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This essay responds to some of the recent publications in the philos-ophy of mathematics which either are or might be connected withthe semantics of set theory. It is concerned particularly with therelevance of forcing to questions in the semantics of set theory orto the question of what axioms should be used for set theory. I amconcerned more specifically to two major recent developments inwhich forcing plays a central role, the work of Woodin and Koellneron the resolution of CH, and the work of Hamkins in connectionwith the multiverse theory, the modal logic of forcing, and set the-oretic geology.
These lines of research are highly technical. It is not intended
to engage with or contribute to the technical developments, butrather to consider the grounds for believing that the technical re-sults are relevant to the issues. Thus, in the case of the resolutionof CH, various technical results are held to provide evidence foror against CH, and we are concerned to understand why this is
thought to be the case. The rational for multiverse theory is basedupon considerations arising from generic extensibility, and the an-cillary investigations into the modal logic and set theoretic geol-ogy investigate set theory by examining the relationship betweenmodels and their forcing extensions. Both the arguments for themultiverse theory and the reasons for giving such a prominent roleto forcing in the study of that multiverse are to be considered.
The analysis is undertaken from a particular philosophical stand-
point which is similar in character to the philosophy of Rudolf Car-nap. Peter Koellner has criticised Carnap’s pluralism and his ownapproach to the resolution of CH (and the more general problem ofdiscovering or justifying new axioms for set theory) is presented byhim in contrast to Carnap’s pluralism. The question of pluralismand the related question of whether there are absolutely undecid-able problems are intimately bound up with his work on CH.
My own understanding of Carnap’s philosophy, and more specif-
ically of his pluralism, differs from that of Koellner, is not open tothe same objections, and furnishes a standpoint from which a fruit-ful analysis and critique of the work of Koellner and Hamkins canbe approached. I therefore begin with a presention of key elementsof that standpoint, contrasting it with the Koellner’s interpretationof Carnap.
In this section I describe a semantic standpoint from which matterspertaining to the foundations of mathematics may be addressed.
This is described by comparison with a selection of alternative
These descriptions are not offered as historically accurate de-
scriptions of the beliefs of philosophers, though various philoso-phers are mentioned as exemplars (or even authors) of the featuresdiscussed.
The closest predecessor to the position I describe is found in the
philosophy of Rudolf Carnap. Like Carnap’s philosophy of mathe-matics, it may be thought of a synthesis of logicism and formalism,
in which the logicism is semantic rather than metaphysical.
To give a fuller description the following varieties of logicism
The notion of formalism I consider here is one which I associatewith Hilbert, but which is distinct from the programme normallyassociated with Hilbert whose purpose was to estabish the consis-tency of the foundations of mathematics.
It is more closely related to the doctrine, against which Frege
argued, to the effect that mathematics is a purely formal disciplinein which symbols are manipulated without assigning to them anymeaning or significance. The
In the first instance I distinguish radical, narrow and broad
Carnap attended lecture courses given by Frege as an undergrad-uate. He was impressed by the precision of the methods and theircontrast with the standard of arguments in philosophy.
1. There are no generic extensions of V.
∃B, G(G is a generic ultrafilter in B over M))))⇒ (ZF C
In the following we are concerned with what is provable in the firstorder theory ZFC, primarily with results about ZFC and its models. Though ZFC is a pure theory of sets, it is customary to simulatetalk about classes using class abstracts as what Russell would havecalled incomplete symbols or Quine, virtual classes. Quantificationof classes is not possible, but theorem schemata can be proven.
If M is a transitive model (set or class), B is an M complete
boolean algebra and G is a generic ultrafilter over B, then thegeneric extension M [G] is the transitive collapse of the quotient ofthe boolean valued model M B by the ultrafilter G (the collapse ofM B/G).
Bell and Jech both prove that under these conditions:
The notion of a generic extension of a transitive model (possibly
a class) M is defined in Jech 14.30 and 14.32 and Bell p97.
There are no generic extensions of V. In this the notion of generic extension is as defined in Jech
14.30 and 14.32 and Bell p97. The notation for this in both casesis M [G] where M is a transitive set or class and G is an M genericultrafilter over some boolean algebra complete in M .
Jech (4.22) and Bell (14.32) both prove:If M[G]
In his book on the continuum hypPaul Cohen pro-ceeds from the assumption that there exists a standard model (SM)of ZFC. In Chapter IV §11 he then explains how this assumptioncan be dispensed with “if one does not care about the constructionof actual models”.
[Bel05] John L. Bell. Set Theory - Boolean-valued Models and
Independence Proofs. Oxford University Press, 2005.
[Coh66] Paul J. Cohen. Set Theory and the Continuum Hypothesis.
[Jec02] Thomas Jech. Set Theory. Springer Verlag, 2002.
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