Dedekind quickly replied that neither could he — but that he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted.
Richard Dedekind had a proof that the algebraic numbers are countable.
Thus if P isthe set of all sets, we can apparently form the set Q = {Ae P| A ¢ A}, leading tothe contradictory Oe Q iff O¢€ Q. This is Russell’s paradox (see Exercise 1A)and can be avoided (in our naive discussion) by agreeing that no aggregate shallbe a set which would be an element of itself.
Russell's paradox (1901) in set theory can be stated as:
If $$P$$ is the set of all sets, one can form the set $$Q = {A \in P | A \notin A}$$ which can lead to the contradiction $$Q \in Q$$ iff $$Q \notin Q$$.
This can be done by dividing P into two non-empty subsets, $$P_1 = {A \in p | A \notin A}$$ and $$P_2={A \in P | A \in A}$$. We then have the contradiction $$P_1 \in P_1$$ iff $$P_1 \notin P_1$$.
The paradox happens when we allow as sets A for which $$A \in A$$. It can be remedied by agreeing that no collection can be a set which would be an element of itself.
Relation to Groucho Marx's quote (earliest 1949) about resigning membership of a club which would have him as a member: https://hypothes.is/a/3_zAfITjEe-H5-PlfOlK8A
Cohen (Independence of the Axiomof Choice; The Independence of the Continuum Hypothesis I, I1) completed theproof of independence for each by showing neither could be deduced from theexisting axioms (by showing the negation of each could consistently be added tothe Zermelo—Fraenkel axiom scheme). See P. J. Cohen (Set Theory and theContinuum Hypothesis) for a discussion of these results and his intuition about thecontinuum hypothesis. Another expository reference is Cohen (IndependenceResults in Set Theory).
In 1963 Paul Cohen completed the work of Gödel by proving the independence of the axiom of choice and the continuum hypothesis from the Zermelo-Fraenkel set theory axioms. He did this by showing that neither could be deduced from the existing axioms and specifically by showing that the negation of each could be added to ZF consistently.
Godel (The Consistency of the Axiom of Choice and of the Generalized Con-tinuum Hypothesis with the Axioms of Set Theory) proved in 1940 that additionof either the axiom of choice or the continuum hypothesis to existing set theoreticaxioms would not produce a contradiction.
Gödel's breakthrough in 1940 was to prove that one could extend the axioms of set theory to include the axiom of choice or the continuum hypothesis without introducing contradictions.
The basis for our intuitive set theory is the Zermelo—Fraenkel set theory developedby Zermelo (Untersuchungen tiber die Grundlagen der Mengenlehre J) andstrengthened by Fraenkel (Zu den Grundlagen der Cantor—Zermeloschen Mengen-lehre). Their work rests on the researches of Cantor in the 1870’s which first putmathematics firmly on a set-theoretic base. Zermelo’s work, in particular, was adirect response to the Russell paradox.
Mary E. Rudin: "Set theory and General Topology" by [[UM-Milwaukee Department of Mathematical Sciences]]
We can use the itertools.combinations function to find all possible subsets of a chord for a given cardinality.
Ha! Found a Ruby method to do the same thing in Sonic Pi. https://in-thread.sonic-pi.net/t/exploring-modes-of-pitch-class-sets-using-chord-invert/5874/10?u=enkerli
Glad this is explicitly mentioned here as it was my initial goal as I got into musical applications of Set Theory!
An Euler diagram (/ˈɔɪlər/, OY-lər) is a diagrammatic means of representing sets and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramming technique, Venn diagrams. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships.
Kuratowski definition of an ordered pair
Willard Van Orman Quine insisted on classical, first-order logic as the true logic, saying higher-order logic was "set theory in disguise".