Here $J_a$ and $M_a$ represent species flux and mobility in the phase $a$ respectively. Eq. (<ref>) is equivalent to the equation responsible for the homogeneous mixture. Therefore, parameters $\alpha''_{a}=0$ and $\kappa_a=0$ can be utilized for the regions exhibiting non-phase separating behaviors, such as electrolyte.

CHAPTER: CALCULATION OF TORTUOSITY

The electrode particles are surrounded by the liquid electrolyte, which conducts species flux in the electrode. The species diffusivity of the electrode is much lower than the electrolyte [209]. Also, the insertion and the extraction of species in the electrode particles takes place at the interface between the electrode and the electrolyte. Therefore, understanding of tortuous pathways in the electrolyte is of scientific interest. In the present study, instead of assuming a Bruggeman relation [205], a computational tool is employed to calculate the tortuosity of the electrolyte microstructure considering infinite resistive electrode particles [206].

Schematic of potential distribution in the electrolyte during the tortuosity calculation.

The tortuosity is calculated in the x-direction, where the species flux propagates, from the separator to the current collector. A potential difference $\phi_{\infty}$ is applied between two ends, which is predefined as 1.0V at separator and 0.0V at the current collector for the present investigation as shown in Figure <ref>. While no-flux boundary conditions applied on the remaining two (four for 3D) boundaries. The potential distribution inside the simulation domain is calculated using the Laplace equation of the form,

\begin{equation}
\boldsymbol{\nabla} \cdot (-\sigma_{\textrm{ety}}\boldsymbol{\nabla} \phi) =0.
\end{equation}

Where $\sigma_{\textrm{ety}}$$\sigma_{\textrm{ety}}$electrical conductivity of the electrolyte is the electrical conductivity of the electrolyte. Afterwards, the current density $i= \sigma_{\textrm{ety}}\nabla \phi$ can be estimated at the each pixels (voxels in 3D). The total current flow from the cross-section perpendicular to the x-direction can be obtained as a summation,

\begin{equation}
I_{\textrm{m}} = \int_{A_{\perp}} i_{\perp} \,\,\,dA_{\perp}.
\end{equation}

Where $A_{\perp}$ is the area of the cross-section surface under consideration$I_{\textrm{m}}$ (or $I_{\textrm{i}}$)measured (or ideal) current flow. Note that, as the no-flux conditions prescribed at the remaining boundaries, due to conservation, the measured current flow at any cross-section perpendicular to the x-direction should be nearly equal. If not then the meaningful conclusions can not be drawn from the presented method. For instance, in the case where there is no path for species to flow across when electrode particles completely block the path by forming an entire blockage in the perpendicular direction to the x-axis.

The tortuosity is a measure of conductive pathways deviated from the ideal straight channels of uniform cross-section. The tortuosity can be calculated based on the actual current flow in the microstructure in contemplation to the current flow in the ideal path without any discontinuities,

\begin{equation}
\tau = \frac{I_{\textrm{i}}}{I_{\textrm{m}}},
\end{equation}

where $I_{\textrm{i}}$ is the ideal current flow, determined from the analytical expression $I_{\textrm{i}}=\sigma_{\textrm{ety}}\phi_{\infty}A_{\perp}/l$ and $l$ is the length of the domain in the direction of the x-axis.

[]List of symbols and abbreviations
tocchapterList of Symbols and Abbreviations

$ \Delta x $, $\Delta y $, and $\Delta z$simulation cell widths in $x$-, $y$- and $z$-directions respectively3.2847

$ \gamma_{a b}$ (or $ \gamma_{a b l}$)interfacial energy density between $a$ and $b$ (or $a$, $b$, and $l$) phasesC.2181

$\alpha$index of a lattice site2.022

$\alpha'$the neighboring interstitial lattice site2.123

$\alpha'_a$ and $\alpha''_a$regular solution parameters of phase $a$3.339

$\beta$conductivity ratio, $\sigma_{\textrm{mat}}/\sigma_{\textrm{icl}}$7.20106

$\boldsymbol{\phi}$phase-field parameter set for Allen-Cahn equationC.1181

$\boldsymbol{\psi}$indicator parameter set3.138

$\boldsymbol{J}$effective diffusional mass fluxes2.2632

$\boldsymbol{J}_A$mass flux of A species2.2832

$\boldsymbol{J}_B$mass flux of B species2.2832

$\boldsymbol{J}_i$flux of the diffusing species $i$4.554

$\boldsymbol{J}_{e}$flux of the charge carrier4.1156

$\boldsymbol{n}$inward pointing unit normal to the surface3.640

$\boldsymbol{q}_{a b}$gradient vector in the normal direction to $a-b$ interfaceC.2181

$\boldsymbol{x}_\alpha$spatial position of the $\alpha$ site2.1528

$\boldsymbol{x}_{\alpha B}$neighbor on the back side of the $\alpha$ site2.1728

$\boldsymbol{x}_{\alpha F}$neighbor on the front side of the $\alpha$ site2.1728

$\boldsymbol{x}_{\alpha L}$neighbor on the left side of the $\alpha$ site2.1728

$\boldsymbol{x}_{\alpha O}$neighbor on the bottom side of the $\alpha$ site2.1728

$\boldsymbol{x}_{\alpha R}$neighbor on the right side of the $\alpha$ site2.1728

$\boldsymbol{x}_{\alpha T}$neighbor on the top side of the $\alpha$ site2.1728

$\boldsymbol{x}_{\alpha'}$spatial position of the $\alpha'$ site2.1628

$\chi$dimensionless number associated with the steady state shapes of an island7.8103

$\chi_0$dimensionless number associated with the stability of circular island/inclusion7.13104

$\chi_{0\textrm{ct}}$critical value of $\chi_0$, above which circular island/inclusion is unstable7.13104

$\delta c_\alpha$ (or $\delta c_{\alpha'}$)fluctuation of $c_\alpha$ (or $c_{\alpha'}$) compared to the average2.624

$\Delta H$change in internal energy of the system2.1325

$\Delta S$change in entropy of the system2.1325

$\Delta {t}$time step increment3.3148

$\delta_s$interface width4.1558

$\epsilon$interface thickness coefficient5.271

$\epsilon_p$parameter related to interface thickness in Allen-Cahn equationB.1179

$\eta$slit characteristic parameter7.48121

$\gamma$aspect ratio of ellipse5.271

$\gamma_s$interfacial energy4.1558

$\hat{\bullet}$normalized quantity of the respective entities3.2546

$\hat{\kappa}_s$non-dimensional curvature of the island surface7.7103

$\hat{M}_{\mathrm{max}}$maximum of dimensionless mobility $\hat{M}$3.3249

$\kappa$gradient energy coefficient2.2330

$\kappa_a$gradient energy coefficient of phase $\alpha$3.439

$\kappa_s$curvature along the island/inclusion surface7.4103

$\lambda'$reference length scale for normalization4.2360

$\Lambda_L$ratio of line width $w$ to the initial inclusion radius $R_i$8.0132

$\mathcal{O}$order of approximation3.2847

$\mathring{a}$lattice parameter2.1928

$\mu$chemical potential2.223

$\mu^{\mathrm{chem}}$chemical potential corresponding to the bulk free energy density3.841

$\mu^{\mathrm{grad}}$chemical potential corresponding to the gradient free energy density3.941

$\mu_A$chemical potentials of the A species2.3032

$\mu_B$chemical potentials of the B species2.3032

$\mu_{a}^{\mathrm{chem}}$chemical potential corresponding to the bulk free energy density of $a$ phase3.841

$\mu_{a}^{\mathrm{grad}}$chemical potential corresponding to the gradient free energy density of $a$ phase3.941

