{"id":43,"date":"2019-07-28T22:17:41","date_gmt":"2019-07-28T19:17:41","guid":{"rendered":"https:\/\/philokatholos.getway.org\/?p=43"},"modified":"2019-07-28T22:17:52","modified_gmt":"2019-07-28T19:17:52","slug":"a-result-from-pcf-theory","status":"publish","type":"post","link":"https:\/\/philokatholos.getway.org\/?p=43","title":{"rendered":"A result from PCF theory"},"content":{"rendered":"\n<p>There is a very curious theorem by Shelah which states that:  <\/p>\n\n\n\n<p> If for every natural number $n$ we have $2^{\\aleph_n} &lt; \\aleph_{\\omega}$ then, $2^{\\aleph_{\\omega}} &lt; \\aleph_{\\omega_4}$. <\/p>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"600\" height=\"801\" src=\"https:\/\/philokatholos.getway.org\/wp-content\/uploads\/2019\/07\/Helen-Frankenthaler-The-Human-Edge-1967-1.jpg\" alt=\"\" class=\"wp-image-44\" srcset=\"https:\/\/philokatholos.getway.org\/wp-content\/uploads\/2019\/07\/Helen-Frankenthaler-The-Human-Edge-1967-1.jpg 600w, https:\/\/philokatholos.getway.org\/wp-content\/uploads\/2019\/07\/Helen-Frankenthaler-The-Human-Edge-1967-1-225x300.jpg 225w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><figcaption> <em>Helen Frankenthaler, The Human Edge, 1967<\/em> <\/figcaption><\/figure>\n\n\n\n<p> An other way to state this is this one: if $\\aleph_{\\omega}$ is a strong limit cardinal then: $$\\prod_{n=0}^{\\infty} \\aleph_n &lt; \\aleph_{\\omega_4} $$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>There is a very curious theorem by Shelah which states that: If for every natural number $n$ we have $2^{\\aleph_n} &lt; \\aleph_{\\omega}$ then, $2^{\\aleph_{\\omega}} &lt; \\aleph_{\\omega_4}$. An other way to state this is this one: if $\\aleph_{\\omega}$ is a strong limit cardinal then: $$\\prod_{n=0}^{\\infty} \\aleph_n &lt; \\aleph_{\\omega_4} $$<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/posts\/43"}],"collection":[{"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=43"}],"version-history":[{"count":2,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/posts\/43\/revisions"}],"predecessor-version":[{"id":46,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/posts\/43\/revisions\/46"}],"wp:attachment":[{"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=43"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=43"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=43"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}