Tootfinder

This https://arxiv.org/abs/2309.06116 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Distinguishing colorings, proper colorings, and covering properties without the Axiom of Choice
We work with simple graphs in ZF (Zermelo--Fraenkel set theory without the Axiom of Choice (AC)) and assume that the sets of colors can be either well-orderable or non-well-orderable to prove that the following statements are equivalent to Kőnig Lemma: (a) Any infinite locally finite connected graph G such that the minimum degree of G is greater than k, has a chromatic number for any fixed integer k greater than or equal to 2. (b) Any infinite locally finite connected graph has a chromatic ind…

On forcing axioms and weakenings of the Axiom of Choice
Diego Lima Bomfim, Charles Morgan, Samuel Gomes da Silva
https://arxiv.org/abs/2404.10736 https://<…

On forcing axioms and weakenings of the Axiom of Choice
We prove forcing axiom equivalents of two families of weakenings of the axiom of choice: a trichotomy principle for cardinals isolated by Lévy, ${\rm H\hskip0.05pt}_κ$, and ${\rm DC}_κ$, the principle of dependent choices generalized to cardinals $κ$, for regular cardinals $κ$. Using these equivalents we obtain new forcing axiom formulations of the axiom of choice. A point of interest is that we use a new template for forcing axioms. For the class of forcings to which we asks that the ax…

This https://arxiv.org/abs/2401.05657 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_eco…

An impossibility theorem concerning positive involvement in voting
In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance propert…

The Thins Ordering on Relations
Ed Voermans, Jules Desharnais, Roland Backhouse
https://arXiv.org/abs/2401.16888 https://arXiv.org/pd…

The Thins Ordering on Relations
Earlier papers \cite{VB2022,VB2023a} introduced the notions of a core and an index of a relation (an index being a special case of a core). A limited form of the axiom of choice was postulated -- specifically that all partial equivalence relations (pers) have an index -- and the consequences of adding the axiom to axiom systems for point-free reasoning were explored. In this paper, we define a partial ordering on relations, which we call the \textsf{thins} ordering. We show that our axiom of ch…

The Axiom of Choice in the $\kappa$-Mantle
Andreas Lietz
https://arxiv.org/abs/2403.09015 https://arxiv.org/pdf/2403.09015

The Axiom of Choice in the $κ$-Mantle
Usuba has asked whether the $κ$-mantle, the intersection of all grounds that extend to $V$ via a forcing of size ${<}κ$, is always a model of ZFC. We give a negative answers by constructing counterexamples where $κ$ is a Mahlo cardinal, $κ=ω_1$ and where $κ$ is the successor of a regular uncountable cardinal.

The Axiom of Choice in the $\kappa$-Mantle
Andreas Lietz
https://arxiv.org/abs/2403.09015 https://arxiv.org/pdf/2403.09015

The Axiom of Choice in the $κ$-Mantle
Usuba has asked whether the $κ$-mantle, the intersection of all grounds that extend to $V$ via a forcing of size ${<}κ$, is always a model of ZFC. We give a negative answers by constructing counterexamples where $κ$ is a Mahlo cardinal, $κ=ω_1$ and where $κ$ is the successor of a regular uncountable cardinal.

Berkeley Cardinals and Vop\v{e}nka's Principle
Marwan Salam Mohammd
https://arxiv.org/abs/2404.10455 https://arxiv.org/pdf/2404.1…

Berkeley Cardinals and Vopěnka's Principle
We introduce "$n$-choiceless" supercompact and extendible cardinals in Zermelo-Fraenkel set theory without the Axiom of Choice. We prove relations between these cardinals and Vopěnka's Principle similar to those of Bagaria's work in his papers "$C^{(n)}$-Cardinals" and "More on the Preservation of Large Cardinals Under Class Forcing." We use these relations to characterize Berkeley cardinals in terms of a restricted form of Vopěnka's Principle. Finally, we establish the equiconsistency of the…

This https://arxiv.org/abs/2006.01077 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the consistency of ZF with an elementary embedding from $V_{λ+2}$ into $V_{λ+2}$
According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal $λ$ and non-trivial elementary embedding $j:V_{λ+2}\to V_{λ+2}$. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone has been discovered. $I_{0,λ}$ is the assertion, introduced by W. Hugh Woodin, that $λ$ is an ordinal and there is an elementary embedding $j:L(V_{λ+1})\to L(V_{λ+1})$ with critical point ${<λ}$. And $I_0$ asserts that $I_{0,λ}$ holds for some $λ$. The axiom $I_0$ is one o…

This https://arxiv.org/abs/2006.01077 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the consistency of ZF with an elementary embedding from $V_{λ+2}$ into $V_{λ+2}$
According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal $λ$ and non-trivial elementary embedding $j:V_{λ+2}\to V_{λ+2}$. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone has been discovered. $I_{0,λ}$ is the assertion, introduced by W. Hugh Woodin, that $λ$ is an ordinal and there is an elementary embedding $j:L(V_{λ+1})\to L(V_{λ+1})$ with critical point ${<λ}$. And $I_0$ asserts that $I_{0,λ}$ holds for some $λ$. The axiom $I_0$ is one o…

Fraenkel--Mostowski models revisited
Jindrich Zapletal
https://arxiv.org/abs/2404.10612 https://arxiv.org/pdf/2404.10612

Fraenkel--Mostowski models revisited
I provide several natural properties of group actions which translate into fragments of axiom of choice in the associated permutation models of choiceless set theory.

This https://arxiv.org/abs/2311.14267 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A note on Łoś's Theorem without the Axiom of Choice
We study some topics about Łoś's theorem without assuming the Axiom of Choice. We prove that Łoś's fundamental theorem of ultraproducts is equivalent to a weak form that every ultrapower is elementary equivalent to its source structure. On the other hand, it is consistent that there is a structure $M$ and an ultrafilter $U$ such that the ultrapower of $M$ by $U$ is elementary equivalent to $M$, but the fundamental theorem for the ultrapower of $M$ by $U$ fails. We also show that weak fragme…

This https://arxiv.org/abs/2401.05657 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_eco…

An impossibility theorem concerning positive involvement in voting
In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance propert…

This https://arxiv.org/abs/2212.11859 has been replaced.
link: https://scholar.google.com/scholar?q=a

Librationist cum classical set theories
A librationist set theory £${}$ is developed. It descends from semantics for truth, initiated by by Kripke, Herzberger, Gupta and others. # extends the system £, of Librationist closures of the paradoxes in Logic and Logical Philosophy 21(4), 323-361, 2012. The focus is upon the paradoxes in theories of sets. A central result is that the extension #$\mathscr{HR}(\mathbf{D})$, of #, accounts for $\mathbf{NBG}$ set theory, with the global Axiom of Choice and Tarski's Axiom. Moreover, # also suc…

This https://arxiv.org/abs/2401.05657 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_eco…

An impossibility theorem concerning positive involvement in voting
In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance propert…

This https://arxiv.org/abs/2401.02215 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs without AC
In set theory without the Axiom of Choice, we study the set-theoretic strength of a generalized version of the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs introduced by Erdős and Rado, concerning their interrelation with several weak choice forms.