Almost free modules, PCF theory bound on $2^{aleph_omega}$












0












$begingroup$


Why here (Almost Free Modules: Set-theoretic Methods by P.C. Eklof, A.H. Mekler), on the page 181, in the 3rd line there is
$$2^{aleph_omega}<aleph_{omega_2}$$? How the index $2$ in the r.h.s. was created?




The notation "pcf" stands for possible cofinality. Shelahintroduced a powerful tool into the study of cardinal arithmetic when he considered (for an infinite set $A$ of regular cardinals) the set $operatorname{pcf}(A)$, defined to be the set of all cardinals $lambda$ such that for some ultrafilter $U$ on $A$, $lambda$ is the cofinality of the ultraproduct of ordered sets $prod_{kappain A} kappa/U$. Among the striking results he proved (in ZFC) are: if $2^{aleph_n}<aleph_omega$ for all $ninomega$ then $2^{aleph_omega}<min{aleph_{(2^{aleph_0})^+},aleph_{omega_4}}$; in particular, if GCH holds below $aleph_omega$, then $2^{aleph_omega}<aleph_{omega_2}$. (See Shelah 1992, Shelah 1994 or Jech 1995 or Burke-Magidor 1990.)




page 180



page 181






set-theory cardinals







share|cite|improve this question











$endgroup$












  • $begingroup$
    could someone tell me what $omega_2$ is supposed to mean? I've never seen this notation before.
    $endgroup$
    – Pink Panther
    Dec 18 '18 at 21:41






  • 1




    $begingroup$
    @PinkPanther: It is the second uncountable cardinal, or the third infinite cardinal.
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '18 at 22:07






  • 2




    $begingroup$
    I don't understand the question; the text you quoted gives the reason for the bound and provides references for the proof. It is a combination of two results of Shelah. One first appeared in the first edition of his "Proper forcing" book (it is not in the revised edition), and the other is in his "Cardinal arithmetic" book.
    $endgroup$
    – Andrés E. Caicedo
    Dec 19 '18 at 12:26










  • $begingroup$
    OK then. I just wondered if $2$ should be read as $4$, but you say that "no".
    $endgroup$
    – user122424
    Dec 19 '18 at 15:03








  • 2




    $begingroup$
    Yes, in fact you do not need the results from "Cardinal arithmetic", just the earlier result in the Proper forcing book.
    $endgroup$
    – Andrés E. Caicedo
    Dec 19 '18 at 16:35
















0












$begingroup$


Why here (Almost Free Modules: Set-theoretic Methods by P.C. Eklof, A.H. Mekler), on the page 181, in the 3rd line there is
$$2^{aleph_omega}<aleph_{omega_2}$$? How the index $2$ in the r.h.s. was created?




The notation "pcf" stands for possible cofinality. Shelahintroduced a powerful tool into the study of cardinal arithmetic when he considered (for an infinite set $A$ of regular cardinals) the set $operatorname{pcf}(A)$, defined to be the set of all cardinals $lambda$ such that for some ultrafilter $U$ on $A$, $lambda$ is the cofinality of the ultraproduct of ordered sets $prod_{kappain A} kappa/U$. Among the striking results he proved (in ZFC) are: if $2^{aleph_n}<aleph_omega$ for all $ninomega$ then $2^{aleph_omega}<min{aleph_{(2^{aleph_0})^+},aleph_{omega_4}}$; in particular, if GCH holds below $aleph_omega$, then $2^{aleph_omega}<aleph_{omega_2}$. (See Shelah 1992, Shelah 1994 or Jech 1995 or Burke-Magidor 1990.)




page 180



page 181






set-theory cardinals







share|cite|improve this question











$endgroup$












  • $begingroup$
    could someone tell me what $omega_2$ is supposed to mean? I've never seen this notation before.
    $endgroup$
    – Pink Panther
    Dec 18 '18 at 21:41






  • 1




    $begingroup$
    @PinkPanther: It is the second uncountable cardinal, or the third infinite cardinal.
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '18 at 22:07






  • 2




    $begingroup$
    I don't understand the question; the text you quoted gives the reason for the bound and provides references for the proof. It is a combination of two results of Shelah. One first appeared in the first edition of his "Proper forcing" book (it is not in the revised edition), and the other is in his "Cardinal arithmetic" book.
    $endgroup$
    – Andrés E. Caicedo
    Dec 19 '18 at 12:26










