In the last constructive gem we studied the exponential `2^NN` and its isomorphic copies. This time we shall compute the double exponential `2^(2^NN)` and even write some Haskell code.
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Constructive gems are usually not about particular results, because all constructive results can be proved classically as well, but rather about the method and the way of thinking. I demonstrate a constructive proof which can be reused in three different settings (set theory, topology, computability) because constructive mathematics has many different interpretations. I promise I will post a constructive gem soon. This constructive stone came up as a reaction to the cardinality of finite sets stone. I show that inhabited sets of natural numbers need not have minima, constructively. Cardinality of sets in constructive mathematics is not as well behaved as in classical mathematics. Cardinalities of finite sets are not natural numbers, and cardinalities are not linearly ordered. Just like in real life, constructive stones are easier to find than constructive gems, so let me start the series with a stone about constructive finite sets. Two girl one cup In various mathematical forums, mostly those of a logical flavor, I regularly see people asking basic questions about constructive mathematics. I also see misconceptions about constructive mathematics. I shall make a series of posts, Constructive Gems and Stones, which will answer basic questions about constructive mathematics, and will hopefully help fix wrong ideas about constructive [...] |
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