$\nu$denominator coefficient3.2245

$\Omega$atomic volume4.2159

$\partial V_\Omega$simulation domain boundary surfaces3.2345

$\phi _\infty $electrical potential boundary condition4.052

$\phi$local electrical potential4.655

$\phi_a$evolving indicator parameter for phase $a$ in Allen-Cahn equationC.1181

$\pi$Archimedes' constant4.1758

$\psi$orthogonal function to the electrical potential7.23107

$\psi_a$stationary indicator parameter for phase $a$3.138

$\rho $porosity6.090

$\rho_{\textrm{mat}}$electrical resistivity of the matrix4.2360

$\sigma$electrical conductivity dependent on order parameter $c$4.1456

$\sigma_{\textrm{ety}}$electrical conductivity of the electrolyteE.1187

$\sigma_{\textrm{icl}}$conductivity of an inclusion4.1456

$\sigma_{\textrm{mat}}$conductivity of the matrix4.1456

$\tau $tortuosity6.090

$\tau'$reference time scale for normalization4.2360

$\tau_p$relaxation coefficient2.3434

$\theta$angle formed by the local tangent at the inclusion surface4.1156

$\Upsilon_a$time-dependent antiforce term for phase $a$C.5182

$\varepsilon$perturbation parameter7.10104

$\varepsilon_t$difference between the A–B bond energy and the average of A-A and B-B bond energies2.1425

$\varepsilon_{AA}$bond energy between A–A species2.1325

$\varepsilon_{AB}$bond energy between A–B species2.1325

$\varepsilon_{BB}$bond energy between B–B species2.1325

$\varkappa$perturbation function7.10104

$\varpi$misorientation angle4.1156

$\xi$discrepancy coefficient7.45120

$\zeta$transformed fraction6.292

$A$strength of anisotropy4.1156

$a$index of the phase3.138

$A_0$amplitude of the perturbation at time $t=0$4.2160

$a_n$surface perturbation coefficient corresponds to n$^{\textrm{th}}$-term7.24107

$A_t$amplitude of the perturbation at time $t$4.2260

$a{_{a b}}$a-b interface anisotropy functionC.2181

$B$Mullins' constant4.2159

$B_{0m}$the internal system energy parameter2.123

$B_{m}$two-body interaction field from neighboring lattices2.123

$c$concentration/order parameter2.523

$c^H$higher concentration level of a phase-separated state5.374

$c^L$lower concentration level of a phase-separated state5.374

$c_\alpha$state of the lattice site $\alpha$2.022

$c_A$concentration of A species2.2832

$c_B$concentration of B species2.2832

$c_i$initial concentration filling5.271

$c_n$Dirichlet concentration boundary condition at the particle surface3.2044

$c_s$concentration at the surface of the particle5.168

$c_{\alpha'}$state of the neighboring interstitial lattice state $\alpha'$2.123

$d$dimensions of the simulation study3.3249

$D_a$diffusion coefficient of phase $a$3.1242

$D_s$surface diffusion coefficient4.1055

$e$electron charge4.1256

$E_t$tangential component of the electric field along the island/inclusion surface7.1103

$E_x, E_y$local electric field components in x and y-directions7.39110

$E_{\infty}$applied electric field7.2103

$F$system free energy2.523

$f$system free energy density2.1225

$f^{\textrm{dw}}$double-well free energy functionB.1179

$f^{\textrm{ob}}$obstacle-type free energy density4.253

$f^{\textrm{S}}$surface energy density3.138

$f^{\theta}$anisotropy function4.1055

$g$grand canonical potential function2.223

$g^{\textrm{ac}}$additional bulk energy densityC.4182

$g^{\theta \beta}$function of shape and conductivity contrast8.1136

$G_a^{\mathrm{chem}}$grand-chemical potential of phase $a$5.374

$H$Hamiltonian of the lattice model2.023

$h$interpolation function3.239

$I$indicator function for the obstacle-type free energy density4.354

$i$complex number7.21106

$i,j,k \in \boldsymbol{x}$spatial location of a simulation grid point in the x-, y-, and z-directions respectively3.2345

$I_{\textrm{m}}$ (or $I_{\textrm{i}}$)measured (or ideal) current flowE.2187

$J_n$Neumann flux boundary condition at the particle surface3.1643

$J_s$surface atomic flux7.1102

$j_{\textrm{Cot}}$Cottrell flux6.191

$J_{ps}$Neumann flux boundary condition at the separator3.2446

$k$frequency of the sinusoidal perturbation4.2160

$k_B$Boltzmann constant2.223

$k_{\textrm{Cot}}$Cottrell flux-time proportionality constant6.191

$L$reference length-scale for normalization3.2546

$M$effective diffusional mobility2.3333

$m$grain symmetry parameter4.1156

$M_m$co-ordination number2.123

$M_{AA}$mobility of the A species, due to the interaction with the A species2.3032

$M_{AB}$mobility of the A species, due to the interaction with the B species2.3032

$M_{BA}$mobility of the B species, due to the interaction with the B species2.3032

$M_{BB}$mobility of the B species, due to the interaction with the B species2.3032

$M_{ee}$mobility of electrons due to interaction with the electrons4.1256

$M_{ei}$mobility of electrons due to interaction with species $i$4.1256

$M_{ie}$mobility of species $i$ due to the interaction with the electrons4.655

$M_{ii}$mobility of species $i$ due to the interaction with the species $i$4.655

$N$total number of phases3.138

$n$discrete time3.2547

$N_B$the total number of specified species, occupied in the lattice sites2.022

$N_c$nucleation rate coefficient6.292

$N_m$total number of lattice sites2.022

$N_x$, $N_y$, and $N_z$number of grid points in x-, y- and z-directions3.2345

$N_{\textrm{ex}}$characteristic extension5.168

$p$transitory perimeter of the inclusion7.39116

$p_0$initial perimeter of the inclusion7.39116

$P_{AB}$total number of A–B bonds2.1425

$R$particle whose area is equivalent to the area of the circular particle of radius $R$6.090

$r, \vartheta$polar coordinates of a point7.13105

$R_0$radius of the reference particle5.168

$R_i$radius of circular island/inclusion7.11104

$S$displacement function7.29108

$s$arc length along the island/inclusion surface7.1103

$S_\Omega$surface of the domain3.038

$T$absolute temperature2.223

$t_\mathrm {PE}$time attained at the end of phase separation5.376

$t_\mathrm {PS}$time at the onset of phase separation5.376

$T_{\textrm{ref}}$reference temperature2.1225

$u$half-slit width7.39112

$V$velocity along the external electric field7.3103

$V_0$steady-state velocity of an island/inclusion7.11104

$V_\Omega$volume of the system or simulation domain2.2029

$V_n$velocity along the surface normal7.2103

$w$line width of the conductor7.39112

$w^{\textrm{ac}}$bulk energy density potentialC.3182

$W_c$spatially dependent weight for Dirichlet concentration boundary condition3.2245

$W_J$spatially dependent weight for Neumann flux boundary condition3.2245

$X_1$first free energy density parameter2.1225

$X_2$second free energy density parameter2.1225

$X_A$barrier height of the obstacle-type free energy density4.354

$Z_i$valence of the diffusing metal species $i$4.655

$Z_s$effective valence7.1103

$Z_w$momentum exchange effect between the electrons and the diffusing species4.655

$Z_{gc}$grand canonical partition function2.123

${\boldsymbol{\eta}_{i,a b}}$position vectors in the Wulff shapeC.3182

${c}_{ps}$Dirichlet concentration boundary condition at the separator3.2345

${f}_{a}^{\mathrm{chem}}$regular solution free energy density of phase $a$3.239

GBgrain boundaries1.213

IMCintermetallic compound7.39115

LFPlithium iron phosphate, LiFePO$_4$3.1141

LMOlithium manganese oxide, LiMn$_2$O$_4$5.270

LTOlithium titanate, Li$_4$Ti$_5$O$_{12}$6.085

MPImessage passing interface5.269

PACE3DParallel Algorithms for Crystal Evolution in 3D4.2361

PFMphase-field model6.090

PITTPotentiostatic Intermittent Titration Technique6.191

REVrepresentative elementary volume3.138

SOCstate of charge5.271

SOC$_\mathrm {PE}$state of charge attained before the end of phase separation5.376

SOC$_\mathrm{PS}$state of charge at the onset of phase separation5.273

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# On forcing axioms and weakenings of the Axiom of Choice

Diego Lima Bomfim, Charles Morgan and Samuel Gomes da Silva

###### Abstract

We prove forcing axiom equivalents of two families of weakenings of the axiom
of choice: a trichotomy principle for cardinals isolated by Lévy, $\mathop{\rm
H\hskip 0.05pt}_{\kappa}$, and $\mathop{\rm DC}_{\kappa}$, the principle of
dependent choices generalized to cardinals $\kappa$, for regular cardinals
$\kappa$. Using these equivalents we obtain new forcing axiom formulations of
$\mathop{\rm AC}$.

