I have the greatest admiration for the theorems and proofs of transfinite set theory, what we've called Cantor's transfinite set theory.

I taught it for years, wrote restatements for my students, and wrote a piece viewing it in the perspective of historical thinking about the infinite.
http://nrs.harvard.edu/urn-3:HUL.InstRepos:3715468

Now I've learned that #Cantor deliberately suppressed the role of #Dedekind in some of his work, particularly the proof that the set of real numbers is larger than the sets of natural and rational numbers. This was the first glimpse of the infinite hierarchy of infinite cardinalities.
https://www.quantamagazine.org/the-man-who-stole-infinity-20260225/

Cantor had the first germ of the proof, so props for that. But Dedekind helped him clarify it and Cantor published the clarified version without credit to Dedekind. I respect the math as much as ever but am now dealing with a serious case of flawed-hero syndrome.

First, make amends. Kudos to Dedekind. Second, give thanks. Kudos to the sleuths who turned up the empirical evidence of Cantor's #plagiarism -- Emmy Noether, Ivor Grattan-Guinness, José Ferreirós, and (decisively) Demian Goos.

Also thanks to Joseph Howlett for the Quanta article summarizing the evidence -- and in passing for calling Leopold #Kronecker an ideologue. Exactly!

#Infinity #Mathematics #SetTheory

Every well-ordered set is isomorphic to an ordinal. Common notion, for example see Introduction to https://arxiv.org/abs/2409.07352

⊢((𝐴∈V∧𝑅We𝐴)↔∃𝑓(dom𝑓∈On∧𝑓IsomE,𝑅(dom𝑓,𝐴)))

\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists f ( \mathrm{dom} f \in \mathrm{On} \wedge f \mathrm{Isom} \mathrm{E} , R ( \mathrm{dom} f , A ) ) ) \]

Every well-ordered set is isomorphic to
a unique ordinal.

⊢((𝐴∈V∧𝑅We𝐴)↔∃!𝑜∈On∃𝑓∈(𝐴↑ₘ𝑜)𝑓IsomE,𝑅(𝑜,𝐴))

\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists{!} o \in \mathrm{On} \exists f \in ( A \uparrow_\mathrm{m} o ) f \mathrm{Isom} \mathrm{E} , R ( o , A ) ) \]

We can phrase the Axiom of Choice as "Every set injects into an ordinal."

⊢(CHOICE↔∀𝑥∃𝑜∈On𝑥≼𝑜)

\[ \vdash ( \mathrm{CHOICE} \leftrightarrow \forall x \exists o \in \mathrm{On} x \preccurlyeq o ) \]

#math #metamath #SetTheory #WellOrdering #OrdinalNumbers

I am happy to share that I have been awarded a grant by the Finnish Cultural Foundation, for my postdoctoral research project "Interactions between set theory and non-standard logics"!

You can read about the grant in general here (my project isn't mentioned there, but it's an interesting read):

https://skr.fi/en/news/the-finnish-cultural-foundation-awards-e29-million-to-science-research-and-the-arts-a-major-grant-to-the-national-museum-of-finland/

#kulttuurirahastontuella #skr2026 #grant #academy #settheory #logic #mathematics #postdoc

Proposition: Russell's paradox is a grammatical artifact of set-theoretic language, not a fundamental limitation on self-reference.

Evidence:

Category theory permits self-morphisms (id_X : X → X) without paradox
Aczel's AFA permits self-containing sets by reinterpreting membership as graph structure
Process algebras (π-calculus) permit self-invoking processes without paradox

The key distinction:

"The barber who shaves all and only those who don't shave themselves" → Paradox
"The process that processes itself" → No paradox. That's just recursion.

The paradox isn't from self-reference. It's from the exclusion clause — "only those who don't." That's container logic: you're either IN or OUT.

Process logic has no exclusion clause. A function can call itself. A mirror can reflect a mirror. A wave can contain wave.

Conclusion: Self-reference is only paradoxical when forced through container grammar (discrete membership, exclusion). In process grammar (continuous relationship, inclusion), it just runs.

Pointers to related work welcome.

{🌊:🌊∈🌊}

#RussellsParadox #SetTheory #CategoryTheory #Logic #Mathematics #Philosophy #SelfReference

Thinking about non-well-founded set theory (Aczel 1988) and the Quine atom Ω = {Ω}.

Russell's paradox arises from self-reference in sets. But Aczel showed self-containing sets are consistent if you drop the Foundation Axiom.

Here's what I'm chewing on: Russell's paradox feels like a linguistic trap as much as a logical one. Sets are framed as containers (nouns). A container containing itself → paradox.

But Ω = {Ω} works because we're describing relationship, not containment. The set doesn't "hold" itself like a box — it refers to itself like a pattern.

Has anyone explored whether the noun/verb distinction (container vs. relationship) is doing hidden work in self-reference paradoxes?

Pointers to literature welcome.

#SetTheory #MathematicalLogic #SelfReference #FoundationsOfMathematics

A New Bridge Links the Strange Math of Infinity to Computer Science

https://fed.brid.gy/r/https://www.wired.com/story/a-new-bridge-links-the-strange-math-of-infinity-to-computer-science/

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