<p>We investigate whether the ultrafilter number function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\kappa \mapsto \mathfrak {u}(\kappa )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>↦</mo> <mi mathvariant="fraktur">u</mi> <mo stretchy="false">(</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on the cardinals is monotone, that is, whether <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathfrak {u}(\lambda ) \le \mathfrak {u}(\kappa )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">u</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi mathvariant="fraktur">u</mi> <mo stretchy="false">(</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> holds for all cardinals <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda < \kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo><</mo> <mi>κ</mi> </mrow> </math></EquationSource> </InlineEquation> or not. We show that monotonicity can fail, but the failure has large cardinal strength. On the other hand, we prove that there are many restrictions of the failure of monotonicity. For instance, if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> is a singular cardinal with countable cofinality or a strong limit singular cardinal, then <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathfrak {u}(\kappa ) \le \mathfrak {u}(\kappa ^+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">u</mi> <mrow> <mo stretchy="false">(</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi mathvariant="fraktur">u</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>κ</mi> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> holds.</p>

Monotonicity of the ultrafilter number function

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2026年3月12日

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Toshimichi Usuba

摘要

We investigate whether the ultrafilter number function $\kappa \mapsto \mathfrak {u}(\kappa )$ $κ \mapsto u (κ)$ on the cardinals is monotone, that is, whether $\mathfrak {u}(\lambda ) \le \mathfrak {u}(\kappa )$ $u (λ) \leq u (κ)$ holds for all cardinals $\lambda < \kappa $ $λ < κ$ or not. We show that monotonicity can fail, but the failure has large cardinal strength. On the other hand, we prove that there are many restrictions of the failure of monotonicity. For instance, if $\kappa $ $κ$ is a singular cardinal with countable cofinality or a strong limit singular cardinal, then $\mathfrak {u}(\kappa ) \le \mathfrak {u}(\kappa ^+)$ $u (κ) \leq u (κ^{+})$ holds.