arXiv subjects

Nam Trang

Publications and source records attributed to Nam Trang.

18 recordsLinked to original sources

arXiv2026

Free Left Distributive Algebras and a Canonical Extension

Assuming a large cardinal hypothesis, Laver gave a representation of the monogenerated free left distributive algebra (LDA) using elementary embeddings and used this representation to prove many algebraic results. Some of these results were later proved by Dehornoy in ZFC, without the large cardinal hypotheses. However, there is an important algebraic result whose consistency strength is unknown. (See Laver (1995) and Dougherty & Jech (1997).) Recent results [arXiv:2508.02244] extend the connection between elementary embeddings of set theory and free LDAs to the many-generated case. Assuming large cardinals, we prove two results. First, we prove that finitely-generated free LDAs with distinct numbers of generators are $\Sigma_1$-elementarily equivalent but not $\Sigma_2$-elementarily equivalent. We also prove a partial structural analogue to Laver's representation of LDAs. We construct an extension of the monogenerated free LDA where application by any fixed element is an elementary embedding of LDAs. We argue that this extension is canonical by demonstrating homogeneity and universality properties. These results also provide additional examples of algebraic properties provable from large cardinals without known proofs from the standard axioms of set theory.

math.LO↗

arXiv2026

More Derived Models in PFA

This paper makes significant progress towards resolving a conjecture relating strong forcing axioms like $PFA$ and the derived model at a limit of Woodin cardinals $\kappa$. In particular, using a concept called Covering Matrices, we show that the $\Theta$ of the derived model at $\kappa$ is strictly less than $\kappa^+$ under various circumstances; in particular, this shows that the conclusion holds under $PFA$ if $\kappa$ is a limit of Woodin cardinals of cofinality $\omega$ and the derived model does not satisfy $LSA$. Assuming a form of mouse capturing, we show that the derived model satisfies $AD_{\mathbb{R}}$ under $PFA$ when $\kappa$ is a regular limit of Woodin cardinals. If $\kappa$ is an indestructibly $(\kappa,\kappa^+)$-weakly compact limit of Woodin cardinals, then the derived model outright satisfies $AD_{\mathbb{R}}$.

math.LO↗

arXiv2025

Partial Tower Sealing

The main result of this paper shows that a weak form of Tower Sealing holds in a generic extension of hod mice with a strong cardinal and a proper class of Woodin cardinals. We show Tower Sealing fails in such extensions in general. We show that this weak form of Tower Sealing (called Partial Tower Sealing) implies Sealing and that its consistency strength is below that of ZFC + there is a Woodin limit of Woodin cardinals.

math.LO↗

arXiv2025

Derived Models in PFA

We discuss a conjecture of Wilson that under the proper forcing axiom, $\Theta_0$ of the derived model at $\kappa$ is below $\kappa^+$. We prove the conjecture holds for the old derived model. Assuming mouse capturing in the new derived model, the conjecture holds there as well. We also show $\Theta < \kappa^+$ in the case of the old derived model, and under additional hypotheses for the new derived model.

math.LO↗

arXiv2023

Preservation of AD via forcings

We show that assuming $\mathsf{ZF}+\mathsf{AD}^+ +$ "$V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)$", any poset which increases $\Theta$ does not preserve the truth of $\mathsf{AD}$. We also show that in $\mathsf{ZF} + \mathsf{AD}$, any non-trivial poset on $\mathbb{R}$ does not preserve the truth of $\mathsf{AD}$. This answers the question of Chan and Jackson. Furthermore, we show that under the assumptions $\mathsf{ZF}+\mathsf{AD}^+ +$ "$V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)$" + "$\Theta \text{ is regular}$", there is a poset on $\Theta$ which adds a new subset of $\Theta$ while preserving the truth of $\mathsf{AD}$. This answers the question of Cunningham.

