Assuming GCH: if $mathrm{cf}(kappa) leq lambda < kappa$, then $kappa^lambda = kappa^+$ (Jech Theorem 5.15)











up vote
1
down vote

favorite












I am trying to fill in the details for part (ii) of Theorem 5.15 in Jech's Set Theory:




Theorem 5.15 If GCH holds and $kappa$ and $lambda$ are infinite cardinals, then



(i) If $kappa leq lambda$, then $kappa^lambda = lambda^+$.



(ii) If $mathrm{cf}(kappa) leq lambda < kappa$, then $kappa^lambda = kappa^+$.



(iii) If $lambda < mathrm{cf}(kappa)$, then $kappa^lambda = kappa$.




In the proof of (ii) he just states that it follows from these two lemmas:




Lemma 5.7 If $|A| = kappa geq lambda$, then the set $[A]^lambda$ has cardinality $kappa^lambda.$




(Here $[A]^lambda = {X subset A : |X| = lambda}.$)




Lemma 5.8 If $lambda$ is an infinite cardinal and $kappa_i > 0$ for each $i < lambda$, then



$$sum_{i<lambda}{kappa_i} = lambdacdotsup_{i<lambda}{kappa_i}.$$




I am struggling to construct an explicit proof using these results and a few facts about cardinal arithmetic such as absorption for infinite cardinals and cardinalities of power sets.



Since, assuming GCH, $kappa^+ = 2^kappa = |mathcal{P}(A)|$ for some $A$ with cardinality $kappa$, I thought I could come up with some sequence ${mu_i : i < mathrm{cf}(kappa)}$ such that



$$kappa^+ = 2^kappa = |mathcal{P}(A)| = sum_{i < mathrm{cf}(kappa)} |[A]^{mu_i}| = sum_{i < mathrm{cf}(kappa)} kappa^{mu_i} = mathrm{cf}(kappa)cdotsup_{i<mathrm{cf}(kappa)}kappa^{mu_i} = mathrm{cf}(kappa)cdotkappa^lambda = kappa^lambda$$



Is this approach a good idea? How would I go about finding the appropriate $mu_i$? I suppose it will have to make use of the assumption that $mathrm{cf}(kappa) leq lambda < kappa$, but I don't see how.



I'd much appreciate any hints on how to proceed (either with my suggestion or another way).






set-theory cardinals







share|cite|improve this question




























    up vote
    1
    down vote

    favorite












    I am trying to fill in the details for part (ii) of Theorem 5.15 in Jech's Set Theory:




    Theorem 5.15 If GCH holds and $kappa$ and $lambda$ are infinite cardinals, then



    (i) If $kappa leq lambda$, then $kappa^lambda = lambda^+$.



    (ii) If $mathrm{cf}(kappa) leq lambda < kappa$, then $kappa^lambda = kappa^+$.



    (iii) If $lambda < mathrm{cf}(kappa)$, then $kappa^lambda = kappa$.




    In the proof of (ii) he just states that it follows from these two lemmas:




    Lemma 5.7 If $|A| = kappa geq lambda$, then the set $[A]^lambda$ has cardinality $kappa^lambda.$




    (Here $[A]^lambda = {X subset A : |X| = lambda}.$)




    Lemma 5.8 If $lambda$ is an infinite cardinal and $kappa_i > 0$ for each $i < lambda$, then



    $$sum_{i<lambda}{kappa_i} = lambdacdotsup_{i<lambda}{kappa_i}.$$




    I am struggling to construct an explicit proof using these results and a few facts about cardinal arithmetic such as absorption for infinite cardinals and cardinalities of power sets.



    Since, assuming GCH, $kappa^+ = 2^kappa = |mathcal{P}(A)|$ for some $A$ with cardinality $kappa$, I thought I could come up with some sequence ${mu_i : i < mathrm{cf}(kappa)}$ such that



    $$kappa^+ = 2^kappa = |mathcal{P}(A)| = sum_{i < mathrm{cf}(kappa)} |[A]^{mu_i}| = sum_{i < mathrm{cf}(kappa)} kappa^{mu_i} = mathrm{cf}(kappa)cdotsup_{i<mathrm{cf}(kappa)}kappa^{mu_i} = mathrm{cf}(kappa)cdotkappa^lambda = kappa^lambda$$



    Is this approach a good idea? How would I go about finding the appropriate $mu_i$? I suppose it will have to make use of the assumption that $mathrm{cf}(kappa) leq lambda < kappa$, but I don't see how.



