<p>Under the assumption of small violations of choice with seed <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1533_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(S\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq111"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1533_Article_IEq111.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">\((SVC(S))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mi>V</mi> <mi>C</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the failure of many choice principles reflect to local properties of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1533_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(S\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation>, which can be a helpful characterisation for preservation proofs. We demonstrate the reflections of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1533_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(DC\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">DC</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1533_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(AC_\lambda\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msub> <mi>C</mi> <mi>λ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1533_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(PP\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">PP</mi> </mrow> </math></EquationSource> </InlineEquation>, and other important forms of choice. As a consequence, we show that if <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1533_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(S\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation> is infinite then <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1533_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(S\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation> can be partitioned into <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1533_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> many non-empty subsets.</p>

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Local reflections of choice

  • C. Ryan-Smith

摘要

Under the assumption of small violations of choice with seed \(S\) S \((SVC(S))\) ( S V C ( S ) ) , the failure of many choice principles reflect to local properties of \(S\) S , which can be a helpful characterisation for preservation proofs. We demonstrate the reflections of \(DC\) DC , \(AC_\lambda\) A C λ , \(PP\) PP , and other important forms of choice. As a consequence, we show that if \(S\) S is infinite then \(S\) S can be partitioned into \(\omega\) ω many non-empty subsets.