Amorphous proper classes in MKWhat sort of structure can amorphous sets support?Splitting infinite setsFor models of ZF, if for some $A$ we have $L[A] = L$, what can we deduce about $A$?What sort of structure can amorphous sets support?Some questions about Ackermann set theoryHartogs number and the three power setsCan $mathbbR$ be partitioned into dedekind-finite sets?How many Dedekind-finite sets can $mathbbR$ be partitioned into?Can ZFC be interpreted in a set theory having finitely many ranks?An axiom for collecting proper classesDo choice principles in all generic extensions imply AC in $V$?

Amorphous proper classes in MK


What sort of structure can amorphous sets support?Splitting infinite setsFor models of ZF, if for some $A$ we have $L[A] = L$, what can we deduce about $A$?What sort of structure can amorphous sets support?Some questions about Ackermann set theoryHartogs number and the three power setsCan $mathbbR$ be partitioned into dedekind-finite sets?How many Dedekind-finite sets can $mathbbR$ be partitioned into?Can ZFC be interpreted in a set theory having finitely many ranks?An axiom for collecting proper classesDo choice principles in all generic extensions imply AC in $V$?













4












$begingroup$


Working in $ZFC$ every cardinal is either finite or in bijection with a proper subset of itself (Dedekind infinite). Without Choice it is consistent that there are infinite sets which can't be partitioned into two infinite subsets (amorphous sets), so the above statement no longer holds since a bijection to a proper subset implies a partition into two disjoint infinite subsets as proven on the wiki -- all of this is discussed in the question and answers here much more succinctly.




Is it consistent in $MK$ without Global Choice that there are amorphous proper classes, meaning proper classes which can't be partitioned into two proper class sized subclasses?




Directly generalizing the argument given on the wiki article for amorphous sets seems to require a notion of transfinite function composition which can be defined in good categorical generality using colimits, but it is not immediately apparent how to generalize the recursive definition of the $S_i$'s for limit ordinal $i$ since the given definitions depend on immediate predecessor steps.










share|cite|improve this question











$endgroup$
















    4












    $begingroup$


    Working in $ZFC$ every cardinal is either finite or in bijection with a proper subset of itself (Dedekind infinite). Without Choice it is consistent that there are infinite sets which can't be partitioned into two infinite subsets (amorphous sets), so the above statement no longer holds since a bijection to a proper subset implies a partition into two disjoint infinite subsets as proven on the wiki -- all of this is discussed in the question and answers here much more succinctly.




    Is it consistent in $MK$ without Global Choice that there are amorphous proper classes, meaning proper classes which can't be partitioned into two proper class sized subclasses?




    Directly generalizing the argument given on the wiki article for amorphous sets seems to require a notion of transfinite function composition which can be defined in good categorical generality using colimits, but it is not immediately apparent how to generalize the recursive definition of the $S_i$'s for limit ordinal $i$ since the given definitions depend on immediate predecessor steps.










    share|cite|improve this question











    $endgroup$














      4












      4








      4





      $begingroup$


      Working in $ZFC$ every cardinal is either finite or in bijection with a proper subset of itself (Dedekind infinite). Without Choice it is consistent that there are infinite sets which can't be partitioned into two infinite subsets (amorphous sets), so the above statement no longer holds since a bijection to a proper subset implies a partition into two disjoint infinite subsets as proven on the wiki -- all of this is discussed in the question and answers here much more succinctly.




      Is it consistent in $MK$ without Global Choice that there are amorphous proper classes, meaning proper classes which can't be partitioned into two proper class sized subclasses?