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 axioms apply, we do not ask that
they apply to all collections of dense sets of a certain cardinality, but
rather only for each particular forcing to a specific family of dense sets of
the cardinality in question.

00footnotetext: 2020 Mathematics Subject Classification: 03E25, 03E57,
03E05.00footnotetext: Keywords: dependent choice, Levy’s trichotomy, forcing
axioms.00footnotetext: Date: 00footnotetext: This research was carried out
whilst the second author was a Professor Visitante at the Universidade Federal
de Bahia (UFBA), under EDITAL PV 001/2019 - PROPG/ UFBA. His grateful for the
support of UFBA under this edital and for the kind hospitality of the
Departamento de Matemática.

##  1 Introduction

This paper had its origins in parts of the first author’s master’s thesis,
[Bomfim22], written under the supervision of the third author. These sections
of [Bomfim22] were inspired by [Parente15] and [Viale19] which, as Viale
writes in the latter article, “focus on the interplay between forcing axioms
and various other non-constructive principles widely used in many fields of
abstract mathematics, such as the axiom of choice and Baire’s category
theorem.”

One of the original objectives of the work recounted in [Bomfim22] was to find
a forcing axiom equivalent of the statement that every infinite set is
Dedekind-infinite. Subsequent discussions between the three authors led to the
broader consideration of the interplay between forcing axioms and weakenings
of the Axiom of Choice which forms the focus of the paper.

One early paper in which various families of weakenings of the axiom of choice
were considered is Lévy’s [Levy64]. Lévy focused, in particular, on what are
nowadays called $\mathop{\rm AC}_{\alpha}$ and $DC_{\alpha}$,111See below for
definitions. for ordinals $\alpha$, and on a family of statements generalizing
the axiom that every infinite set is Dedekind-infinite.

We remind the reader of the standard notation surrounding injective embeddings
of one set into another. Doing so allows us to introduce this generalization
immediately.

######  Notation 1.1.

Suppose $X$ and $Y$ are sets. We write $X\preccurlyeq Y$ if there is an
injection of $X$ into $Y$. We write $X\prec Y$ if $X\preccurlyeq Y$ and
$Y\not\preccurlyeq X$. We write $X\approx Y$ if $X\preccurlyeq Y$ and
$Y\preccurlyeq X$.

######  Definition 1.2.

Let $X$, $Y$ be sets. We say _trichotomy_ (for cardinals) holds for $X$ and
$Y$ if $X\prec Y$ or $X\approx Y$ or $Y\prec X$, or, equivalently,
$X\preccurlyeq Y$ or $Y\preccurlyeq X$, or $X\prec Y$ or $Y\preccurlyeq X$.

The statement ‘every infinite set is Dedkind-infinite’ can be rephrased as
‘every set is either finite or there is an injection of a $\omega$ into it’.
That is, for every set $X$ trichotomy holds between $X$ and $\omega$ (in the
form $X\prec\omega$ or $\omega\preccurlyeq X$). As noted above, Lévy
considered a generalization of this statement.

######  Definition 1.3.

(_cf._ [Levy64]) Let $\alpha$ be an ordinal. $\mathop{\rm H\hskip
0.05pt}_{\alpha}$ is the principle:

$\forall X\hskip 2.0pt\hskip 2.0pt(X\prec\alpha\hbox{ or }\alpha\preccurlyeq
X)$

that is, for every set $X$ trichotomy holds between $X$ and $\alpha$:

######  Remark 1.4.

[Herrlich06] writes $\mathop{\hbox{Fin}}$ for $\mathop{\rm H\hskip
0.05pt}_{\omega}$, a notation used in [Bomfim22].

As we recall below in Proposition (2.6), Hartogs ([Hartogs1915]) proved
$\mathop{\rm AC}\Longleftrightarrow$ for all ordinals $\alpha$ we have
$\mathop{\rm H\hskip 0.05pt}_{\alpha}$. We speculate that Lévy used the letter
‘H’ as a glancing reference to Hartogs’s work.

Our main results are to prove forcing axiom equivalents of $\mathop{\rm
H\hskip 0.05pt}_{\kappa}$ (and consequently of $\mathop{\rm H\hskip
0.05pt}_{\omega}$ – $\mathop{\hbox{Fin}}$) and $\mathop{\rm DC}_{\kappa}$ for
regular cardinals $\kappa$. Using these equivalents we obtain new forcing
axiom formulations of $\mathop{\rm AC}$.

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 axioms apply, we do not ask that
they apply to all collections of dense sets of a certain cardinality, but
rather only for each particular forcing to a specific family of dense sets of
the cardinality in question.

We now introduce some further notation, definitions and results, for the most
part standard for set theory, which will be used in the course of the paper.

As the paper concerns weakenings of the axiom of choice our base theory is ZF
and all results, unless otherwise stated, are ZF-results.

######  Definition 1.5.

A cardinal which is an ordinal is an _intial ordinal_ or an _aleph_.

######  Notation 1.6.

Let $X$ be a set. We write  $\hfil\displaystyle{X}\hfil$ for the cardinality
of $X$. If $X$ is a set we write $h(X)$ for its Hartogs number, the cardinal
which is the least ordinal $\alpha$ for which there is no injection of
$\alpha$ into $X$.

######  Remark 1.7.

In the ensuing we will be particularly interested in the statement
$X\not\prec\kappa$ for initial ordinals $\kappa$. This says that there is no
injective embedding of $X$ into any well-ordered set of cardinality less than
$\kappa$.

The following lemma is trivial, however we make heavy use of the negation of
the right hand side of the conclusion as an equivalent of the negation of the
left hand side.

######  Lemma 1.8.

Let $X$ be a set and $\kappa$ an initial ordinal. Then $X\prec\kappa$ if and
only if there is some $\beta<\kappa$ and an injection from $\beta$ into $X$
which is surjective.

###### Proof.

If $X\prec\kappa$ it is well-orderable and $\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{X}\hfil$\crcr}}}\hfil$\crcr}}<\kappa$,
so there is a bijection from some cardinal less than $\kappa$ to $X$. If there
is a bijection from some cardinal less than $\kappa$ to $X$ then the inverse
demonstrates that $X\prec\kappa$. ∎

######  Notation 1.9.

Let $X$ be a set. Then $[X]^{\kappa}$ is $\\{\hskip 1.0ptY\subseteq X\hskip
2.0pt|\hskip 2.0pt\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{Y}\hfil$\crcr}}}\hfil$\crcr}}=\kappa\hskip
1.0pt\\}$ and $[X]^{<\kappa}$ is $\\{\hskip 1.0ptY\subseteq X\hskip
2.0pt|\hskip 2.0pt\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{Y}\hfil$\crcr}}}\hfil$\crcr}}<\kappa\hskip
1.0pt\\}$. If $\alpha\in\mathop{\rm On}$ we write ${{}^{\alpha}X}$ for
$\\{\hskip 1.0ptf\hskip 2.0pt|\hskip 2.0ptf:\alpha\longrightarrow X\hskip
1.0pt\\}$ and ${{}^{<\alpha}X}$ for $\\{\hskip 1.0ptf\hskip 2.0pt|\hskip
2.0pt\exists\beta<\alpha\hskip 2.0pt\hskip 2.0ptf:\beta\longrightarrow X\hskip
1.0pt\\}$.

We introduce two _non-standard_ pieces of notation.

######  Notation 1.10.

We write $(X)^{\alpha}$ for $\\{\hskip 1.0ptf\hskip 2.0pt|\hskip
2.0ptf:\alpha\longrightarrow X\hbox{ and }f\hbox{ is an injection}\hskip
1.0pt\\}$ and $(X)^{<\alpha}$ for $\\{\hskip 1.0ptf\hskip 2.0pt|\hskip
2.0pt\exists\beta<\alpha\hskip 2.0pt\hskip 2.0ptf:\beta\longrightarrow X\hbox{
such that }f\hbox{ is an injection}\hskip 1.0pt\\}$.