math.LO↗

arXiv2022

Ideals and Strong Axioms of Determinacy

We show that the following two theories are equiconsistent: (T) ZFC, CH and "There is a dense ideal on the first uncountable cardinal such that if j is the generic embedding associated with it then its restriction on ordinals is independent of the generic object is". (S) ZF, ADR and "Theta is a regular cardinal." The main result of this paper is that T implies that the minimal model of S exists. Woodin, in unpublished work, showed that the consistency of S implies the consistency of T. We will also give a proof of this result, which, together with our main theorem, establishes the equiconsistency of T and S. Our main result partially resolves a well-known conjecture of Woodin, and completely solves one of the main Core Model Induction problems dating back to 90s.

math.LO↗

arXiv2022

Condensation for Mouse Pairs

In this paper, we prove a fine condensation theorem. This is quite similar to condensation theorems for pure extender mice in the literature, except that condensation for iteration strategies has been added to the mix.

math.LO↗

arXiv2021

The Largest Suslin Axiom

We develop the basic fine structure theory of the minimal model of the Largest Suslin Axiom. In particular, we prove that that the minimal model of the Largest Suslin Axiom satisfies the Mouse Set Conjecture, and that the Proper Forcing Axiom implies the minimal model of the Largest Suslin Axiom exists.

math.LO↗

arXiv2021

Sealing from Iterability

We obtain sealing by forcing over a self-iterable model. The proof is fine-structure free and uses only basic ideas from iteration theory. We believe that such fine-structure free proofs will make the subject more accessible to the general set theoretic community.

math.LO↗

arXiv2021

The exact strength of generic absoluteness for the universally Baire sets

A set of reals is \textit{universally Baire} if all of its continuous preimages in topological spaces have the Baire property. $\sf{Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The $\sf{Largest\ Suslin\ Axiom}$ ($\sf{LSA}$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let $\sf{LSA-over-uB}$ be the statement that in all (set) generic extensions there is a model of $\sf{LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory, $\sf{Sealing}$ is equiconsistent with $\sf{LSA-over-uB}$. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see \rdef{dfn:hod_pm}). As a consequence, we obtain that $\sf{Sealing}$ is weaker than the theory $``\sf{ZFC} + $there is a Woodin cardinal which is a limit of Woodin cardinals". A variation of $\sf{Sealing}$, called $\sf{Tower \ Sealing}$, is also shown to be equiconsistent with $\sf{Sealing}$ over the same large cardinal theory. The result is proven via Woodin's $\sf{Core\ Model\ Induction}$ technique, and is essentially the ultimate equiconsistency that can be proven via the current interpretation of $\sf{CMI}$ as explained in the paper.

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arXiv2019

On supercompactness of $ω_1$

This paper studies structural consequences of supercompactness of $ω_1$ under $\sf{ZF}$. We show that the Axiom of Dependent Choice $(\sf{DC})$ follows from "$ω_1$ is supercompact". "$ω_1$ is supercompact" also implies that $\sf{AD}^+$, a strengthening of the Axiom of Determinacy $(\sf{AD})$, is equivalent to $\sf{AD}_\mathbb{R}$. It is shown that "$ω_1$ is supercompact" does not imply $\sf{AD}$. The most one can hope for is Suslin co-Suslin determinacy. We show that this follows from "$ω_1$ is supercompact" and Hod Pair Capturing $(\sf{HPC})$, an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. "$ω_1$ is supercompact" on its own implies that every Suslin co-Suslin set is the projection of a determined (in fact, homogenously Suslin) set. "$ω_1$ is supercompact" also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.

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arXiv2017

On a class of maximality principles

We study various classes of maximality principles, $\rm{MP}(κ,Γ)$, introduced by J.D. Hamkins, where $Γ$ defines a class of forcing posets and $κ$ is a cardinal. We explore the consistency strength and the relationship of $\textsf{MP}(κ,Γ)$ with various forcing axioms when $κ\in\{ω,ω_1\}$. In particular, we give a characterization of bounded forcing axioms for a class of forcings $Γ$ in terms of maximality principles MP$(ω_1,Γ)$ for $Σ_1$ formulas. A significant part of the paper is devoted to studying the principle MP$(κ,Γ)$ where $κ\in\{ω,ω_1\}$ and $Γ$ defines the class of stationary set preserving forcings. We show that MP$(κ,Γ)$ has high consistency strength; on the other hand, if $Γ$ defines the class of proper forcings or semi-proper forcings, then by Hamkins, it is shown that MP$(κ,Γ)$ is consistent relative to $V=L$.