      Directly generalizing the argument given on the wiki article for amorphous sets seems to require a notion of transfinite function composition which can be defined in good categorical generality using colimits, but it is not immediately apparent how to generalize the recursive definition of the $S_i$'s for limit ordinal $i$ since the given definitions depend on immediate predecessor steps.










      share|cite|improve this question











      $endgroup$




      Working in $ZFC$ every cardinal is either finite or in bijection with a proper subset of itself (Dedekind infinite). Without Choice it is consistent that there are infinite sets which can't be partitioned into two infinite subsets (amorphous sets), so the above statement no longer holds since a bijection to a proper subset implies a partition into two disjoint infinite subsets as proven on the wiki -- all of this is discussed in the question and answers here much more succinctly.




      Is it consistent in $MK$ without Global Choice that there are amorphous proper classes, meaning proper classes which can't be partitioned into two proper class sized subclasses?




      Directly generalizing the argument given on the wiki article for amorphous sets seems to require a notion of transfinite function composition which can be defined in good categorical generality using colimits, but it is not immediately apparent how to generalize the recursive definition of the $S_i$'s for limit ordinal $i$ since the given definitions depend on immediate predecessor steps.







      set-theory lo.logic axiom-of-choice






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 1 hour ago









      David Roberts

      17.5k463177




      17.5k463177










      asked 4 hours ago









      Alec RheaAlec Rhea

      1,3331819




      1,3331819




















          1 Answer
          1






          active

          oldest

          votes


















          5












          $begingroup$

          Unless I'm missing something, the answer is no: we have a surjection $s$ from a given proper class to the class of ordinals - sending each element to its rank and then "collapsing" appropriately - and this lets us partition the original class into two proper classes, for example $s^-1(limits)$ versus $s^-1(successors)$.






          share|cite|improve this answer









          $endgroup$








          • 1




            $begingroup$
            @Alec: In that case the answer is positive. Just do Fraenkel's model over a proper class of atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            @Alec: That's the OG model for amorphous sets. Just remember that ZFA (or ZFU) is equivalent to ZF-Foundation with Quine atoms for the atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            Tsk tsk tsk. TSK. TSK. TSK.
            $endgroup$
            – Asaf Karagila
            2 hours ago






          • 3




            $begingroup$
            @Noah Asaf is calling you uncool for not knowing.
            $endgroup$
            – David Roberts
            1 hour ago







          • 1




            $begingroup$
            Hahah, it’s an abbreviation for the american colloquialism “original gangster” meaning a member of the original older generation of badasses in a given gang/discipline of mathematics.
            $endgroup$
            – Alec Rhea
            33 mins ago











          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: "504"
          ;
          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%2fmathoverflow.net%2fquestions%2f325796%2famorphous-proper-classes-in-mk%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









          5












          $begingroup$

          Unless I'm missing something, the answer is no: we have a surjection $s$ from a given proper class to the class of ordinals - sending each element to its rank and then "collapsing" appropriately - and this lets us partition the original class into two proper classes, for example $s^-1(limits)$ versus $s^-1(successors)$.






          share|cite|improve this answer









          $endgroup$








          • 1




            $begingroup$
            @Alec: In that case the answer is positive. Just do Fraenkel's model over a proper class of atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            @Alec: That's the OG model for amorphous sets. Just remember that ZFA (or ZFU) is equivalent to ZF-Foundation with Quine atoms for the atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            Tsk tsk tsk. TSK. TSK. TSK.
            $endgroup$
            – Asaf Karagila
            2 hours ago






          • 3




            $begingroup$
            @Noah Asaf is calling you uncool for not knowing.
            $endgroup$
            – David Roberts
            1 hour ago







          • 1




            $begingroup$
            Hahah, it’s an abbreviation for the american colloquialism “original gangster” meaning a member of the original older generation of badasses in a given gang/discipline of mathematics.
            $endgroup$
            – Alec Rhea
            33 mins ago
















          5












          $begingroup$

          Unless I'm missing something, the answer is no: we have a surjection $s$ from a given proper class to the class of ordinals - sending each element to its rank and then "collapsing" appropriately - and this lets us partition the original class into two proper classes, for example $s^-1(limits)$ versus $s^-1(successors)$.






