A naturalistic case in favour of the Generic Multiverse with a core

Abstract

In this paper, I compare the Generic Multiverse with a core (henceforth, GMH) with the classical set theory ZFC, making use of the principle MAXIMIZE introduced by Maddy (1997). I argue that the classic set theory ZFC is restrictive over the GMH , that is, the GMH strongly maximizes over ZFC in the sense that it provides a wide range of isomorphism types that are not available in ZFC.

Full text

A naturalistic case in favour of the Generic
Multiverse with a core
May 26, 2018
In this paper, I compare the Generic Multiverse with a core (henceforth,
GMH
) with the classical set theory
ZF C
, making use of the principle MAX-
IMIZE introduced by Maddy (1997). This principle states that, since the
aim of set theory is to represent all the known mathematics within a single
theory, it should
maximize
the range of available isomorphism types. This
is particular important for mathematics, since isomorphisms make it is pos-
sible to import methods and results from a mathematical eld to another. I
argue that the classic set theory
ZF C
is
restrictive
over the
GMH
, that is,
the
GMH
strongly maximizes
over
ZF C
in the sense that it provides a wide
range of isomorphism types that are not available in
ZF C
.
I briey dene the
GMH
as the multiverse with a common core of truths,
shared between all the universes of the multiverse. A universe in this mul-
tiverse is a model of a certain set of axioms of set theory (for example
ZF C +V=L
or
ZF +AD
), while the core is a set of propositions sat-
ised in every universe of the multiverse. Obviously in the multiverse there
is also the universe that satises only the propositions in the core (that is,
the core has a model that is part of the multiverse). All the other universes
are extensions of this core: they satisfy all that is true in it, and more. For
example, if the core is the the intended model of
ZF C
, the multiverse in-
cludes a model of
ZF C +V=L
and a model of
ZF C+

0#
exists (see
Steel (2014)).
The
GMH
thus dened strongly maximizes over
ZF C
: there is no theory
T
extending
ZF C
that properly maximizes over the
GMH
and the
GMH
inconsistently maximizes over
ZF C
. This means that the
GMH
provides
structures that cannot be satised by
ZF C
, even if properly extended. To
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see this, assume that the core of the
GMH
is
ZF −
(set theory minus the
Axiom of Foundation). From this core we can build a multiverse in which,
among others, there is a universe for
ZF C
and a universe for
ZF +AD
. In
this multiverse one can have
both
the Axiom of Choice (provided by
ZF C
)
and
a full Axiom of Determinacy (provided by
ZF +AD
). Determinacy and
Choice are actually incompatible, but they can coexist in the
GMH
. Hence,
the
GMH
, unlike the intended model of
ZF C
, can include all the structures
based on Determinacy. That is, the
GMH
provides a new isomorphism type,
i.e. it proves the existence of a structure that is not isomorphic to anything
in
ZF C
.
Furthermore, the
GMH
also provides what Maddy calls a
fair interpreta-
tion
of
ZF C
, i.e. the
GMH
validates all the axioms of
ZF C
(this is because
ZF C
is part of the multiverse) and one can build natural models, inner mod-
els, and truncations of proper class of inner models at inaccessible levels of
ZF C
in the
GMH
.
I conclude that, assuming MAXIMIZE, the
GMH
is better justied than
ZF C
, since it provides more isomorphism types and it can fairly interpret
ZF C
itself.
References
Maddy, Penelope (1997).
Naturalism in Mathematics
. Oxford: Oxford Uni-
versity Press.
 (1998). V= L and maximize. In:
Logic colloquium
. Vol. 95, pp. 918.
Steel, John R. (2014). Gödel's program. In:
Interpreting Gödel
. Ed. by
Juliette Kennedy. Cambridge University Press.
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