Toby Meadows (Aberdeen): What's so natural about generic extensions?

Abstract: Since Cohen's celebrated proof of the independence of CH  from ZFC, forcing has been part of the essential furniture of set theoretic research. More recently, a number of set theorists have taken up the view that forcing provides us with genuinely alternative conceptions of set theoretic  reality. In particular, we shall be concerned with John Steel's conception of the generic multiverse. But just how natural are these generic extensions? This paper will explore some of the logical fundamentals of forcing and some its pleasing properties with a view to obtaining an answer to our question.