<p>We study the generalized dominating number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathfrak {d}_{\mu }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">d</mi> <mi>μ</mi> </msub> </math></EquationSource> </InlineEquation> at a singular cardinal <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> of cofinality <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>. We prove two basic lower bounds: in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\text {ZFC}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>ZFC</mtext> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\text {cf}\left( {[\mu ]^\kappa ,\subseteq }\right) \le \mathfrak {d}_{\mu }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>cf</mtext> <mfenced close=")" open="("> <mrow> <msup> <mrow> <mo stretchy="false">[</mo> <mi>μ</mi> <mo stretchy="false">]</mo> </mrow> <mi>κ</mi> </msup> <mo>,</mo> <mo>⊆</mo> </mrow> </mfenced> <mo>≤</mo> <msub> <mi mathvariant="fraktur">d</mi> <mi>μ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, and under mild cardinal-arithmetic assumptions, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$2^{<\mu } \le \mathfrak {d}_{\mu }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mrow> <mo><</mo> <mi>μ</mi> </mrow> </msup> <mo>≤</mo> <msub> <mi mathvariant="fraktur">d</mi> <mi>μ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. We also clarify when <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathfrak {d}_{\mu }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">d</mi> <mi>μ</mi> </msub> </math></EquationSource> </InlineEquation> can differ from <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$2^\mu $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>μ</mi> </msup> </math></EquationSource> </InlineEquation>: assuming <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\text {GCH}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>GCH</mtext> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\kappa = \text {cf}\left( {\mu }\right) > \omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>=</mo> <mtext>cf</mtext> <mfenced close=")" open="("> <mi>μ</mi> </mfenced> <mo>></mo> <mi>ω</mi> </mrow> </math></EquationSource> </InlineEquation>, a finite-support iteration of Cohen forcing of length <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mu ^{++}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>μ</mi> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> yields <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathfrak {d}_{\mu }< 2^\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="fraktur">d</mi> <mi>μ</mi> </msub> <mo><</mo> <msup> <mn>2</mn> <mi>μ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. On the other hand, for <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\kappa = \text {cf}\left( {\mu }\right) = \omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>=</mo> <mtext>cf</mtext> <mfenced close=")" open="("> <mi>μ</mi> </mfenced> <mo>=</mo> <mi>ω</mi> </mrow> </math></EquationSource> </InlineEquation>, natural <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-cc posets force <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\mathfrak {d}_{\mu }= 2^\mu .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="fraktur">d</mi> <mi>μ</mi> </msub> <mo>=</mo> <msup> <mn>2</mn> <mi>μ</mi> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

Dominating numbers at singular cardinals

期刊论文
发表日期：

2026年3月19日

查看全文

Yusuke Hayashi

摘要

We study the generalized dominating number $\mathfrak {d}_{\mu }$ $d_{μ}$ at a singular cardinal $\mu $ $μ$ of cofinality $\kappa $ $κ$ . We prove two basic lower bounds: in $\text {ZFC}$ $ZFC$ , $\text {cf}\left( {[\mu ]^\kappa ,\subseteq }\right) \le \mathfrak {d}_{\mu }$ $cf ({[μ]}^{κ}, \subseteq) \leq d_{μ}$ , and under mild cardinal-arithmetic assumptions, $2^{<\mu } \le \mathfrak {d}_{\mu }$ $2^{< μ} \leq d_{μ}$ . We also clarify when $\mathfrak {d}_{\mu }$ $d_{μ}$ can differ from $2^\mu $ $2^{μ}$ : assuming $\text {GCH}$ $GCH$ and $\kappa = \text {cf}\left( {\mu }\right) > \omega $ $κ = cf (μ) > ω$ , a finite-support iteration of Cohen forcing of length $\mu ^{++}$ $μ^{+ +}$ yields $\mathfrak {d}_{\mu }< 2^\mu $ $d_{μ} < 2^{μ}$ . On the other hand, for $\kappa = \text {cf}\left( {\mu }\right) = \omega $ $κ = cf (μ) = ω$ , natural $\mu $ $μ$ -cc posets force $\mathfrak {d}_{\mu }= 2^\mu .$ $d_{μ} = 2^{μ} .$