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Register a new userDescriptive set theory and forcing; How to prove theorems about Borel sets the hard way
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Arnold Miller
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These are lecture notes from a course I gave at the University of Wisconsin during the Spring semester of 1993. Part 1 is concerned with Borel hierarchies. Section 13 contains an unpublished theorem of Fremlin concerning Borel hierarchies and MA. Section 14 and 15 contain new results concerning the lengths of Borel hierarchies in the Cohen and random real model. Part 2 contains standard results on the theory of Analytic sets. Section 25 contains Harringtons Theorem that it is consistent to have $Pi^1_2$ sets of arbitrary cardinality. Part 3 has the usual separation theorems. Part 4 gives some applications of Gandy forcing. We reverse the usual trend and use forcing arguments instead of Baire category. In particular, Louveaus Theorem on $Pi^0_alpha$ hyp-sets has a simpler proof using forcing.
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Descriptive set theory was originally developed on Polish spaces. It was later extended to $omega$-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending descriptive set theory and its effective counterpart to general represented spaces, including non-countably-based spaces has been started in [Pauly, de Brecht 2015]. We study the spaces $mathcal{O}(mathbb{N}^mathbb{N})$, $mathcal{C}(mathbb{N}^mathbb{N},2)$ and the Kleene-Kreisel spaces $mathbb{N}langlealpharangle$. We show that there is a $Sigma^0_2$-subset of $mathcal{O}(mathbb{N}^mathbb{N})$ which is not Borel. We show that the open subsets of $mathbb{N}^{mathbb{N}^mathbb{N}}$ cannot be continuously indexed by elements of $mathbb{N}^mathbb{N}$ or even $mathbb{N}^{mathbb{N}^mathbb{N}}$, and more generally that the open subsets of $mathbb{N}langlealpharangle$ cannot be continuously indexed by elements of $mathbb{N}langlealpharangle$. We also derive effecti
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A major challenge in applying machine learning to automated theorem proving is the scarcity of training data, which is a key ingredient in training successful deep learning models. To tackle this problem, we propose an approach that relies on training with synthetic theorems, generated from a set of axioms. We show that such theorems can be used to train an automated prover and that the learned prover transfers successfully to human-generated theorems. We demonstrate that a prover trained exclusively on synthetic theorems can solve a substantial fraction of problems in TPTP, a benchmark dataset that is used to compare state-of-the-art heuristic provers. Our approach outperforms a model trained on human-generated problems in most axiom sets, thereby showing the promise of using synthetic data for this task.
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