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

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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 $$MM^{++}$$.
Original language English >Computational prospects of infinity Part II: Presented Talks Singapore World Scientific Publishing Co. Pte Ltd 23-46 23 Lecture Notes Series 15 978-981-4471-52-7 978-981-279-654-7 Published - 2008