  • $begingroup$
    OK then. I just wondered if $2$ should be read as $4$, but you say that "no".
    $endgroup$
    – user122424
    Dec 19 '18 at 15:03








  • 2




    $begingroup$
    Yes, in fact you do not need the results from "Cardinal arithmetic", just the earlier result in the Proper forcing book.
    $endgroup$
    – Andrés E. Caicedo
    Dec 19 '18 at 16:35














0












0








0





$begingroup$


Why here (Almost Free Modules: Set-theoretic Methods by P.C. Eklof, A.H. Mekler), on the page 181, in the 3rd line there is
$$2^{aleph_omega}<aleph_{omega_2}$$? How the index $2$ in the r.h.s. was created?




The notation "pcf" stands for possible cofinality. Shelahintroduced a powerful tool into the study of cardinal arithmetic when he considered (for an infinite set $A$ of regular cardinals) the set $operatorname{pcf}(A)$, defined to be the set of all cardinals $lambda$ such that for some ultrafilter $U$ on $A$, $lambda$ is the cofinality of the ultraproduct of ordered sets $prod_{kappain A} kappa/U$. Among the striking results he proved (in ZFC) are: if $2^{aleph_n}<aleph_omega$ for all $ninomega$ then $2^{aleph_omega}<min{aleph_{(2^{aleph_0})^+},aleph_{omega_4}}$; in particular, if GCH holds below $aleph_omega$, then $2^{aleph_omega}<aleph_{omega_2}$. (See Shelah 1992, Shelah 1994 or Jech 1995 or Burke-Magidor 1990.)




page 180



page 181






set-theory cardinals







share|cite|improve this question











$endgroup$




Why here (Almost Free Modules: Set-theoretic Methods by P.C. Eklof, A.H. Mekler), on the page 181, in the 3rd line there is
$$2^{aleph_omega}<aleph_{omega_2}$$? How the index $2$ in the r.h.s. was created?




The notation "pcf" stands for possible cofinality. Shelahintroduced a powerful tool into the study of cardinal arithmetic when he considered (for an infinite set $A$ of regular cardinals) the set $operatorname{pcf}(A)$, defined to be the set of all cardinals $lambda$ such that for some ultrafilter $U$ on $A$, $lambda$ is the cofinality of the ultraproduct of ordered sets $prod_{kappain A} kappa/U$. Among the striking results he proved (in ZFC) are: if $2^{aleph_n}<aleph_omega$ for all $ninomega$ then $2^{aleph_omega}<min{aleph_{(2^{aleph_0})^+},aleph_{omega_4}}$; in particular, if GCH holds below $aleph_omega$, then $2^{aleph_omega}<aleph_{omega_2}$. (See Shelah 1992, Shelah 1994 or Jech 1995 or Burke-Magidor 1990.)




page 180



page 181






set-theory cardinals




set-theory cardinals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 19 '18 at 18:10









Lord_Farin

15.6k636108




15.6k636108










asked Dec 18 '18 at 21:13









user122424user122424

1,1232716




1,1232716












  • $begingroup$
    could someone tell me what $omega_2$ is supposed to mean? I've never seen this notation before.
    $endgroup$
    – Pink Panther
    Dec 18 '18 at 21:41






  • 1




    $begingroup$
    @PinkPanther: It is the second uncountable cardinal, or the third infinite cardinal.
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '18 at 22:07






  • 2




    $begingroup$
    I don't understand the question; the text you quoted gives the reason for the bound and provides references for the proof. It is a combination of two results of Shelah. One first appeared in the first edition of his "Proper forcing" book (it is not in the revised edition), and the other is in his "Cardinal arithmetic" book.
    $endgroup$
    – Andrés E. Caicedo
    Dec 19 '18 at 12:26










  • $begingroup$
    OK then. I just wondered if $2$ should be read as $4$, but you say that "no".
    $endgroup$
    – user122424
    Dec 19 '18 at 15:03








  • 2




    $begingroup$
    Yes, in fact you do not need the results from "Cardinal arithmetic", just the earlier result in the Proper forcing book.
    $endgroup$
    – Andrés E. Caicedo
    Dec 19 '18 at 16:35


















