Cardinal Arithmetic, foundations and constructive math - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:18:59Z http://mathoverflow.net/feeds/question/111421 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111421/cardinal-arithmetic-foundations-and-constructive-math Cardinal Arithmetic, foundations and constructive math Najdorf 2012-11-04T01:55:51Z 2012-11-04T11:01:02Z <p>This is not my area but a question occurred to me that I can not find the answer to. There is a very strong <strong>axiom of constructibility</strong> which <em>ironically</em> gives us highly non-constructive math (<strong>GCH</strong> is one of its implications). What would be an equally strong axiom in the opposite direction? And I mean direction in a philosophical sense, so what would be the strongest axiom that constructivists/intuitionists would approve of?</p> <p>My first idea was to find the largest $\kappa$ such that $2^{\aleph_0} = \aleph_{\kappa}$ is consistent with <strong>ZF</strong> but this set is unbounded ($\kappa$ can be any finite number) and $2^{\aleph_0} &lt; \aleph_{\omega}$. Which brings up the question, how much fundamental difference are there between <strong>CH</strong> and $2^{\aleph_0} = \aleph_{118}$ for example?</p> http://mathoverflow.net/questions/111421/cardinal-arithmetic-foundations-and-constructive-math/111425#111425 Answer by Andrej Bauer for Cardinal Arithmetic, foundations and constructive math Andrej Bauer 2012-11-04T04:15:09Z 2012-11-04T04:15:09Z <p>Cardinal arithmetic is the wrong thing to think about, constructively speaking. Here are some facts about constructive mathematics (when I say "may" that means there is a model validating the fact):</p> <ul> <li>Cardinals cannot be shown to be linearly ordered.</li> <li>The ordinals may form a set.</li> <li>A subset of a finite set need not be finite.</li> <li>A subset of a countable set need not be countable.</li> <li>It is conistent to assume that there is an embedding $\mathbb{R} \to \mathbb{N}$.</li> <li>It is consistent to assume that every set is a quotient of a subset of $\mathbb{N}$, for example $\mathcal{P}(\mathcal{P}(\mathbb{N}))$.</li> </ul> <p>So, I think you're going in the wrong direction. Size is simply not measured the same way intuitionistically. "Extra" axioms considered by constructive mathematics can broadly be divided into several groups:</p> <ol> <li><strong>Fragments of the law of excluded middle</strong>, such as Limited Principle of Omniscience and Markov Principle, which bring us closer to classical mathematics.</li> <li><strong>Choice principles</strong>, such as countable choice, dependent choice, function choice, which bring in some amount of the axiom of choice. Another such axiom is Aczel's presentation axiom, which states that every set is covered by one for which choice holds.</li> <li><strong>Continuity principles</strong>, such as "every function between complete separable spaces is continuous", which are typically incompatible with the law of excluded middle. Another such axiom is the Fan principle, whose important consequence is that the closed interval is compact .</li> <li><strong>Induction principles</strong>, which guarantee existence of sets defined by various induction schemata.</li> <li><strong>Computability principles</strong>, stating that "everything is computable" in some form. The best known such principle is the formal Church's thesis.</li> </ol> <p>The induction principles vaguely correspond to large cardinal axioms, and so they are perhaps closest to what you are asking for. A <em>predicative</em> constructivist will worry about existence of powersets, so he is going to consider various other axioms that bring in powersets in a limited form, but let us not get into that.</p>