math.LO↗

arXiv2016

Determinacy from strong compactness of $ω_1$

In the absence of the Axiom of Choice, the "small" cardinal $ω_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that $ω_1$ is $X$-strongly compact (where $X$ is any set) if there is a fine, countably complete measure on $\mathcal{P}_{ω_1}(X)$. Working in $\mathsf{ZF} + \mathsf{DC}$, we prove that the $\mathcal{P}(ω_1)$-strong compactness and $\mathcal{P}(\mathbb{R})$-strong compactness of $ω_1$ are equiconsistent with $\mathsf{AD}$ and $\mathsf{AD}_\mathbb{R} + \mathsf{DC}$ respectively, where $\mathsf{AD}$ denotes the Axiom of Determinacy and $\mathsf{AD}_\mathbb{R}$ denotes the Axiom of Real Determinacy. The $\mathcal{P}(\mathbb{R})$-supercompactness of $ω_1$ is shown to be slightly stronger than $\mathsf{AD}_\mathbb{R} + \mathsf{DC}$, but its consistency strength is not computed precisely. An equiconsistency result at the level of $\mathsf{AD}_\mathbb{R}$ without $\mathsf{DC}$ is also obtained.

math.LO↗

arXiv2016

PFA and guessing models

This paper explores the consistency strength of The Proper Forcing Axiom ($\textsf{PFA}$) and the theory (T) which involves a variation of the Viale-Wei$ß$ guessing hull principle. We show that (T) is consistent relative to a supercompact cardinal. The main result of the paper implies that the theory "$\sf{AD}$$_\mathbb{R} + Θ$ is regular" is consistent relative to (T) and to $\textsf{PFA}$. This improves significantly the previous known best lower-bound for consistency strength for (T) and $\textsf{PFA}$, which is roughly "$\sf{AD}$$_\mathbb{R} + \textsf{DC}$".

math.LO↗

arXiv2016

Scales in hybrid mice over $\mathbb{R}$

We develop a general theory of strategic mice, prove their condensation properties, and analyze the scales pattern in the stack of $Θ$-g-organized $\mathcal{F}$-mice over $\mathbb{R}$, Lp$^{G\mathcal{F}}(\mathbb{R})$, for a class of nice operators $\mathcal{F}$.

math.LO↗

arXiv2016

The fine structure of operator mice

We develop the fine structure theory of operator-premice. These are a generalization of standard premice, in which an abstract operator $F$ is used to form the successor steps in the internal hierarchy of the premouse, instead of Jensen's $J$-operator (which computes rudimentary closure). Such notions have seen applications in core model induction arguments, but their theory has not previously been developed in detail. We define fine condensation for operators $F$ and show that fine condensation and iterability together ensure that $F$-mice have the fundamental fine structural properties including universality and solidity of the standard parameter.

math.LO↗

arXiv2013

Determinacy in L(R,μ)

Assume L(\mathbb{R},μ) satisfies ZF+DC+Θ>ω_2 + μis a normal fine measure on \powerset_{ω_1}(\mathbb{R}). The main result of this paper is the characterization theorem of L(\mathbb{R},μ) which states that L(\mathbb{R},μ) satisfies Θ>ω_2 if and only if L(\mathbb{R},μ) satisfies AD^+. As a result, we obtain the equiconsistency between the two theories: "ZFC + there are ω^2 Woodin cardinals" and "ZF+DC+μis a normal fine measure on \powerset_{ω_1}(\mathbb{R}) + Θ>ω_2".

math.LO↗

arXiv2013

HOD in natural models of AD^+

This paper analyzes full HOD of natural models of AD^+ under a certain smallness assumption of the models. This assumption is made to utilize Sargsyan's work on the theory of hod mice. We show that HOD is a fine-structural model and in particular satisfies GCH.

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