(Note that typically in set theory $(X)^{\alpha}$ denotes the collection of
_increasing_ sequences of length $\alpha$ of elements of a set $X$ which is
equipped with a strict partial ordering – equivalently the collection of
strictly increasing injections from $\alpha$ to $X$. Here we do not assume any
such ordering exists.)

######  Notation 1.11.

For a sequence of sets $t$ we write $\mathop{\rm lh}(t)$ for $\mathop{\rm
dom}(t)$ and if we are thinking of $t$ as an element of a tree of sequences
we, further, write $0pt(t)$ for $\mathop{\rm lh}(t)$.

######  Definition 1.12.

If $Y$ is a set of ordinals, $\mathop{\rm ssup}(Y)$ (the _strong supremum_ of
$Y$) is the least ordinal greater than all the elements of $Y$: if $Y$ has a
maximal element then $\mathop{\rm ssup}(Y)=\bigcup Y+1$, and if not then
$\mathop{\rm ssup}(Y)=\bigcup Y$.

Later in the paper we will need the notion of (possibly infinite)
concatenation.

######  Definition 1.13.

Let $\tau$ be an ordinal (possibly infinite) and let
$\langle\tau_{i}:i<\tau\rangle$ be a sequence of non-zero ordinals. For
convenience, define $\tau^{*}=\tau+1$ if $\tau$ is a sucessor ordinal, and
$=\tau$ if $\tau$ is a limit.

From $\langle\tau_{i}:i<\tau\rangle$ we define a sequence of ordinals
$\langle\sigma_{i}:i<\tau^{*}\rangle$ by induction on $\tau^{*}$. Let
$\sigma_{0}=0$. Given $\sigma_{i}$, let $\sigma_{i+1}=\sigma_{i}+\tau_{i}$. If
$i$ is a limit ordinal let $\sigma_{i}=\bigcup_{j<i}\sigma_{j}$.

Now let $X$ be a set and suppose, for each $i<\tau$ and $j<\tau_{i}$, that
$t_{ij}$ is an element of $X$. Let $t_{i}=\langle t_{ij}:j<\tau_{i}\rangle$
and let $t=\langle t_{i}:i<\tau\rangle$.

Now we define a sequence $t^{c}$ of length $\mathop{\rm
ssup}(\\{\sigma_{i}:i<\tau^{*}\\})$ by $t^{c}_{\sigma_{i}+j}=t_{ij}$ for every
$i<\tau$ and $j<\tau_{i}$. We call $t^{c}$ _the concatenation of_ $t$.

Finally we state $\mathop{\rm AC}_{\alpha}$ for the reader’s convenience. We
define and discuss $\mathop{\rm DC}_{\alpha}$ in the following section,
immediately below.

######  Definition 1.14.

($\mathop{\rm AC}_{\alpha}$) Let $\alpha$ be an ordinal. $\mathop{\rm
AC}_{\alpha}$ is the axiom that whenever ${\mathcal{X}}=\\{\hskip
1.0ptX_{\beta}\hskip 2.0pt|\hskip 2.0pt\beta<\alpha\hskip 1.0pt\\}$ is a
collection of sets such that for all $\beta<\alpha$ we have
$X_{\beta}\neq\emptyset$, there is a function
$f:\alpha\longrightarrow\bigcup{\mathcal{X}}$ such that for all $\beta<\alpha$
we have $f(\beta)\in X_{\beta}$.

######  Definition 1.15.

($\mathop{\rm DC}_{\alpha}$) Let $\alpha$ be an ordinal. $\mathop{\rm
DC}_{\alpha}$ is the axioms that

##  2 Variants of the Axiom of Dependent Choices

We recall the Axiom of Dependent Choice ($DC$) and the axiom $\mathop{\rm
DC}_{\alpha}$.

######  Definition 2.1.

$\mathop{\rm DC}$ is the _axiom of dependent choice_ : for every set $X$ and
every binary relation $R$ on $X$ for which for all $x\in X$ there is some
$y\in X$ with $x\hskip 2.0ptR\hskip 2.0pty$ there is a function
$f:\omega\longrightarrow X$ such that for all $i<\omega$ we have $f(i)\hskip
2.0ptR\hskip 2.0ptf(i+1)$.

######  Definition 2.2.

If $\alpha$ is an ordinal then $\mathop{\rm DC}_{\alpha}$ is the axiom: if
$F:{{}^{<\alpha}X}\longrightarrow{\mathcal{P}}(X)\setminus\\{\hskip
1.0pt\emptyset\hskip 1.0pt\\}$ there is a function $g:\alpha\longrightarrow X$
such that for all $i<\alpha$ we have $g(i)\in F(g\upharpoonright i)$.

It is well known that $\mathop{\rm DC}$ and $\mathop{\rm DC}_{\omega}$ are
equivalent (under ZF). However, the two equivalents inspire separate ways to
weaken the axiom’s force.

For $\mathop{\rm DC}$ itself we can consider restrictions on the class of
binary relations. In order to give examples of these we need a little more
notation.

######  Definition 2.3.

Let $P=(\mathbb{P},<_{\mathbb{P}})$ be a pre-order. $P$ has the _finite
predecessor property (fpp)_ if each element of $\mathbb{P}$ has finitely many
predecessors under $<_{\mathbb{P}}$. That is, for $p\in\mathbb{P}$ we have
$\\{\hskip 1.0ptq\in\mathbb{P}\hskip 2.0pt|\hskip 2.0ptp<_{\mathbb{P}}q\hskip
1.0pt\\}$ is finite. An element $p$ of $\mathbb{P}$ is _maximal_ if it has no
predecessors and _minimal_ if there is no $q\in\mathbb{P}$ such that
$q<_{\mathbb{P}}p$.

######  Definition 2.4.

Let $P=(\mathbb{P},<_{\mathbb{P}})$ be a partial order. $P$ is _strict_ if for
all $x$ we have $x\not<_{\mathbb{P}}x$. $P$ is _reticulated_ or _a lattice_ if
every pair of elements has a sup and an inf: for all $p$, $q\in\mathbb{P}$,
there are $p\wedge q$, $p\vee q$ where if $r<p$, $q$ then $r<p\wedge q$ or
$r=p\wedge q$ and if $p$, $q<r$ then $p\vee q<r$ or $p\vee q=r$.

Diener ([Diener94]) showed that $\mathop{\rm DC}$ is equivalent to the formal
restriction of $\mathop{\rm DC}$ to the class of relations that are (strict)
partial orders. Gomes de Silva [Silva12] considered the following (true)
weakening of $\mathop{\rm DC}$.

######  Notation 2.5.

  * $\mathop{\rm DC}_{l,fp}$ is the axiom of dependent choice restricted to strict lattices with the fpp. More explicitly, $\mathop{\rm DC}_{l,fp}$ is the principle that for every set $X$ and strict lattice relation $R$ on $X$ with the fpp, if additionally for all $x\in X$ there is some $y\in X$ such that $x\hskip 2.0ptR\hskip 2.0pty$ there is a function $f:\omega\longrightarrow X$ such that for all $i<\omega$ we have $f(i)\hskip 2.0ptR\hskip 2.0ptf(i+1)$.

However, the formulation of $\mathop{\rm DC}_{\alpha}$ also naturally leads to
weakenings given by restricting the class of functions $F$ for which the axiom
applies. One example example of such restriction is the collection of axioms
$\mathop{\rm H\hskip 0.05pt}_{\alpha}$ introduced in §1.

