Kurepa tree

<h2 id="specializing-a-kurepa-tree">Specializing a Kurepa tree</h2>
<p>A Kurepa tree can be "killed" by <a href="/facts/Forcing_(mathematics)/3vMqqKcq">forcing</a> the existence of a function whose value on any non-root node is an ordinal less than the rank of the node, such that whenever three nodes, one of which is a lower bound for the other two, are mapped to the same ordinal, then the three nodes are comparable.  This can be done without <a href="/facts/Ordinal_collapsing_function/YEjT7QzV">collapsing</a> ℵ1, and results in a tree with exactly ℵ1 branches.
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Aronszajn_tree/GASB35JW">Aronszajn tree</a></li>
<li><a href="/facts/Suslin_tree/h1LrlFhB">Suslin tree</a></li></ul>

<ul><li><a href="/facts/Thomas_Jech/E4JQu4Mk">Jech, Thomas J.</a> (1971), "Trees", <i><a href="/facts/Journal_of_Symbolic_Logic/lfXVXcoy">Journal of Symbolic Logic</a></i>, 36: 1–14, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2271510">10.2307/2271510</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2271510">2271510</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0284331">0284331</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0245.02054">0245.02054</a></li>
<li>Jech, Thomas (2002). <i>Set Theory</i>. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-44085-2.</li>
<li>Kurepa, G. (1935), <a href="http://elibrary.matf.bg.ac.rs/handle/123456789/326">"Ensembles ordonnés et ramifiés"</a>, <i>Publ. math. Univ. Belgrade</i>, 4: 1–138, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:61.0980.01">61.0980.01</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0014.39401">0014.39401</a></li>
<li>Silver, Jack (1971), "The independence of Kurepa's conjecture and two-cardinal conjectures in model theory", <i>Axiomatic Set Theory</i>, Proc. Sympos. Pure Math., vol. XIII, Providence, R.I.: Amer. Math. Soc., pp. 383–390, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0277379">0277379</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0255.02068">0255.02068</a></li></ul>

Kurepa tree open-in-new

Kurepa tree