    I'd much appreciate any hints on how to proceed (either with my suggestion or another way).






    set-theory cardinals







    share|cite|improve this question


























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I am trying to fill in the details for part (ii) of Theorem 5.15 in Jech's Set Theory:




      Theorem 5.15 If GCH holds and $kappa$ and $lambda$ are infinite cardinals, then



      (i) If $kappa leq lambda$, then $kappa^lambda = lambda^+$.



      (ii) If $mathrm{cf}(kappa) leq lambda < kappa$, then $kappa^lambda = kappa^+$.



      (iii) If $lambda < mathrm{cf}(kappa)$, then $kappa^lambda = kappa$.




      In the proof of (ii) he just states that it follows from these two lemmas:




      Lemma 5.7 If $|A| = kappa geq lambda$, then the set $[A]^lambda$ has cardinality $kappa^lambda.$




      (Here $[A]^lambda = {X subset A : |X| = lambda}.$)




      Lemma 5.8 If $lambda$ is an infinite cardinal and $kappa_i > 0$ for each $i < lambda$, then



      $$sum_{i<lambda}{kappa_i} = lambdacdotsup_{i<lambda}{kappa_i}.$$




      I am struggling to construct an explicit proof using these results and a few facts about cardinal arithmetic such as absorption for infinite cardinals and cardinalities of power sets.



      Since, assuming GCH, $kappa^+ = 2^kappa = |mathcal{P}(A)|$ for some $A$ with cardinality $kappa$, I thought I could come up with some sequence ${mu_i : i < mathrm{cf}(kappa)}$ such that



      $$kappa^+ = 2^kappa = |mathcal{P}(A)| = sum_{i < mathrm{cf}(kappa)} |[A]^{mu_i}| = sum_{i < mathrm{cf}(kappa)} kappa^{mu_i} = mathrm{cf}(kappa)cdotsup_{i<mathrm{cf}(kappa)}kappa^{mu_i} = mathrm{cf}(kappa)cdotkappa^lambda = kappa^lambda$$



      Is this approach a good idea? How would I go about finding the appropriate $mu_i$? I suppose it will have to make use of the assumption that $mathrm{cf}(kappa) leq lambda < kappa$, but I don't see how.



      I'd much appreciate any hints on how to proceed (either with my suggestion or another way).






      set-theory cardinals







      share|cite|improve this question















      I am trying to fill in the details for part (ii) of Theorem 5.15 in Jech's Set Theory:




      Theorem 5.15 If GCH holds and $kappa$ and $lambda$ are infinite cardinals, then



      (i) If $kappa leq lambda$, then $kappa^lambda = lambda^+$.



      (ii) If $mathrm{cf}(kappa) leq lambda < kappa$, then $kappa^lambda = kappa^+$.



      (iii) If $lambda < mathrm{cf}(kappa)$, then $kappa^lambda = kappa$.




      In the proof of (ii) he just states that it follows from these two lemmas:




      Lemma 5.7 If $|A| = kappa geq lambda$, then the set $[A]^lambda$ has cardinality $kappa^lambda.$




      (Here $[A]^lambda = {X subset A : |X| = lambda}.$)




      Lemma 5.8 If $lambda$ is an infinite cardinal and $kappa_i > 0$ for each $i < lambda$, then



      $$sum_{i<lambda}{kappa_i} = lambdacdotsup_{i<lambda}{kappa_i}.$$




      I am struggling to construct an explicit proof using these results and a few facts about cardinal arithmetic such as absorption for infinite cardinals and cardinalities of power sets.



      Since, assuming GCH, $kappa^+ = 2^kappa = |mathcal{P}(A)|$ for some $A$ with cardinality $kappa$, I thought I could come up with some sequence ${mu_i : i < mathrm{cf}(kappa)}$ such that



      $$kappa^+ = 2^kappa = |mathcal{P}(A)| = sum_{i < mathrm{cf}(kappa)} |[A]^{mu_i}| = sum_{i < mathrm{cf}(kappa)} kappa^{mu_i} = mathrm{cf}(kappa)cdotsup_{i<mathrm{cf}(kappa)}kappa^{mu_i} = mathrm{cf}(kappa)cdotkappa^lambda = kappa^lambda$$



      Is this approach a good idea? How would I go about finding the appropriate $mu_i$? I suppose it will have to make use of the assumption that $mathrm{cf}(kappa) leq lambda < kappa$, but I don't see how.



      I'd much appreciate any hints on how to proceed (either with my suggestion or another way).






