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Hanul Jeon

Hanul Jeon

@hanuljeon95.bsky.social

46 followers 42 following 48 posts

A doctoral student at Cornell. Interested in logic and set theory. he/him. Homepage: https://hanuljeon95.github.io

hanuljeon95.github.io

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Reposted by Hanul Jeon

Kivol @kivol.net · 6d

'아무튼 공황과 침체는 안된다고' 이러면서 유동성을 공급한 결과는 그 유동성으로 소시오패스를 세계 최고의 부자로 만들었고, 이 스노우볼이 굴러가서 수억 수십억명이 고통받고 수십 수백만명이 사망하는 세상을 만들었다. 비둘기들은 본인들이 벌인 일에 대한 자각이 있는지 궁금하다.

자본주의도 민주주의도 이제 황혼을 지나고 있다는 생각이 계속 든다.

Hanul Jeon @hanuljeon95.bsky.social · 12d

That means, when we ask "If ZFC + {n inaccessibles | n natural} and CIC + LEM + (some choice) prove the same statements," we should formulate it as sth like

If F is an interpretation from type-theoretic language to set theory, then whether "type theory proves P" iff "set theory proves F(P)."

Hanul Jeon @hanuljeon95.bsky.social · 12d

The debate on this question reminded me of what I thought before: Type theory can better force a meaning of a sentence through its syntax, so when we ask about the conservativity between set theory and type theory, we should ask about the conservativity of type-theoretic statements.

t.co/cNGFQBJLoO

https://mathoverflow.net/a/91055/48041

Hanul Jeon @hanuljeon95.bsky.social · 12d

Junk theorems in set theory are not a bug but a feature.

Hanul Jeon @hanuljeon95.bsky.social · 12d

Good googologists must be logicians (the converse does not hold)

Hanul Jeon @hanuljeon95.bsky.social · 12d

I think Girard's beta cut elimination theorem is essentially cut elimination via operator controlled derivations since we can think of a dilator as an operator taking an ordinal and returning the closure under operators the dilator encodes.

Hanul Jeon @hanuljeon95.bsky.social · 15d

The entrance door of logician's hell has a plaque with a single word "Π¹₃"

Hanul Jeon @hanuljeon95.bsky.social · 15d

Googologists should refrain from making and naming random large countable ordinals for no reason.

I am quite skeptical of how many googologists have even tried to learn Gentzen's ordinal analysis, where the story about recursive ordinal systems started.

Hanul Jeon @hanuljeon95.bsky.social · 18d

I learned that every non-standard model of IE1 contains an initial segment satisfying PA. It really sounds like a low-level version of Ville's lemma, which says every ill-founded ω-model of weak set theory (probably Ø-provident set theory) contains an initial segment satisfying KP.

Hanul Jeon @hanuljeon95.bsky.social · 21d

Team Cherry demands the hardcore user to really be a hardworking hardcore user. Alas, we are already too busy to be a hardcore user in math.

Hanul Jeon @hanuljeon95.bsky.social · 21d

Doing every achievement sounds really tough...

Hanul Jeon @hanuljeon95.bsky.social · 21d

Just for curiosity, to a month-old post: Did you finally free Phaloom?

Hanul Jeon @hanuljeon95.bsky.social · 22d

I hated infinitary types in the proof of projective determinacy when I learned it first time.

Now I think infinitary types are an unavoidable feature of Woodin cardinals.

Hanul Jeon @hanuljeon95.bsky.social · 22d

OTOH, Dorais' equiconsistency proof between ACA0+ and Provident set theory seems to give a bi-interpretation between these two theories in a way to preserve β-models. (A β-model of ACA0+ should be an ω-model of ACA0+ that is correct about recursible well-orders.)

Hanul Jeon @hanuljeon95.bsky.social · 22d

BTW I found my idea about connecting rudimentary set theory and ACA0 turned out to be wrong. I re-checked Taranovsky's claimed proof, and his idea only produces how to generate an ω-model of one from the other.

Hanul Jeon @hanuljeon95.bsky.social · 22d

I believe type theory also has some cumulative nature when we start to talk about large-cardinal-like axioms (like the level of universes). Set theory has better control on the hierarchy, so replacing set theory with type theory as a consistency-strength calibrator may be hard.

Hanul Jeon @hanuljeon95.bsky.social · 22d

One of the main features of set theory as a foundational theory is its cumulative nature. Set theory, in particular, combined with fine-structural machinery, allows fine control in each cumulative hierarchy that often has a role in calibrating the strength of theories.

Hanul Jeon @hanuljeon95.bsky.social · Sep 17

I believe it is somehow relevant to understanding the iterated Bachmann-Howard collapsing of a dilator.

Hanul Jeon @hanuljeon95.bsky.social · Sep 17

I am thinking about the possibility that the number of arguments of a predilator can be… functorially variable, so it takes a linear order A first, then takes an A-indexed family of linear orders, then returns a linear order.

Hanul Jeon @hanuljeon95.bsky.social · Sep 14

You should protect your time from Silksong first (by a person who lost 40+ hours to it).

Hanul Jeon @hanuljeon95.bsky.social · Apr 19

I think, we should apply the same POV to zero sharp. Adding zero sharp means, we add bunch of ordinals from the least height model of ZFC so that at some height of L-hierarchy, we have an L-ultrafilter, giving an iterable mouse.

Hanul Jeon @hanuljeon95.bsky.social · Apr 19

We can enhance this theory by adding the axiom "There are two cardinals." The most natural model-theoretic POV for adding the axiom is end-extending from H(ω₁) to H(ω₂).

We do not think something like, "Add the axiom, then the (fixed) largest cardinal now has an additional cardinal below."

Hanul Jeon @hanuljeon95.bsky.social · Apr 19

One misconception(?) about the zero sharp is that it "makes" ω₁ inaccessible in L. I think it is not a good POV.

To see why, let us go below: Lat us think of ZFC⁻ (= ZFC without powerset) plus "There is the largest cardinal."

Hanul Jeon @hanuljeon95.bsky.social · Mar 31

We also proved that the supremum of β-ranks of recursive theories is δ^1_2. We also calculated the β-rank of some theories, and for sufficiently strong theories, it coincides with the height of the least transitive model of the theory.

Hanul Jeon @hanuljeon95.bsky.social · Mar 31

In this paper, we defined an alternative way to rank the strength of theories via β-models called the β-rank, and established its basic properties.

We proved that the supremum of the β-rank of theories in the language of set theory is ω1, and the same for the language of SOA if we assume V=L.

arxiv math.LO @arxiv-math-lo.bsky.social · Mar 27

Hanul Jeon, Patrick Lutz, Fedor Pakhomov, James Walsh
Ranking theories via encoded $\beta$-models
https://arxiv.org/abs/2503.20470