Happy birthday of set theory, for all those who celebrate!

On December 7, 1873, Georg Cantor (https://en.wikipedia.org/wiki/Georg_Cantor) wrote a letter (https://www.aleph1.info/?call=Puc&permalink=cd1_Briefe_Z4) to Richard Dedekind in which he showed that there are more real numbers than integers and that therefore different kinds of infinity exist. Cantor's proof at this time is not the “diagonalisation” proof that is now usually given.

December 7, 1873 is also the 50th birthday of Leopold Kronecker (https://en.wikipedia.org/wiki/Leopold_Kronecker), which is ironic, given the heavy conflicts they would have about set theory.

The birthday of set theory is usually celebrated with a birthday cake that has ℵ₀ candles on it, but you can take fewer if you don't have the space for them. 🕯️

#SetTheory #Mathematics #HistoryOfMathematics #HistoryOfScience #GeorgCantor #LeopoldKronecker

Any #mathematics professors want to take a shot at explaining to someone with say a typical math major undergrad background what the deal is with the recent introduction of exacting and ultraexacting cardinals?

#math #maths #settheory #zfc

https://arxiv.org/abs/2411.11568

Readings shared November 24, 2025. https://jaalonso.github.io/vestigium/posts/2025/11/25-readings_shared_11-24-25 #CompSci #CoqProver #ITP #LeanProver #Logic #Math #Physics #SetTheory #TypeTheory

Set theory with types. ~ Lawrence Paulson. https://lawrencecpaulson.github.io//2025/11/21/Typed_Set_Theory.html #Logic #Math #CompSci #SetTheory #TypeTheory

"Like parsley in Greek food: Elementary set theory and the case for DM1" by Siddharth Bhaskar (https://dblp.org/pid/170/0077.html) got accepted at SIGCSE TS 2026.

https://sigcse2026.sigcse.org/details/sigcse-ts-2026-Papers/164/Like-parsley-in-Greek-food-Elementary-set-theory-and-the-case-for-DM1

#SIGCSE #setTheory #CSEducation #math #discreteMath #computerScience

Readings shared October 23, 2025. https://jaalonso.github.io/vestigium/posts/2025/10/24-readings_shared_10-23-25 #AI #Agda #ITP #LLM #LeanProver #Math #SetTheory

And here's a "general audience" (or "popular math") description of the general theme in this paper, and much of my work in general. Don't hesitate to ask for more details!

In this paper I explore the boundaries of what is mathematically possible when constructing universes of set theory. A "universe of set theory" can be thought of as a model for all of mathematics, and what we, set theorists, like to do, is build different kinds of such universes.
We have two main tools:
1. Take a universe, and add stuff, to create a larger one;
2. Take a universe, and define inside of it a smaller one.
In my paper I use both methods: I extend a given universe and build a larger one, in which I use a specific definition to construct a smaller universe.
The cool thing is, that I can use that definition again - to get an even smaller universe. And again, and again - infinitely many times!

#math #SetTheory #multiverse #PopularMath #PopularScience

My paper has now been officially published!

https://www.worldscientific.com/doi/10.1142/S0219061325500229

#math #paper #publication #Logic #SetTheory

Constructive ordinal exponentiation. ~ Tom de Jong, Nicolai Kraus, Fredrik Nordvall Forsberg, Chuangjie Xu. https://arxiv.org/abs/2501.14542 #ITP #Agda #SetTheory

Readings shared October 14, 2025. https://jaalonso.github.io/vestigium/posts/2025/10/15-readings_shared_10-14-25 #FunctionalProgramming #Haskell #ITP #IsabelleHOL #LeanProver #Logic #Math #Python #RustLang #SetTheory

ZF style set theory in Knuckledragger I. ~ Philip Zucker. https://www.philipzucker.com/zf_knuckle1/ #Python #Logic #SetTheory

A gentle introduction to the axiom of choice. ~ Andreas Blass, Dhruv Kulshreshtha. https://arxiv.org/abs/2509.01830 #Math #SetTheory

Set theory as a unified framework

" #settheory, the ranges of binders are the sets. Thus, beyond its simplifying advantage of removing types, set theory will get more power by its strengthening axioms which amount to accept more classes as sets."
https://settheory.net/foundations/unif#:~:text=set%20theory%2C%20the%20ranges%20of%20binders%20are%20the%20sets.%20Thus%2C%20beyond%20its%20simplifying%20advantage%20of%20removing%20types%2C%20set%20theory%20will%20get%20more%20power%20by%20its%20strengthening%20axioms%20which%20amount%20to%20accept%20more%20classes%20as%20sets.