  • $begingroup$
    could someone tell me what $omega_2$ is supposed to mean? I've never seen this notation before.
    $endgroup$
    – Pink Panther
    Dec 18 '18 at 21:41






  • 1




    $begingroup$
    @PinkPanther: It is the second uncountable cardinal, or the third infinite cardinal.
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '18 at 22:07






  • 2




    $begingroup$
    I don't understand the question; the text you quoted gives the reason for the bound and provides references for the proof. It is a combination of two results of Shelah. One first appeared in the first edition of his "Proper forcing" book (it is not in the revised edition), and the other is in his "Cardinal arithmetic" book.
    $endgroup$
    – Andrés E. Caicedo
    Dec 19 '18 at 12:26










  • $begingroup$
    OK then. I just wondered if $2$ should be read as $4$, but you say that "no".
    $endgroup$
    – user122424
    Dec 19 '18 at 15:03








  • 2




    $begingroup$
    Yes, in fact you do not need the results from "Cardinal arithmetic", just the earlier result in the Proper forcing book.
    $endgroup$
    – Andrés E. Caicedo
    Dec 19 '18 at 16:35
















$begingroup$
could someone tell me what $omega_2$ is supposed to mean? I've never seen this notation before.
$endgroup$
– Pink Panther
Dec 18 '18 at 21:41




$begingroup$
could someone tell me what $omega_2$ is supposed to mean? I've never seen this notation before.
$endgroup$
– Pink Panther
Dec 18 '18 at 21:41




1




1




$begingroup$
@PinkPanther: It is the second uncountable cardinal, or the third infinite cardinal.
$endgroup$
– Asaf Karagila♦
Dec 18 '18 at 22:07




$begingroup$
@PinkPanther: It is the second uncountable cardinal, or the third infinite cardinal.
$endgroup$
– Asaf Karagila♦
Dec 18 '18 at 22:07




2




2




$begingroup$
I don't understand the question; the text you quoted gives the reason for the bound and provides references for the proof. It is a combination of two results of Shelah. One first appeared in the first edition of his "Proper forcing" book (it is not in the revised edition), and the other is in his "Cardinal arithmetic" book.
$endgroup$
– Andrés E. Caicedo
Dec 19 '18 at 12:26




$begingroup$
I don't understand the question; the text you quoted gives the reason for the bound and provides references for the proof. It is a combination of two results of Shelah. One first appeared in the first edition of his "Proper forcing" book (it is not in the revised edition), and the other is in his "Cardinal arithmetic" book.
$endgroup$
– Andrés E. Caicedo
Dec 19 '18 at 12:26












$begingroup$
OK then. I just wondered if $2$ should be read as $4$, but you say that "no".
$endgroup$
– user122424
Dec 19 '18 at 15:03






$begingroup$
OK then. I just wondered if $2$ should be read as $4$, but you say that "no".
$endgroup$
– user122424
Dec 19 '18 at 15:03






2




2




$begingroup$
Yes, in fact you do not need the results from "Cardinal arithmetic", just the earlier result in the Proper forcing book.
$endgroup$
– Andrés E. Caicedo
Dec 19 '18 at 16:35




$begingroup$
Yes, in fact you do not need the results from "Cardinal arithmetic", just the earlier result in the Proper forcing book.
$endgroup$
– Andrés E. Caicedo
Dec 19 '18 at 16:35










1 Answer
1






active

oldest

votes


















1












$begingroup$

The $aleph_{omega_2}$ arises because under the assumption of GCH below $aleph_omega$, we have:



$$(2^{aleph_0})^+ = aleph_1^+ = omega_2$$



so that the minimum $min{aleph_{(2^{aleph_0})^+},aleph_{omega_4}}$ is actually not $aleph_{omega_4}$, but the other argument, which evaluates to $aleph_{omega_2}$.