In order to see why we think of $\mathop{\rm H\hskip 0.05pt}_{\alpha}$ as a
variant of $\mathop{\rm DC}_{\alpha}$ note it can be rewritten as the
statement that for every $X$ such that $X\not\prec\alpha$ there is a function
$g:\alpha\longrightarrow X$ such that for all $i<\omega$ we have $g(i)\in
X\setminus\mathop{\rm rge}(g\upharpoonright i)$. We can think of this
formulation as $\mathop{\rm DC}_{\alpha}$ restricted to the special cases (one
for each set $X$) of the functions
$F^{X}_{seq}:{{}^{<\alpha}X}\longrightarrow{\mathcal{P}}(X)\setminus\\{\hskip
1.0pt\emptyset\hskip 1.0pt\\}$ given by $F^{X}_{seq}(t)=X\setminus\mathop{\rm
rge}(t)$. The requirement that $X\not\prec\alpha$ is (using Lemma 1.8) exactly
what is necessary to show that $F^{X}_{seq}$ is well defined. For
$F^{X}_{seq}$ is only not defined everywhere if there is some
$t\in{{}^{<\alpha}X}$ with $F^{X}_{seq}(t)=\emptyset$, which in turn says that
$t$ is an injection of some ordinal less than $\alpha$ into $X$ which is
surjective.

Perhaps more obviously, for ordinals $\alpha$ one can also see $\mathop{\rm
H\hskip 0.05pt}_{\alpha}$ as an instance of trichotomy for between pais
consisting of an ordinal and an arbitrary set.

######  Proposition 2.6.

[Hartogs1915] $\mathop{\rm AC}\Longleftrightarrow\forall\hbox{ regular initial
ordinals }\kappa\hskip 2.0pt\hskip 2.0pt\mathop{\rm H\hskip 0.05pt}_{\kappa}$.

###### Proof.

“$\Longrightarrow$”: $\mathop{\rm AC}$ is equivalent to WO, the principle that
every set can be well ordered. This immediately gives us that $\mathop{\rm
H\hskip 0.05pt}_{\alpha}$ holds for every ordinal $\alpha$ by the trichotomy
theorem for ordinals. “$\Longleftarrow$”: for any set $X$ apply $\mathop{\rm
H\hskip 0.05pt}_{h(X)^{+}}$, where $h(X)$ is the Hartog’s number for $X$. This
gives an injection of $X$ into the regular cardinal $h(X)^{+}$ and hence a
well-ordering of $X$. ∎

The translation between these two types of weakening of $\mathop{\rm DC}$ is
not necessarily completely straightforward.

We first of all observe that, as shown by Lévy ([Levy64]), we may restrict our
attention to the cases where $\alpha$ is a cardinal. We give proofs as the
proofs in [Levy64] are indirect.

######  Proposition 2.7.

[Levy64] For any ordinal $\alpha$ we have $\mathop{\rm H\hskip
0.05pt}_{\alpha}$ if and only if $\mathop{\rm H\hskip
0.05pt}_{\vbox{\halign{#\cr\cleaders\hrule height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\scriptstyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\scriptstyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}}}$.

###### Proof.

For any ordinal $\alpha$ we have a bijection between $\alpha$ and
$\hfil\displaystyle{\alpha}\hfil$ . Hence if we have for every $X$ that
$X\prec\alpha$ or $\alpha\preccurlyeq X$ then for every $X$ we have
$X\prec\alpha\preccurlyeq\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}}$
or $\vbox{\halign{#\cr\cleaders\hrule height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}}\preccurlyeq\alpha\preccurlyeq
X$. Similarly if for all $X$ we have $X\prec\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}}$
or $\vbox{\halign{#\cr\cleaders\hrule height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}}\preccurlyeq
X$ then for all $X$ we have $X\prec\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}}\preccurlyeq{\alpha}$
or ${\alpha}\preccurlyeq\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}}\preccurlyeq
X$ ∎

######  Proposition 2.8.

[Levy64] Let $\alpha$ be a singular limit ordinal. Then

$ZF\vdash(\forall\beta<\alpha\hskip 2.0pt\hskip
2.0ptDC_{\beta})\Longrightarrow DC_{\alpha}.$

###### Proof.

Let $\langle\alpha_{\xi}:\xi<\mathop{\rm cf}(\alpha)\rangle$ be an increasing,
continuous sequence of ordinals with $\alpha_{0}=0$ and
$\bigcup_{\xi<\mathop{\rm cf}(\alpha)}\alpha_{\xi}=\alpha$. For
$\xi<\mathop{\rm cf}(\alpha)$ let $\gamma_{\xi}=\mathop{\rm
otp}([\alpha_{\xi},\alpha_{\xi+1}))$.

Suppose
$F:{}^{<\alpha}X\longrightarrow{\mathcal{P}}(X)\setminus\\{\emptyset\\}$.
Define $G:{}^{<\mathop{\rm
cf}(\alpha)}({}^{<\alpha}X)\longrightarrow{\mathcal{P}}({}^{<\alpha}X)\setminus\\{\emptyset\\}$
by $G(t)=\\{s\in{{}^{\gamma_{\xi}}X}:\forall j<\gamma_{\xi}\;s_{j}\in
F(t^{c}\kern-0.25pt\raise 4.0pt\hbox{$\frown$}\kern-0.25pts\upharpoonright
j)\\}$ if $\forall i<\xi(\mathop{\rm otp}(t_{i})=\gamma_{i})$ and
$G(t)=\\{\langle\rangle\\}$ (the set whose only element is the empty sequence)
otherwise. (Recall, here, the notation that $t^{c}$ is the concatenation of
$t$.)

Note: the definition of $G$ uses $DC_{\beta}$ for $\beta\leq\alpha_{\xi+1}$ to
ensure that the $G(t)$ are non-empty in the substantive cases.

Let $h:\mathop{\rm cf}(\alpha)\longrightarrow{}^{<\alpha}X$ be given by
$DC_{\mathop{\rm cf}(\alpha)}$ applied to $G$ and let $g=h^{c}$.

Then $g:\alpha\longrightarrow X$ is a function and
$\forall\beta<\alpha\;g(\beta)\in F(g\upharpoonright\beta)$. ∎

######  Corollary 2.9.

[Levy64] If $\alpha$ is an ordinal then $\mathop{\rm DC}_{\alpha}$ holds if
and only if $\mathop{\rm DC}_{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\scriptstyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\scriptstyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}}}$
does. If $\alpha$ is a singular initial ordinal then $\mathop{\rm
DC}_{\alpha}$ holds if and only if for all regular $\beta<\alpha$ we have that
$\mathop{\rm DC}_{\beta}$ holds.

###### Proof.

For the first statement use induction on $[\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}},\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\alpha}\hfil$\crcr}}}\hfil$\crcr}}^{+})$,
applying Proposition (2.8) at limit ordinals. The second statement follows
from Proposition (2.8) and the first. ∎

######  Remark 2.10.

It is not clear that we have a result for $\mathop{\rm H\hskip
0.05pt}_{\alpha}$ for singular cardinals $\alpha$ similar to that for
$\mathop{\rm DC}_{\alpha}$, although it is the case that if we have for all
regular cardinals $\beta<\alpha$ that $\mathop{\rm H\hskip 0.05pt}_{\beta}$
holds and we have (the full) $\mathop{\rm DC}_{\mathop{\rm cf}(\alpha)}$ we
then have $\mathop{\rm H\hskip 0.05pt}_{\alpha}$.

As a consequence of these results, we restrict attention here to $\mathop{\rm
DC}_{\kappa}$ and $\mathop{\rm H\hskip 0.05pt}_{\kappa}$ for cardinals
$\kappa$.

We now take the opportunity to introduce some notation relevant to forcing
axioms.

######  Definition 2.11.

Let $\mathbb{P}$ be a partial order, and ${\mathcal{D}}$ a collection dense
subsets of $\mathbb{P}$. A filter $G$ on $\mathbb{P}$ is _$\mathbb{P}$
-generic for ${\mathcal{D}}$_ if for every $D\in{\mathcal{D}}$ we have $G\cap
D\neq\emptyset$. We say $\mathop{\rm FA}(\mathbb{P},{\mathcal{D}})$ holds if
there is $\mathbb{P}$-generic filter for ${\mathcal{D}}$.

######  Definition 2.12.

Let ${\mathcal{P}}$ be a class of partial orders and $\kappa$ a cardinal. We
say $\mathop{\rm FA}_{\kappa}({\mathcal{P}})$ holds if for every
$\mathbb{P}\in{\mathcal{P}}$ and collection ${\mathcal{D}}$ of $\kappa$-many
dense subsets of $\mathbb{P}$ we have that $\mathop{\rm
FA}(\mathbb{P},{\mathcal{D}})$ holds.