      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 4 at 16:39









      Asaf Karagila♦

      301k32422753




      301k32422753










      asked Dec 4 at 16:35









      ryan221b

      9510




      9510






















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          You fell into a trap that also caught me when I was just starting to read mathematical books. When you see a reference to (5.7) it does not mean Lemma 5.7, but rather the displayed formula whose tag is 5.7, which in this case is the formula:



          $$kappaleqkappa^lambdaleq 2^kappa.$$ Similarly, (5.8) refers to the inequality $$kappa<kappa^lambdaquadtext{ if }lambdageqoperatorname{cf}kappa.$$



          And indeed these are the inequalities needed to prove (i) and (ii).






          share|cite|improve this answer





















          • Oh! That makes a lot more of sense. Thanks! I think it was more confusing as the line immediately above cites Lemma 5.6, then it cites (5.7) and (5.8).
            – ryan221b
            Dec 4 at 16:55











          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',
          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%2f3025799%2fassuming-gch-if-mathrmcf-kappa-leq-lambda-kappa-then-kappa-lamb%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








          up vote
          1
          down vote



          accepted










          You fell into a trap that also caught me when I was just starting to read mathematical books. When you see a reference to (5.7) it does not mean Lemma 5.7, but rather the displayed formula whose tag is 5.7, which in this case is the formula:



          $$kappaleqkappa^lambdaleq 2^kappa.$$ Similarly, (5.8) refers to the inequality $$kappa<kappa^lambdaquadtext{ if }lambdageqoperatorname{cf}kappa.$$



          And indeed these are the inequalities needed to prove (i) and (ii).






          share|cite|improve this answer





















          • Oh! That makes a lot more of sense. Thanks! I think it was more confusing as the line immediately above cites Lemma 5.6, then it cites (5.7) and (5.8).
            – ryan221b
            Dec 4 at 16:55















          up vote
          1
          down vote



          accepted










          You fell into a trap that also caught me when I was just starting to read mathematical books. When you see a reference to (5.7) it does not mean Lemma 5.7, but rather the displayed formula whose tag is 5.7, which in this case is the formula:



          $$kappaleqkappa^lambdaleq 2^kappa.$$ Similarly, (5.8) refers to the inequality $$kappa<kappa^lambdaquadtext{ if }lambdageqoperatorname{cf}kappa.$$



          And indeed these are the inequalities needed to prove (i) and (ii).






          share|cite|improve this answer





















          • Oh! That makes a lot more of sense. Thanks! I think it was more confusing as the line immediately above cites Lemma 5.6, then it cites (5.7) and (5.8).
            – ryan221b
            Dec 4 at 16:55













          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          You fell into a trap that also caught me when I was just starting to read mathematical books. When you see a reference to (5.7) it does not mean Lemma 5.7, but rather the displayed formula whose tag is 5.7, which in this case is the formula:



          $$kappaleqkappa^lambdaleq 2^kappa.$$ Similarly, (5.8) refers to the inequality $$kappa<kappa^lambdaquadtext{ if }lambdageqoperatorname{cf}kappa.$$



          And indeed these are the inequalities needed to prove (i) and (ii).






          share|cite|improve this answer












          You fell into a trap that also caught me when I was just starting to read mathematical books. When you see a reference to (5.7) it does not mean Lemma 5.7, but rather the displayed formula whose tag is 5.7, which in this case is the formula:



          $$kappaleqkappa^lambdaleq 2^kappa.$$ Similarly, (5.8) refers to the inequality $$kappa<kappa^lambdaquadtext{ if }lambdageqoperatorname{cf}kappa.$$



          And indeed these are the inequalities needed to prove (i) and (ii).







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 4 at 16:44









          Asaf Karagila♦

          301k32422753




          301k32422753












          • Oh! That makes a lot more of sense. Thanks! I think it was more confusing as the line immediately above cites Lemma 5.6, then it cites (5.7) and (5.8).
            – ryan221b
            Dec 4 at 16:55


















          • Oh! That makes a lot more of sense. Thanks! I think it was more confusing as the line immediately above cites Lemma 5.6, then it cites (5.7) and (5.8).
            – ryan221b
            Dec 4 at 16:55
















          Oh! That makes a lot more of sense. Thanks! I think it was more confusing as the line immediately above cites Lemma 5.6, then it cites (5.7) and (5.8).
          – ryan221b
          Dec 4 at 16:55




          Oh! That makes a lot more of sense. Thanks! I think it was more confusing as the line immediately above cites Lemma 5.6, then it cites (5.7) and (5.8).
          – ryan221b
          Dec 4 at 16:55


















          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.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • 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.


          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%2f3025799%2fassuming-gch-if-mathrmcf-kappa-leq-lambda-kappa-then-kappa-lamb%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

          Bressuire

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more

          Cabo Verde

          Image
            Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
          Read more

          Gyllenstierna

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more