Zermelo–Fraenkel set theory (abbreviated ZF) is a system of axioms used to describe set theory. When the axiom of choice is added to ZF, the system is called ZFC. It is the system of axioms used in set theory by most mathematicians today.
After Russell’s paradox was found in the 1901, mathematicians wanted to find a way to describe set theory that did not have contradictions. Ernst Zermelo proposed a theory of set theory in 1908. In 1922, Abraham Fraenkel proposed a new version based on Zermelo’s work. [1]
Axioms
An axiom is a statement which is accepted without question, and which has no proof. ZF contains eight axioms. [1]
- The axiom of extension says that two sets are equal if and only if they have the same elements. For example, the set {1,3} and the set {3,1} are equal.
- The axiom of foundation says that every set S (other than the empty set) contains an element that is disjoint (shares no members) with S.
- The axiom of specification says that given a set S, and a predicate F (a function that is either true or false), that a set exists that contains exactly those elements of S where F is true. For example, if S ={1,2,3,5,6}, and F is “this is an even number”, then the axiom says that the set {2,6} exists.
- The axiom of pairing says that given two sets, there is a set whose members are exactly the two given sets. So, given the two sets {0,3} and {2,5}, this axiom says that the set {{0,3},{2,5}} exists.
- The axiom of union says that for any set, there exists a set that consists of just the elements of the elements of that set. For example, given the set {{0,3},{2,5}}, this axiom says that the set {0,3,2,5} exists.
- The axiom of replacement says that for any set S and a function F, that the set consisting of the results of calling S on all the members of S exists. For example, if S = {1,2,3,5,6} and F is “add ten to this number”, then the axiom says that the set {11,12,13,15,16} exists.
- The axiom of infinity says that the set of all integers (as defined by the Von Neumann construction) exists. This is the set {0,1,2,3,4,…}
- The axiom of power set says that the power set (the set of all subsets) of any set exists. For example, the power set of {2,5} is {{},{2},{5},{2,5}}
Axiom of choice
The axiom of choice says that it is possible to take one object out of each of the elements of a set and make a new set. For example, given the set {{0,3},{2,5}}, the axiom of choice would show that a set such as {3,5} exists. For finite sets, this axiom can be proved from the other axioms, but not for infinite sets. [1]
References
[1] “Zermelo–Fraenkel set theory Facts for Kids | KidzSearch.com<“. 2023. wiki.kidzsearch.com. https://wiki.kidzsearch.com/wiki/Zermelo%E2%80%93Fraenkel_set_theory.
Additional Reading
“Chapter 13 The Axioms of Set Theory ZFC”. 2023. people.math.ethz.ch. https://people.math.ethz.ch/~halorenz/4students/LogikGT/Ch13.pdf.
Ferretti, Andrea, Carl Mummert and Timothy Chow. 2010. “Can We Prove Set Theory Is Consistent?”. MathOverflow. https://mathoverflow.net/questions/22635/can-we-prove-set-theory-is-consistent/24919#24919.
“Zermelo-Fraenkel Axioms — From Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html.
“Zermelo–Fraenkel Set Theory – Simple English Wikipedia, The Free Encyclopedia”. 2023. Wikipedia. https://simple.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory.
“Zermelo–Fraenkel Set Theory – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory.
“Zermelo–Fraenkel Set Theory Facts For Kids | KidzSearch”. 2023. wiki.kidzsearch.com. https://wiki.kidzsearch.com/wiki/Zermelo%E2%80%93Fraenkel_set_theory.
“ZFC | Brilliant Math & Science Wiki”. 2023. brilliant.org. https://brilliant.org/wiki/zfc/.
“ZFC – Encyclopedia of Mathematics”. 2023. encyclopediaofmath.org. https://encyclopediaofmath.org/wiki/ZFC.
Videos
“Axiomatic Set Theory”. 2023. youtube.com. https://www.youtube.com/@td904587/videos.
Very clear and understandable explanations that help with gaining an intuition in axiomatic set theory.
What if the foundation that all of mathematics is built upon isn’t as firm as we thought it was?
Note: The natural numbers sometimes include zero and sometimes don’t — it depends on how you define it. Within logic, zero is always included as a natural number.
Correction – The image shown at 8:15 is of Newton’s Principia and not Russell and Whitehead’s Principia Mathematica as was intended.
Does every set – or collection of numbers – have a size: a length or a width? In other words, is it possible for a set to be sizeless? This in an updated version of our September 8th video. We found an error in our previous video and corrected it within this version.
One of the Zermelo-Fraenkel axioms, called axiom of choice, is remarkably controversial. It links to linear algebra and several paradoxes- find out what is so strange about it!
Naïve set theory and the axiom of unrestricted comprehension have a massive flaw, which is that they allow Russell’s paradox; a serious logical inconsistency. Zermelo-Fraenkel set theory is a fresh definition of set theory that makes sure to not allow Russell’s paradox within it. This video covers 6 out of the 9 ZF axioms (axiom of extensionality, power set axiom, pairing axiom, union axiom, axiom of infinity and the axiom of subsets/restricted comprehension). The further 3 axioms are far more complex and are not covered and instead are left for subsequent videos. The video also briefly discusses the axiom of choice (the 10th axiom in ZFC) and compares and contrasts it to the axiom of restricted comprehension.
The featured image on this page is from the ZFC page of the Brilliant website.
Mathematical Mysteries 