An intermediate notion, essentially a uniform version of the first definition
across a class of partial orders, is the following.

######  Definition 2.13.

Let ${\mathcal{P}}$ be a class of partial orders and let ${\mathfrak{D}}$ be a
uniformly definable Function such that there is some cardinal $\kappa$ such
that if $\mathbb{P}\in{\mathcal{P}}$ then
${\mathfrak{D}}(\mathbb{P})=\langle\hskip
1.0pt{\mathfrak{D}}(\mathbb{P})_{i}\hskip 2.0pt|\hskip 2.0pti<\kappa\hskip
1.0pt\rangle$ is a collection of dense subsets of $\mathbb{P}$. We say
$\mathop{\rm FA}({\mathcal{P}},{\mathfrak{D}})$ holds if for every
$\mathbb{P}\in{\mathcal{P}}$ we have that $\mathop{\rm
FA}(\mathbb{P},{\mathfrak{D}}(\mathbb{P}))$ holds.

This is a provisional definition. We could make a more limited definition,
oriented to the specific use we make below, but we tried to formulate
something more open, at the cost of not being completely explicit about what
‘uniformly definable’ might mean.

##  3 Results from [Bomfim22]

.

######  Definition 3.1.

$\mathop{\hbox{Fin}}$ is the statement that every infinite set is Dedekind-
infinite. That is, that whenever $X$ is infinite there is an injection
$f:\omega\longrightarrow X$, or equivalently that $X$ has a countable,
infinite subset.

As remarked in the introduction, one of the original motivations of the work
recounted in [Bomfim22] was to find a forcing axiom equivalent of
$\mathop{\hbox{Fin}}$. We give here a sketch of the principal results of
[Bomfim22] on this problem.

We introduce some relevant weakenings of the axiom of choice.

  * $\mathop{\rm AC}_{\omega}(\mathop{\hbox{Fin}})$ is the axiom of choice for countably families of finite sets.

  * $\mathop{\rm CUT}(\mathop{\hbox{Fin}})$ is: every countable union of finite sets is countable.

We also introduce notation for a specific class of forcings.

######  Notation 3.2.

Let $\Gamma$ be the class of pre-orders which are countable unions of finite
pre-orders.

Previously, Shannon ([Shannon90]), Herrlich ([Herrlich06] and Gomes da Silva
([Silva12]) had showed the following.

######  Proposition 3.3.

([Shannon90], Corollary 2) $\mathop{\rm
AC}_{\omega}(\mathop{\hbox{Fin}})\Longleftrightarrow\mathop{\rm
FA}_{\omega}(\Gamma)$.

######  Proposition 3.4.

([Herrlich06]) $\mathop{\rm
AC}_{\omega}(\mathop{\hbox{Fin}})\Longleftrightarrow\mathop{\rm
CUT}(\mathop{\hbox{Fin}})$.

######  Proposition 3.5.

([Silva12]) $\mathop{\hbox{Fin}}\Longleftrightarrow\mathop{\rm
DC}_{l,fp}+\mathop{\rm AC}_{\omega}(\mathop{\hbox{Fin}})$.

[Bomfim22] revisited the forcing axiom $\mathop{\rm FA}_{\omega}(\Gamma)$ used
in Shannon ([Shannon90]) and showed $\mathop{\hbox{Fin}}$ is a consequence of
the combination of this forcing axiom and another forcing axiom. In order to
state this result it is convenient to introduce notation for a another class
of forcings.

######  Notation 3.6.

Let $\Lambda$ be the class of trees $T$ such that there is a strict lattice
${\mathcal{L}}$ with fpp and no minimal elements and the elements of $T$ are
finite, strictly decreasing sequences from ${\mathcal{L}}$, ordered by
extension.

######  Theorem 3.7.

([Bomfim22], Theorem 3.2.5) $\mathop{\rm FA}_{\omega}(\Lambda)+\mathop{\rm
FA}_{\omega}(\Gamma)\Longrightarrow\mathop{\hbox{Fin}}$.

##  4 Principal results

######  Notation 4.1.

Let $\kappa$ be a regular cardinal. Let $X$ be an infinite set with
$X\not\prec\kappa$. $\mathop{\rm Coll}(\kappa,X)$ is the set $\\{\hskip
1.0ptf\in{{{}^{<\kappa}}X}\hskip 2.0pt|\hskip 2.0ptf\hbox{ is injective}\hskip
1.0pt\\}$, partially ordered by extension: $g<f\Longleftrightarrow\mathop{\rm
dom}(f)<\mathop{\rm dom}(g)\hskip 2.0pt\hskip 2.0pt\&\hskip 2.0pt\hskip
2.0ptg\upharpoonright\mathop{\rm dom}(f)=f$.

For $i<\kappa$ set $L^{X}_{i}=\\{\hskip 1.0ptf\in{{{}^{<\kappa}}X}\hskip
2.0pt|\hskip 2.0pti\leq\mathop{\rm dom}(f)\hskip 1.0pt\\}$. Write $L_{i}$ for
$L^{X}_{i}$ when the context is clear.

######  Note 4.2.

Given a regular cardinal $\kappa$ and an infinite set $X$ with
$X\not\prec\kappa$, for each $i<\kappa$ clearly $L^{X}_{i}\cap\mathop{\rm
Coll}(\kappa,X)$ is a dense, open subset of $\mathop{\rm Coll}(\kappa,X)$.

######  Notation 4.3.

Let $\kappa$ be a regular cardinal. Let ${\mathit{C}oll}_{\kappa}$ be the
class of partial orders of the form $\mathop{\rm Coll}(\kappa,X)$ for $X$
infinite with $X\not\prec\kappa$. Let ${\mathfrak{L}}^{\kappa}$ be the
Function such that for each infinite set $X$ with $X\not\prec\kappa$ we have
${\mathfrak{L}}^{\kappa}(\mathop{\rm Coll}(\kappa,X))=\langle\hskip
1.0ptL^{X}_{i}\hskip 2.0pt|\hskip 2.0pti<\kappa\hskip 1.0pt\rangle$.

Now we can state our first observations.

######  Proposition 4.4.

$\mathop{\rm
FA}({\mathit{C}oll_{\omega}},{\mathfrak{L}^{\omega}})\Longleftrightarrow\mathop{\hbox{Fin}}$.

###### Proof.

“$\Longrightarrow$”. We show that if $X$ is an infinite set then

$\mathop{\rm FA}(\mathop{\rm Coll}(\omega,X),\langle\hskip
1.0ptL^{X}_{n}\hskip 2.0pt|\hskip 2.0ptn<\omega\hskip
1.0pt\rangle)\Longrightarrow\hbox{ there is an injection of }\omega\hbox{ into
}X.$

This is immediate from the definitions once we work through them: if $G$ is
$\mathop{\rm Coll}(\omega,X)$-generic for $\langle\hskip 1.0ptL^{X}_{n}\hskip
2.0pt|\hskip 2.0ptn<\omega\hskip 1.0pt\rangle$ then $g=\bigcup
G:\omega\longrightarrow X$ is injective.

“$\Longleftarrow$”. Suppose $X$ is an infinite set. By $\mathop{\hbox{Fin}}$,
let $g:\omega\longrightarrow X$ be an injection. Set $G=\\{\hskip
1.0ptg\upharpoonright n\hskip 2.0pt|\hskip 2.0ptn\in\omega\hskip 1.0pt\\}$.
Then $G$ is $\mathop{\rm Coll}(\omega,X)$-generic for $\langle\hskip
1.0ptL^{X}_{n}\hskip 2.0pt|\hskip 2.0ptn<\omega\hskip 1.0pt\rangle$. ∎

The following is also immediate by a very similar proof.

######  Proposition 4.5.

If $\kappa$ is a regular cardinal, then $\mathop{\rm H\hskip
0.05pt}_{\kappa}\Longleftrightarrow\mathop{\rm
FA}({\mathit{C}oll_{\kappa}},{\mathfrak{L}}^{\kappa})$.

