version I use is the following: $\forall A \, \exists B \, \forall C, (C \in B \leftrightarrow \exists D \,(D \in A \wedge P(D,C)))$ I noticed other people had a version with $\exists! B$. The thing that puzzles me is that if you take "my" version of the axiom and instantiate B with $X$ and then with $Y$, it is v

Cites WillG · 2020-01 · Mathematics Stack Exchange · on “Is the "Axiom on $\in$-Relation" equivalent to the Axiom of Extensionality?”

In Frederic Schuller's lecture series [LINK] Lectures on Geometrical Anatomy of Theoretical Physics , he declares ( here [this video] ) the first axiom of ZFC set theory to be the Axiom on $\in$ -Relation , stated as follows: Informal: " $x\in y$ " is a proposition if and only if $x$ and $y$ are both sets. Formal: $\forall x:\forall y:(x\in y)\veebar \neg(x\in y)$ . The "formal" version above (where $\veebar$ means "exclusive or") is not given in the lecture, but is stated in Simon Rea's transcription of the lectures, found here [LINK] (p. 8). Schuller does not include the traditional Axiom of Extensionality [LINK] that I have seen in every other ZFC book, which states that two sets are equal if

Cites Quantumwhisp · 2020-08 · Mathematics Stack Exchange · on “Are there elements of sets that are no sets in Zermelo Fraenkel Set theory?”

I've seen the ZFC formalised in a lecture where the lecturer introduced, part by part, propositional logic, 1. order logic, and then zermelo-fraenkel-set theory. The lecturer didn't introduce any notion of identity ("=") in the part about 1. order logic, and defined identity in the part about set theory. There, two sets where defined equal when from something being an element of the first set followed that it must be an element of the 2nd set as well, and vice versa (the definition captures what I read would be the "axiom of extensionality", later on). However, in the further proceeding of the lecturer (here: [this video] ), the lecturer used the "=" sign, and the notion of identity, not only for

Cites Clemens Bartholdy · 2021-05 · Mathematics Stack Exchange · on “In ZFC, why was it chosen to not define what exactly a set or $\epsilon$ relation is?”

From about the first twenty minutes of this lecture [this video] by Frederic Schuller, what I understood was that the philosophy of ZFC was that we define the idea of a set and $\epsilon$ relation by their properties rather than stating what they are literally. However, why exactly do we not define these objects? What exactly is the difficulty to do this?

Cites Suraaj K S · 2021-10 · Mathematics Stack Exchange · on “A question about Axiomatic Set Theory: Axiom on $\in$ relation”

I am exploring Axiomatic set theory out of curiosity, and was referring to [this video] as an introduction to the subject. My question is here (time in video 10:47) :- The first axiom of the theory is stated as this: Axiom on $\in$ -relation: $x \ \in\ y$ is a proposition if and only if x and y are sets. This does not seem like an axiom that is in an axiomatic system . An axiom in an axiomatic system is a formula that is not a tautology in that logic system, but is assumed to be a universal truth (Please correct me if I am incorrect). To me, this "axiom" seems to just be saying this - $\in$ is a (binary) predicate. The variables on which the truth value of $\in$ depends are called sets. If I reme

Cites Clemens Bartholdy · 2022-02 · Mathematics Stack Exchange · on “How to find a 'set' after knowing the axioms of ZFC?”

I don't believe this answer. Take this lecture [this video] from frederic schuller , he clearly remarks there is no clear definition on what a set or ethe epsilon relation is but rather we only know what it is through it's behavior and axiom it follows