The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters. Issue 4 (5th August 2013)

• Record Type: Journal Article
• Authors: Hall, Eric J.; Keremedis, Kyriakos; Tachtsis, Eleftherios
• Is Part Of: Mathematical logic quarterly. Volume 59:Issue 4/5(2013)
• Page Start: 258
• Page End: 267
• Publication Date: 2013-08-05
• Subjects: Mathematics -- Periodicals; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1002/malq.201100092
• Languages: English
• ISSNs: 0942-5616

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>Let <italic>X</italic> be an infinite set and let <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z3d2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="sans-serif">BPI</mml:mi><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2zc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="sans-serif">UF</mml:mi><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> denote the propositions "<italic>every filter on</italic><italic>X</italic><italic>can be extended to an ultrafilter</italic>" and "<italic>X</italic><italic>has a free ultrafilter</italic>", respectively. We denote by <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2w8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> the <italic>Stone space</italic> of the Boolean algebra of all subsets of <italic>X</italic>. We show: <list id="malq201100092-list-0001" list-type="order"><list-item><p>For every well‐ordered cardinal number ℵ, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2vq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="sans-serif">UF</mml:mi></mml:math></alternatives>(ℵ) iff <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2sm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="sans-serif">UF</mml:mi></mml:math></alternatives>(2<sup>ℵ</sup>).</p></list-item><list-item><p><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z379" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="sans-serif">UF</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> iff "<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z356" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn mathvariant="bold">2</mml:mn><mml:mi>ω</mml:mi></mml:msup></mml:math></alternatives><italic>is a continuous image of</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z34n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives>" iff "<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z310" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives><italic>has a free open ultrafilter</italic> " iff "<italic>every countably infinite subset of</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2bf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives><italic>has a limit point</italic>".</p></list-item><list-item><p><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2dj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0011" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="sans-serif">BPI</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> implies "<italic>every open filter on</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2c0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0012" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn mathvariant="bold">2</mml:mn><mml:mi>ω</mml:mi></mml:msup></mml:math></alternatives><italic>extends to an open ultrafilter</italic>" implies "<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z1x9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0013" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn mathvariant="bold">2</mml:mn><mml:mi>ω</mml:mi></mml:msup></mml:math></alternatives><italic>has an open ultrafilter</italic>" implies <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z1tn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0014" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="sans-serif">UF</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mo>)</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></p></list-item><list-item><p>It is relatively consistent with <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z221" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0015" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="sans-serif">ZF</mml:mi></mml:math></alternatives> that <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z20x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0016" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="sans-serif">UF</mml:mi></mml:math></alternatives>(ω) holds, whereas <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z267" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0017" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="sans-serif">BPI</mml:mi></mml:math></alternatives>(ω) fails. In particular, none of the statements given in (2) implies <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z244" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100092:equation:malq201100092-math-0018" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="sans-serif">BPI</mml:mi></mml:math></alternatives>(ω).</p></list-item></list></p> </abstract>