###### Proof.

Suppose $X\not\prec\kappa$. “$\Longrightarrow$”. If $g:\kappa\longrightarrow
X$ is as given by $\mathop{\rm H\hskip 0.05pt}_{\kappa}$, then $\\{\hskip
1.0ptg\upharpoonright i\hskip 2.0pt|\hskip 2.0pti<\kappa\hskip 1.0pt\\}$ is
$\mathop{\rm Coll}(\kappa,X)$-generic for $\langle\hskip 1.0ptL^{X}_{i}\hskip
2.0pt|\hskip 2.0pti<\kappa\hskip 1.0pt\rangle$. “$\Longleftarrow$”. Let $G$ be
$\mathop{\rm Coll}(\kappa,X)$-generic for $\langle\hskip 1.0ptL^{X}_{i}\hskip
2.0pt|\hskip 2.0pti<\kappa\hskip 1.0pt\rangle$. Then $\bigcup G$ is as
required for the instance of $\mathop{\rm H\hskip 0.05pt}_{\kappa}$ for
$F^{X}_{seq}$. ∎

######  Corollary 4.6.

$\mathop{\rm AC}\Longleftrightarrow\forall\hbox{ regular }\kappa\hskip
2.0pt\hskip 2.0pt\mathop{\rm
FA}({\mathit{C}oll_{\kappa}},{\mathfrak{L}}^{\kappa})$.

###### Proof.

Immediate from the previous proposition and Proposition (2.6). ∎

######  Corollary 4.7.

$\mathop{\rm AC}\Longleftrightarrow\forall\hbox{ regular }\kappa\hskip
2.0pt\hskip 2.0pt\mathop{\rm FA}_{\kappa}(\kappa\hbox{-closed})$

###### Proof.

“$\Longleftarrow$” is immediate from the previous corollary.
“$\Longrightarrow$” appears with a proof in [Viale19], Theorem (1.8), where it
is seemingly attributed as an unpublished result due to Todorčević.222“The
following formulation of the axiom of choice in terms of forcing axioms _has
been handed to me_ by Todorčević, I’m not aware of any published reference.”
([Viale19], p.7.). Our emphasis.  ∎

######  Definition 4.8.

Let $\kappa$ be a regular cardinal and $X$ a set such that $X\not\prec\kappa$.
Let $\mathbb{Q}^{X}_{\kappa}$ be the tree of sequences of length less than
$\kappa$ from $[X]^{<\omega}$ which are strictly increasing under $\subseteq$
and such that for $t\in\mathbb{Q}^{X}_{\kappa}$ we have $\mathop{\rm lh}(t)$
is a successor ordinal, for all limit $\lambda<\mathop{\rm lh}(t)$ we have
$t_{\lambda}=\bigcup{t_{i}}{i<\lambda}$ and for all $i+1<\mathop{\rm lh}(t)$
we have $\vbox{\halign{#\cr\cleaders\hrule height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{\vbox{\halign{#\cr\cleaders\hrule
height=0.2pt\hfill\crcr\kern
1.0pt\nointerlineskip\cr$\hfil\displaystyle{t_{i+1}\setminus
t_{i}}\hfil$\crcr}}}\hfil$\crcr}}=1$. Let $L^{X,\kappa,*}_{i}=\\{\hskip
1.0ptt\in\mathbb{Q}^{X}_{\kappa}\hskip 2.0pt|\hskip 2.0pti\leq\mathop{\rm
dom}(f)\hskip 1.0pt\\}$ and $L^{X,\kappa,*}=\langle\hskip
1.0ptL^{X,\kappa,*}_{i}\hskip 2.0pt|\hskip 2.0pti<\kappa\hskip 1.0pt\rangle$.
Let $\mathbb{Q}_{\kappa}=\\{\hskip 1.0pt\mathbb{Q}^{X}_{\kappa}\hskip
2.0pt|\hskip 2.0ptX\not\prec\kappa\hskip 1.0pt\\}$. Let
${\mathfrak{L}}^{\kappa,*}$ be the Function such that if $X$ a set with
$X\not\prec\kappa$ then ${\mathfrak{L}}^{\kappa,*}(X)=L^{X,\kappa,*}$.

######  Lemma 4.9.

Let $\kappa$ be a regular cardinal and $X$ a set such that $X\not\prec\kappa$.
There is a definable bijection between $\mathop{\rm Coll}(\kappa,X)$ to
$\mathbb{Q}^{X}_{\kappa}$.

###### Proof.

Given $f\in\mathbb{P}$ define $t^{f}=\langle\hskip 1.0pt\mathop{\rm
rge}(f\upharpoonright i)\hskip 2.0pt|\hskip 2.0pti\leq\mathop{\rm
dom}(f)\hskip 1.0pt\rangle$ and if $t\in\mathbb{Q}$ and $\mathop{\rm
lh}(t)=\beta+1$ define $f^{t}\in{{{}^{<\kappa}}X}$ by $f(i)$ is the unique
element of $t(i+1)\setminus t(i)$ for $i+1<\mathop{\rm lh}(t)$. ∎

######  Corollary 4.10.

If $\kappa$ is a regular cardinal, $\mathop{\rm
FA}(\mathbb{Q}_{\kappa},{\mathfrak{L}^{\kappa,*}})\Longleftrightarrow\mathop{\rm
FA}({\mathit{C}oll_{\kappa}},{\mathfrak{L}^{\kappa}})$.

######  Corollary 4.11.

If $\kappa$ is a regular cardinal then $\mathop{\rm H\hskip
0.05pt}_{\kappa}\Longleftrightarrow\mathop{\rm
FA}(\mathbb{Q}_{\kappa},{\mathfrak{L}^{\kappa,*}})$.

We next show how this result allows us to improve on Theorem (3.7).

######  Proposition 4.12.

$\mathop{\rm FA}_{\omega}(\Lambda)\Longrightarrow\mathop{\rm
FA}({\mathit{C}oll_{\omega}},{\mathfrak{L}}^{\omega})$.

###### Proof.

Let $X$ be an infinite set. Clearly $\mathbb{Q}^{X}_{\omega}\in\Lambda$, and
so $\mathop{\rm FA}_{\omega}(\Lambda)\Longrightarrow\mathop{\rm
FA}(\mathbb{Q}_{\omega},{\mathfrak{L}}^{\omega,*})$.

Consequently, by Lemma 4.9, $\mathop{\rm
FA}_{\omega}(\Lambda)\Longrightarrow\mathop{\rm
FA}({\mathit{C}oll}_{\omega},{\mathfrak{L}}^{\omega})$. ∎

######  Corollary 4.13.

$\mathop{\rm FA}_{\omega}(\Lambda)\Longrightarrow\mathop{\hbox{Fin}}$.

######  Corollary 4.14.

$\mathop{\rm FA}_{\omega}(\Lambda)\Longrightarrow\mathop{\rm
FA}_{\omega}(\Gamma)$.

Finally we show $\mathop{\rm
FA}_{\omega}(\Lambda)\Longleftrightarrow\mathop{\rm DC}$ and generalize this
to arbitrary regular $\kappa$ in place of $\omega$.

######  Definition 4.15.

Let $\kappa$ be an initial ordinal. If $X$ is a set with $X\not\prec\kappa$
let ${\mathcal{T}}^{X}_{\kappa}$ be the collection of $\kappa$-closed sub
partial orders of $\mathop{\rm Coll}(\kappa,X)$. Let ${\mathcal{T}}_{\kappa}$
be the class of all such partial orders.

######  Definition 4.16.

Let $\kappa$ be a initial ordinal. Let $\mathop{\rm DC}^{*}_{\kappa}$ be the
assertion that for any $X$ with $X\not\prec\kappa$ and any function
$F:(X)^{<\kappa}\longrightarrow{\mathcal{P}}(X)\setminus\\{\hskip
1.0pt\emptyset\hskip 1.0pt\\}$ such that for all $t\in(X)^{<\kappa}$ and all
$x\in F(t)$ we have $t\kern-0.25pt\raise
4.0pt\hbox{$\frown$}\kern-0.25ptx\in(X)^{\kappa}$ there is some
$g:\kappa\longrightarrow X$ such that for all $i<\kappa$ we have $g(i)\in
F(g\upharpoonright i)$.