share|cite|improve this answer









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3045703%2falmost-free-modules-pcf-theory-bound-on-2-aleph-omega%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    The $aleph_{omega_2}$ arises because under the assumption of GCH below $aleph_omega$, we have:



    $$(2^{aleph_0})^+ = aleph_1^+ = omega_2$$



    so that the minimum $min{aleph_{(2^{aleph_0})^+},aleph_{omega_4}}$ is actually not $aleph_{omega_4}$, but the other argument, which evaluates to $aleph_{omega_2}$.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      The $aleph_{omega_2}$ arises because under the assumption of GCH below $aleph_omega$, we have:



      $$(2^{aleph_0})^+ = aleph_1^+ = omega_2$$



      so that the minimum $min{aleph_{(2^{aleph_0})^+},aleph_{omega_4}}$ is actually not $aleph_{omega_4}$, but the other argument, which evaluates to $aleph_{omega_2}$.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        The $aleph_{omega_2}$ arises because under the assumption of GCH below $aleph_omega$, we have:



        $$(2^{aleph_0})^+ = aleph_1^+ = omega_2$$



        so that the minimum $min{aleph_{(2^{aleph_0})^+},aleph_{omega_4}}$ is actually not $aleph_{omega_4}$, but the other argument, which evaluates to $aleph_{omega_2}$.






        share|cite|improve this answer









        $endgroup$



        The $aleph_{omega_2}$ arises because under the assumption of GCH below $aleph_omega$, we have:



        $$(2^{aleph_0})^+ = aleph_1^+ = omega_2$$



        so that the minimum $min{aleph_{(2^{aleph_0})^+},aleph_{omega_4}}$ is actually not $aleph_{omega_4}$, but the other argument, which evaluates to $aleph_{omega_2}$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 19 '18 at 18:10









        Lord_FarinLord_Farin

        15.6k636108




        15.6k636108






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3045703%2falmost-free-modules-pcf-theory-bound-on-2-aleph-omega%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Wiesbaden

            Image
            Wappen Deutschlandkarte 50.0825 8.24 117 Koordinaten: 50° 5′  N , 8° 14′  O Basisdaten Bundesland: Hessen Regierungsbezirk: Darmstadt Höhe: 117 m ü. NHN Fläche: 203,93 km 2 Einwohner: 278.654 (31. Dez. 2017) [1] Bevölkerungsdichte: 1366 Einwohner je km 2 Postleitzahlen: 65183–65207, 55246, 55252 Vorlage:Infobox Gemeinde in Deutschland/Wartung/PLZ enthält Text Vorwahlen: 0611, 06122, 06127, 06134 Kfz-Kennzeichen: WI Gemeindeschlüssel: 06 4 14 000 LOCODE: DE WIB NUTS: DE714 Stadtgliederung: 26 Ortsbezirke Adresse der Stadtverwaltung: Schlossplatz 6 65183 Wiesbaden Webpräsenz: www.wiesbaden.de Oberbürgermeister: Sven Gerich (SPD) Lage der Landeshauptstadt Wiesbaden in Hessen und im Regierungsbezirk Darmstadt Offizielles Logo der Landeshauptstadt Wiesbaden Flagge der Landeshauptstadt Wiesbaden Wiesbaden is
            Read more

            Marschland

            Image
            Kilometerweite Marsch in den Vier- und Marschlanden am Elbdeich Die Wedeler Marschlandschaft in Schleswig-Holstein Der Hadelner Kanal zur Entwässerung der Marsch Die tief gelegene Marsch und ein Entwässerungskanal Das Schöpfwerk Otterndorf im Land Hadeln zur Entwässerung der Medem verfügte damals bereits über die größte Kreiselpumpe Europas. Das Schöpfwerk in Neuhaus für die Aue und des Neuhaus-Bülkauer Kanals Als Marsch(land) (v. niederdt., altsächs. mersc ) – auch Masch , Mersch oder Schwemmland genannt – bezeichnet man eine nacheiszeitlich entstandene geomorphologische Landform im Gebiet der nordwestdeutschen Küsten und Flüsse sowie vergleichbare Landformen weltweit. Inhaltsverzeichnis 1 Landschaft 2 Entstehung 3 Marschböden 4 Nutzung 5 Marschgebiete in Deutschland 6 Siehe auch 7 Literatur 8 Weblinks 9 Einzelnachweise Landschaft | Marschen sind generell flache Landstriche ohne natürliche Erhebungen. Si
            Read more

            Dieringhausen

            Image
            Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
            Read more