          share|cite|improve this answer









          $endgroup$








          • 1




            $begingroup$
            @Alec: In that case the answer is positive. Just do Fraenkel's model over a proper class of atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            @Alec: That's the OG model for amorphous sets. Just remember that ZFA (or ZFU) is equivalent to ZF-Foundation with Quine atoms for the atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            Tsk tsk tsk. TSK. TSK. TSK.
            $endgroup$
            – Asaf Karagila
            2 hours ago






          • 3




            $begingroup$
            @Noah Asaf is calling you uncool for not knowing.
            $endgroup$
            – David Roberts
            1 hour ago







          • 1




            $begingroup$
            Hahah, it’s an abbreviation for the american colloquialism “original gangster” meaning a member of the original older generation of badasses in a given gang/discipline of mathematics.
            $endgroup$
            – Alec Rhea
            33 mins ago














          5












          5








          5





          $begingroup$

          Unless I'm missing something, the answer is no: we have a surjection $s$ from a given proper class to the class of ordinals - sending each element to its rank and then "collapsing" appropriately - and this lets us partition the original class into two proper classes, for example $s^-1(limits)$ versus $s^-1(successors)$.






          share|cite|improve this answer









          $endgroup$



          Unless I'm missing something, the answer is no: we have a surjection $s$ from a given proper class to the class of ordinals - sending each element to its rank and then "collapsing" appropriately - and this lets us partition the original class into two proper classes, for example $s^-1(limits)$ versus $s^-1(successors)$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          Noah SchweberNoah Schweber

          19.5k349146




          19.5k349146







          • 1




            $begingroup$
            @Alec: In that case the answer is positive. Just do Fraenkel's model over a proper class of atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            @Alec: That's the OG model for amorphous sets. Just remember that ZFA (or ZFU) is equivalent to ZF-Foundation with Quine atoms for the atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            Tsk tsk tsk. TSK. TSK. TSK.
            $endgroup$
            – Asaf Karagila
            2 hours ago






          • 3




            $begingroup$
            @Noah Asaf is calling you uncool for not knowing.
            $endgroup$
            – David Roberts
            1 hour ago







          • 1




            $begingroup$
            Hahah, it’s an abbreviation for the american colloquialism “original gangster” meaning a member of the original older generation of badasses in a given gang/discipline of mathematics.
            $endgroup$
            – Alec Rhea
            33 mins ago













          • 1




            $begingroup$
            @Alec: In that case the answer is positive. Just do Fraenkel's model over a proper class of atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            @Alec: That's the OG model for amorphous sets. Just remember that ZFA (or ZFU) is equivalent to ZF-Foundation with Quine atoms for the atoms.
            $endgroup$
            – Asaf Karagila
            3 hours ago






          • 1




            $begingroup$
            Tsk tsk tsk. TSK. TSK. TSK.
            $endgroup$
            – Asaf Karagila
            2 hours ago






          • 3




            $begingroup$
            @Noah Asaf is calling you uncool for not knowing.
            $endgroup$
            – David Roberts
            1 hour ago







          • 1




            $begingroup$
            Hahah, it’s an abbreviation for the american colloquialism “original gangster” meaning a member of the original older generation of badasses in a given gang/discipline of mathematics.
            $endgroup$
            – Alec Rhea
            33 mins ago








          1




          1




          $begingroup$
          @Alec: In that case the answer is positive. Just do Fraenkel's model over a proper class of atoms.
          $endgroup$
          – Asaf Karagila
          3 hours ago




          $begingroup$
          @Alec: In that case the answer is positive. Just do Fraenkel's model over a proper class of atoms.
          $endgroup$
          – Asaf Karagila
          3 hours ago




          1




          1




          $begingroup$
          @Alec: That's the OG model for amorphous sets. Just remember that ZFA (or ZFU) is equivalent to ZF-Foundation with Quine atoms for the atoms.
          $endgroup$
          – Asaf Karagila
          3 hours ago




