Thursday, February 01, 2018

The Problem with Prefaces

Suppose that it says, in the preface to some non-fiction book, that there is bound to be some false statement in the book, even though each statement in the book is believed to be true by the author. That is the Preface Paradox. A first analytical thought might be to regard belief as sufficiently high credence, so that we would not believe large conjunctions of our beliefs (cf. how you will probably not get a 1 or a 6 with one throw of a die, but with three dice the chance is less than 30%); but of course, we very often do (and for other reasons belief is not simply high credence, as recently posted). A more sophisticated response might discover contextualist aspects (e.g. self-reflection upon one's fallibility); but what I am interested in here is how we naturally overlook the following absurd response: Why not say that sets are simply such that sets of beliefs are like that? We could then have all that we want, and nothing that we do not want: we simply put precisely that much into the axioms of set theory! But of course, that would be too easy; what about what sets of beliefs really are? The thing is, that is essentially the mainstream response to Cantor's Paradox (which I shall be posting on in my next few posts). The Preface Paradox therefore shows how absurd the mainstream foundations of mathematics are.

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