ordered fields with the bounded value property, without choice - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:46:33Z http://mathoverflow.net/feeds/question/73201 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73201/ordered-fields-with-the-bounded-value-property-without-choice ordered fields with the bounded value property, without choice James Propp 2011-08-19T04:11:58Z 2011-08-20T16:09:38Z <p>In his answer to my question <a href="http://mathoverflow.net/questions/71432/ordered-fields-with-the-bounded-value-property" rel="nofollow">http://mathoverflow.net/questions/71432/ordered-fields-with-the-bounded-value-property</a>, Ali Enayat showed that if one assumes the countable axiom of choice, then there exists a non-Archimedean ordered field $F$ with the bounded value property, by which I mean: for all $a &lt; b$ in $F$ and for every continuous function $f$ from $[a,b]_F :=${$x \in F: a \leq x \leq b$} to $F$, there exists $B$ in $F$ such that $-B \leq f(x) \leq B$ for all $x$ in $[a,b]_F$. (Here we say $f$ is continuous if it satisfies the usual $\epsilon,\delta$ definition of continuity, where all quantification is over $F$.)</p> <p>In the absence of $AC_\omega$, what can one prove? E.g., could we perhaps prove the assertion using an explicit subfield of the Field of surreal numbers, such as the set of surreal numbers created prior to day $\omega_1$? (I'm not a logician, so it's possible that such notions as "the set of surreal numbers created prior to day $\omega_1$" intrinsically depend on $AC_\omega$ in ways I'm not seeing.)</p> http://mathoverflow.net/questions/73201/ordered-fields-with-the-bounded-value-property-without-choice/73250#73250 Answer by Ali Enayat for ordered fields with the bounded value property, without choice Ali Enayat 2011-08-19T22:58:37Z 2011-08-20T16:09:38Z <p>In my answer to the <a href="http://mathoverflow.net/questions/71432/ordered-fields-with-the-bounded-value-property" rel="nofollow">related question</a>, $AC_{\omega}$ was <strong>only</strong> used to ensure that one can get hold of a regular uncountable cardinal (i.e., $\omega_1$). And of course Gitik's remarkable theorem assures us that, assuming the consistency of a proper class of strongly compact cardinals, there is a model of $ZF$ with no uncountable regular cardinals. </p> <blockquote> <p>But despite the limitations imposed by Gitik's theorem, we can construct, in $ZF$ alone, a proper class field $F$ such that the statement "$F$ has the bounded value property" is provable in $NBG$ (i.e., <a href="http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" rel="nofollow">von-Neumann-Bernays-Gödel theory of classes</a>). </p> </blockquote> <p><strong>Explanation:</strong> $NBG$ is a "conservative" extension of $ZF$, designed to handle "large objects" such as the class <strong>V</strong> of sets, the class <strong>Ord</strong> of ordinals, and the field <strong>No</strong> of surreal numbers. $NBG$ can prove that the class of ordinals <strong>Ord</strong> is regular in the sense that every function from <strong>Ord</strong> to <strong>Ord</strong> with bounded range is constant on an unbounded subclass of <strong>Ord</strong>. On the other hand, Schmerl's proof of Sikorski's theorem, when implemented in $NBG$ shows that for any regular uncountable cardinal $\kappa \leq$ <strong>Ord</strong> there is an ordered field $F$ of cardinality and cofinality $\kappa$ that satisfies $BW(\kappa)$. Moreover, The analysis of Schmerl's proof reveals that $F$ can be arranged to be <em>well-orderable</em>, which, coupled with THE LEMMA established in <a href="http://mathoverflow.net/questions/71432/ordered-fields-with-the-bounded-value-property" rel="nofollow">my answer</a>, shows that $NBG$ proves that $F$ has the bounded value property.</p> <p>I will close this note by relating the discussion to surreals <strong>No</strong>.</p> <p>In the presence of $CH$ (the continuum hypothesis), and $AC$, every ordered field of cardinality at most $\aleph_1$ is isomorphic to a subfield of <strong>No</strong>($&lt;\omega_1$), where <strong>No</strong>($&lt;\omega_1$) is the collection of surreals "born" before $\omega_1$. Coupled with my answer to the other question, this shows that in the presence of $AC+CH$, we have:</p> <blockquote> <p><strong>(1)</strong> There is a subfield of <strong>No</strong>($&lt;\omega_1$) with the bounded value property.</p> </blockquote> <p>Moreover, using the "resplendence" property of saturated models, one can show:</p> <blockquote> <p><strong>(2)</strong> $ZFC+CH$ proves that <strong>No</strong>($&lt;\omega_1$) does NOT have the bounded value property.</p> <p><strong>(3)</strong> $NBG$ plus global choice proves that <strong>No</strong> does NOT have the bounded value property. </p> </blockquote> <p>My proofs of (2) and (3) using resplendence are nonconstructive, but there might be explicit failures of the bounded value property for <strong>No</strong> and <strong>No</strong>($&lt;\omega_1$) already in $ZF$.</p>