          $begingroup$
          @Alec: That's the OG model for amorphous sets. Just remember that ZFA (or ZFU) is equivalent to ZF-Foundation with Quine atoms for the atoms.
          $endgroup$
          – Asaf Karagila
          3 hours ago




          1




          1




          $begingroup$
          Tsk tsk tsk. TSK. TSK. TSK.
          $endgroup$
          – Asaf Karagila
          2 hours ago




          $begingroup$
          Tsk tsk tsk. TSK. TSK. TSK.
          $endgroup$
          – Asaf Karagila
          2 hours ago




          3




          3




          $begingroup$
          @Noah Asaf is calling you uncool for not knowing.
          $endgroup$
          – David Roberts
          1 hour ago





          $begingroup$
          @Noah Asaf is calling you uncool for not knowing.
          $endgroup$
          – David Roberts
          1 hour ago





          1




          1




          $begingroup$
          Hahah, it’s an abbreviation for the american colloquialism “original gangster” meaning a member of the original older generation of badasses in a given gang/discipline of mathematics.
          $endgroup$
          – Alec Rhea
          33 mins ago





          $begingroup$
          Hahah, it’s an abbreviation for the american colloquialism “original gangster” meaning a member of the original older generation of badasses in a given gang/discipline of mathematics.
          $endgroup$
          – Alec Rhea
          33 mins ago


















          draft saved

          draft discarded
















































          Thanks for contributing an answer to MathOverflow!


          • 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%2fmathoverflow.net%2fquestions%2f325796%2famorphous-proper-classes-in-mk%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

          How should I use the fbox command correctly to avoid producing a Bad Box message?How to put a long piece of text in a box?How to specify height and width of fboxIs there an arrayrulecolor-like command to change the rule color of fbox?What is the command to highlight bad boxes in pdf?Why does fbox sometimes place the box *over* the graphic image?how to put the text in the boxHow to create command for a box where text inside the box can automatically adjust?how can I make an fbox like command with certain color, shape and width of border?how to use fbox in align modeFbox increase the spacing between the box and it content (inner margin)how to change the box height of an equationWhat is the use of the hbox in a newcommand command?

          Doxepinum Nexus interni Notae | Tabula navigationis3158DB01142WHOa682390"Structural Analysis of the Histamine H1 Receptor""Transdermal and Topical Drug Administration in the Treatment of Pain""Antidepressants as antipruritic agents: A review"

          inputenc: Unicode character … not set up for use with LaTeX The Next CEO of Stack OverflowEntering Unicode characters in LaTeXHow to solve the `Package inputenc Error: Unicode char not set up for use with LaTeX` problem?solve “Unicode char is not set up for use with LaTeX” without special handling of every new interesting UTF-8 characterPackage inputenc Error: Unicode character ² (U+B2)(inputenc) not set up for use with LaTeX. acroI2C[I²C]package inputenc error unicode char (u + 190) not set up for use with latexPackage inputenc Error: Unicode char u8:′ not set up for use with LaTeX. 3′inputenc Error: Unicode char u8: not set up for use with LaTeX with G-BriefPackage Inputenc Error: Unicode char u8: not set up for use with LaTeXPackage inputenc Error: Unicode char ́ (U+301)(inputenc) not set up for use with LaTeX. includePackage inputenc Error: Unicode char ̂ (U+302)(inputenc) not set up for use with LaTeX. … $widehatleft (OA,AA' right )$Package inputenc Error: Unicode char â„¡ (U+2121)(inputenc) not set up for use with LaTeX. printbibliography[heading=bibintoc]Package inputenc Error: Unicode char − (U+2212)(inputenc) not set up for use with LaTeXPackage inputenc Error: Unicode character α (U+3B1) not set up for use with LaTeXPackage inputenc Error: Unicode characterError: ! Package inputenc Error: Unicode char ⊘ (U+2298)(inputenc) not set up for use with LaTeX