@LeoTsai14 While they do not actually call it so, set theoreticians do a lot of work in a category in which the objects are the models of set theory and the arrows are the elementary embeddings (https://en.wikipedia.org/wiki/Elementary_equivalence#Elementary_embeddings) between them.
Models of (ZFC-like) set theories have the interesting property that the maps between them are to some amount determined by the mappings between their classes of ordinals: If this map is an isomorphism, the whole map is one (https://en.wikipedia.org/wiki/Critical_point_(set_theory)).
You may also have a look at inner model theory (https://en.wikipedia.org/wiki/Inner_model_theory), I think.

#SetTheory #ModelTheory #MathematicalLogic #Categories

What is the difference between intensional and extensional logic? - Philosophy Stack Exchange
"The predicates "having a kidney" and "having a heart" clearly have different senses, but as a matter of contingent fact everything that has a kidney has a heart: hence we case the two terms are co-referential. The set of the renates = the set of the cordates, because set membership is extensional. First order logics, including set theory are all extensional in this sense. It doesn't matter whether we are talking about the set of the renates or the set of the cordates, because they are the same set, and we know they're the same set, because we define a set extensionally--i.e. purely by the reference of the terms. We don't need to know what the terms "renate" and "cordate" really mean, we just need to know which things they refer to."
#settheory
/frege https://philosophy.stackexchange.com/questions/16164/what-is-the-difference-between-intensional-and-extensional-logic#:~:text=The%20predicates%20%22having,they%20refer%20to.

Yet another academic paper that dives into the rabbit hole of Zermelo’s "Axiom of Choice" 🐇🎩, as if it was the Da Vinci Code of set theory. 😴🔍 100 years and counting, and we're still trying to decipher what the heck the problem was—spoiler alert, it's probably just a bunch of mathematicians arguing over who gets the last slice of infinity pie. 🍰♾️
https://research.mietek.io/mi.MartinLof2006.html #AxiomOfChoice #SetTheory #InfinityPie #MathHumor #AcademicPapers #HackerNews #ngated

A brain dump from Matrix Dreams: A better-defined series of sets

I’ve been pondering the weird collection of sets I cooked up in this post. In this post, I’d like to take a crack at defining the sets.

I’m going to bow to my ego and call these the MattSets. MattSet Prime is the set of all integers found in all the other MattSets.

MattSet0 is the set of all powers of two. As such, it is an even set. All members of MattSet0 are even.

There are two types of MattSets – even and not even (odd) sets. Each set is generated from the set before it.

MattSet1 is the first odd set. It is the set of all the numbers when multiplied by three that the numbers in the previous set is one more than. Another way to say this is – make an intermediary set of all members of MattSet0 (MattSetn-1 in general form) that are one more than a multiple of three. Take all the members of this intermediary set and apply the function (x-1)/3. This will give you MattSet1 which is an odd set.

MattSet2 is two be an even set. MattSet2 is the set of all integers that are twice a member of MattSet1 (MattSetn-1 in general form).

Each subsequent set is generated in the same way such that MattSets with even indexes are even numbers and MattSets with odd indexes are odd.

As far as I have, so far, tested each odd MattSet has half the number of members as the preceding even MattSet.

In theory: As n approaches infinity, the size of the MattSetn approaches 0. At least I think that it does.

The question I am looking to answer is does MattSet Prime contain all the positive integers? If I can’t prove that one way or the other, is there anything interesting about these sets that can be proved?

A better-defined series of sets, 8th October 2024, by Matt

Note: I’m not sure I was all that clear the first time, but only integer values can be members of these sets.

You could add some other series of (empty) sets for completion. MattSetnx. A member of the MattSetnx is an even number not found in MattSet0 that is twice the value of a MattSet0 member. MattSetn+1x. Would be all the even numbers not in the preceding sets that are twice the value of one of the previous nx sets. In theory, all MattSetnx sets are empty, as twice a power of two is another power of two.

Prove me wrong.

#maths #MatrixDreams #MattSet #RSVP #setTheory #syndicated

Hey #SetTheory #lazyweb, if I have a set A and its proper subset B, what is the name that describes the elements in A that are not in B?

I _think_ it's 'symmetric difference', but that to me implies that B could be something other than a proper subset. Is there a more exact name?

0 can only be
- AOC dabbler
#settheory #logic