######  Remark 4.17.

Let $\kappa$ be a cardinal. $\mathop{\rm DC}^{*}_{\kappa}$ is clearly
equivalent to the assertion that for any $X$ with $X\not\prec\kappa$ and any
function $F:{{}^{<\kappa}X}\longrightarrow{\mathcal{P}}(X)\setminus\\{\hskip
1.0pt\emptyset\hskip 1.0pt\\}$ such that for all $t\in{{}^{<\kappa}X}$ we have
$F(t)\subseteq F^{X}_{seq}(t)$ there is some $g:\kappa\longrightarrow X$ such
that for all $i<\kappa$ we have $g(i)\in F(g\upharpoonright i)$.

######  Proposition 4.18.

$\mathop{\rm DC}_{\kappa}\Longleftrightarrow\mathop{\rm
FA}({\mathcal{T}}_{\kappa},{\mathfrak{L}}^{\kappa})\Longleftrightarrow\mathop{\rm
DC}^{*}_{\kappa}$.

###### Proof.

In the course of the proof of [Viale19] Theorem 1.8, Viale shows $\mathop{\rm
DC}_{\kappa}$ implies $\mathop{\rm FA}_{\kappa}(\kappa\hbox{-closed})$,
apparently attributing the result to Todorˇcević, as discussed above (see
Footnote (2)). Consequently, $\mathop{\rm DC}_{\kappa}$ implies $\mathop{\rm
FA}({\mathcal{T}}_{\kappa},{\mathfrak{L}}^{\kappa})$.

Given $F$ as in the statement of $\mathop{\rm DC}^{*}_{\kappa}$ consider the
forcing obtained by considering the tree of sequences conforming to $F$: those
$t\in(X)^{<\kappa}$ such that for all $i<\mathop{\rm lh}(t)$ $t_{i}\in
F(t\upharpoonright i)$. By the hypothesis of $\mathop{\rm DC}^{*}_{\kappa}$
this gives a $\kappa$-closed sub-partial order of $\mathop{\rm Coll}_{\kappa}$
to which we can apply the forcing axiom $\mathop{\rm
FA}({\mathcal{T}}_{\kappa},{\mathfrak{L}}^{\kappa})$. The union of the generic
filter will, as in the proof of Porposition (4.5), give us an injection as
required for $\mathop{\rm DC}^{*}_{\kappa}$.

Finally, we need to show that $\mathop{\rm DC}^{*}_{\kappa}$ implies
$\mathop{\rm DC}_{\kappa}$.

The idea is to take sequences from ${{}^{<\kappa}X}$ and replace them by
sequences from $(X\times\kappa)^{<\kappa}$, where the second coordinate
contains a markers keeping score of how many times the element in the first
coordinate has already appeared in the original sequence. (Notice that this is
one of the few places where we do not have a “set-by-set” equivalence.)

Suppose $F:{{}^{<\kappa}X}\longrightarrow{\mathcal{P}}(X)\setminus\\{\hskip
1.0pt\emptyset\hskip 1.0pt\\}$. Fix $x_{0}\in X$.

For $\alpha<\kappa$ and $t\in{{}^{\alpha}X}$ define
$u^{t}\in(X\times\kappa)^{\alpha}$ by $u^{t}_{i}=\langle\hskip
1.0ptt_{i},\gamma^{t}_{i}\hskip 1.0pt\rangle$ where
$\gamma^{t}_{i}=\mathop{\rm otp}(\langle\hskip 1.0ptj<i\hskip 2.0pt|\hskip
2.0ptt_{j}=t_{i}\hskip 1.0pt\rangle)$.

We define
$G:(X\times\kappa)^{<\kappa}\longrightarrow{\mathcal{P}}(X\times\kappa)\setminus\\{\hskip
1.0pt\emptyset\hskip 1.0pt\\}$ by $G(u)=\\{\hskip 1.0pt\langle\hskip
1.0ptx,\delta^{t}_{x}\hskip 1.0pt\rangle\hskip 2.0pt|\hskip 2.0ptx\in
F(t)\hskip 2.0pt\hskip 2.0pt\&\hskip 2.0pt\hskip
2.0pt\delta^{t}_{x}=\mathop{\rm otp}(\langle\hskip 1.0ptj<i\hskip 2.0pt|\hskip
2.0ptt_{j}=x\hskip 1.0pt\rangle)\hskip 1.0pt\\}$ if $u=u^{t}_{i}$, and
$G(u)=\\{\hskip 1.0pt\langle\hskip 1.0ptx_{0},0\hskip 1.0pt\rangle\hskip
1.0pt\\}$ otherwise.

We can thus apply $\mathop{\rm DC}^{*}_{\kappa}$ to $G$ giving us a function
$g:\kappa\longrightarrow X\times\kappa$ such that for all $i<\kappa$ we have
$g(i)\in G(g\upharpoonright i)$.

Now we can define $f:\kappa\longrightarrow X$ by $f(i)=g(i)_{0}$, the first
coordinate of the pair $g(i)$. By induction we have that $f(i)\in
F(f\upharpoonright i)$ as required. ∎

######  Corollary 4.19.

$\mathop{\rm DC}\Longleftrightarrow\mathop{\rm FA}_{\omega}(\Lambda)$.

###### Proof.

“$\Longrightarrow$”. $\mathop{\rm DC}$ implies $\mathop{\rm
FA}_{\omega}(\\{\hskip 1.0pt\mathbb{P}\hskip 2.0pt|\hskip
2.0pt\mathbb{P}\hbox{ is a po}\hskip 1.0pt\\})$, hence implies $\mathop{\rm
FA}_{\omega}(\Lambda)$. “$\Longleftarrow$” $\mathop{\rm FA}_{\omega}(\Lambda)$
implies $\mathop{\rm FA}({\mathit{C}oll_{\omega}},{\mathfrak{L}^{\omega}})$,
by Corollary (4.11), and this in turn implies $\mathop{\rm DC}$ by the
previous proposition in the case $\kappa=\omega$. ∎

######  Corollary 4.20.

$\mathop{\rm AC}\Longleftrightarrow\hbox{ for all regular }\kappa\hbox{ we
have }\mathop{\rm FA}({\mathcal{T}}_{\kappa},{\mathfrak{L}}^{\kappa})$.

## References

  * [Bomfim22] Diego Lima Bomfim, _Axiomas de Forcing e Princípios de Escolha_ , Master’s Dissertation, Universidade Federal de Bahia, (2022).
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  * [Hartogs1915] Friedrich Hartogs, ‘Uber das Problem der Wohlordnung’, Math. Annalen, 76, 436-443, (1915).
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  * [Levy64] Azriel Lévy, ‘ The interdependence of certain consequences of the axiom of choice’, Fundamenta mathematicae, 54(2), 135–157, (1964).
  * [Parente15] Francesco Parente, _Boolean valued models, saturation, forcing axioms_ , Ph.D. thesis, University of Pisa, (2015).
  * [RubinRubin63] Herman Rubin and Jean Rubin, _Equivalents of the Axiom of Choice_. North Holland, (1963).
  * [Silva12] Samuel Gomes da Silva, ‘A topological statement and its relation to certain weak choice principles’, Questions and Answers in General Topology, 30(1), 1-8, (2012).
  * [Shannon90] Gary Shannon, ‘Provable forms of Martin’s axiom’, Notre Dame Journal of Formal Logic, 31(3), 382-388, (1990).
  * [Viale19] Matteo Viale, ‘Useful axioms’, https://arxiv.org/abs/1610.02832v3, (2019).

Departmento de Matemática, Universidade Federal de Bahia, Av. Adhemar de
Barros, S/N, Ondina, Salvador, Bahia, Brazil. CEP: 40170-110.

E-mail address, Diego Lima Bomfim: bomfim.diego<EMAIL_ADDRESS>E-mail
address, Charles Morgan<EMAIL_ADDRESS>E-mail address, Samuel Gomes da
Silva<EMAIL_ADDRESS>