# Coding into $$H(\omega_2)$$ together (or not) with forcing axioms. A survey

Aspero, David (2008) Coding into $$H(\omega_2)$$ together (or not) with forcing axioms. A survey. In: >Computational prospects of infinity. World Scientific Publishing Co Pte Ltd, Singapore, pp. 23-46. ISBN 978-981-279-654-7

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## Abstract

This paper is mainly a survey of recent results concerning the possibility of building forcing extensions in which there is a simple definition, over the structure $$\langle H(\omega_2), \in\rangle$$ and without parameters, of a prescribed member of $H(omega_2)$ or of a well--order of $$H(\omega_2)$$. Some of these results are in conjunction with strong forcing axioms like $$PFA^{++}$$ or $$MM$$, some are not. I also observe (Corollary 4.4) that the existence of certain objects of size $$\aleph_1$$ follows outright from the existence of large cardinals. This observation is motivated by an attempt to extend the $$PFA^{++}$$ result to a result mentioning $$MM^{++}$$.

Item Type: Book Section Faculty of Science > School of Mathematics Pure Connector 09 Jul 2014 13:06 23 Mar 2019 00:50 https://ueaeprints.uea.ac.uk/id/eprint/49361