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本文深入探讨了集合论中宇宙V的概念，利用大基数理论中的鼠标迭代和独立性理论中的强迫方法，介绍了一种名为Mighty Mouse的绝对鼠标模型。该模型能够捕捉到V，即V是通过定义良好的迭代过程从Mighty Mouse得到的可定义内模型的类泛型扩展。文章还讨论了Stable Core的重要工具，并展示了Mighty Mouse如何强烈地捕捉到Stable Core。通过这些方法，作者进一步描述了Enriched Stable Core中的实数，并证明了Stable Core不是刚性的，从而V可以是某个可定义且非刚性内模型的类泛型扩展。这为理解集合论宇宙提供了一个全新的视角。

• 文件名称：Sy David Friedman,Capturing the Universe.pdf

• 文件类型：PDF文档

• 文件标签：集合论、大基数、强迫方法

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Capturing the Universe

Sy-David Friedman (KGRC Vienna)

November 2, 2022

Abstract

We describe the universe V of sets using the ideas of mouse iter-
ation from large cardinal theory and forcing from the theory of inde-
pendence. We introduce Mighty Mouse, an absolute mouse of modest
strength and show that it captures V in the sense that V is a class-
generic extension of a definable inner model resulting from a definable
iteration of Mighty Mouse via a forcing that is definable and whose
antichains are sets.

A key tool is the Stable Core of [7]. We show that Mighty Mouse
strongly captures the Stable Core in the sense that the Stable Core
is definable over a definable inner model resulting from a definable
iteration of Mighty Mouse. Applying the methods used to prove this
result, we characterise the reals of the Enriched Stable Core of [8]
(assuming Ord is Mahlo) and show that the Stable Core is not rigid
and therefore V is class-generic over a definable, non-rigid inner model
(assuming the existence of a satisfaction predicate for V).

Introduction

The universe of sets V is important for the foundations of mathematics
as it provides an arena in which virtually all mathematical constructions can
be carried out. But what does V look like? Can it be described using the
tools that set-theorists have for building universes of set theory?

G¨odel [13] provided us with the universe L of constructible sets, an im-
portant inner model (subuniverse) of V with remarkable combinatorial prop-
erties and clear internal structure ([14]). Cohen [3] later produced a method

1

for creating new universes from old, the forcing method, which can be used
to obtain generic extensions of L which are larger than L. Is V simply a
generic extension of L? If so, then V can be described using just the methods
of constructibility and forcing.

However further work of Scott [19] and Silver [21] revealed that V cannot
be a generic extension of L if large cardinals exist. Large cardinals are es-
sential to set theory as they are needed to show that important set-theoretic
phenomena are consistent with the traditional axioms for set theory.

So to achieve our goal of describing V we need something more, and this
is the notion of mouse (first introduced in [4]). To explain mice we take a
closer look at the type of large cardinal that Scott considered, a measurable
cardinal. Let us say that U is a measure on a set X if U is a collection of
subsets of X such that for any subset Y of X, either Y or X \ Y belongs to
U and whenever U0 is a subcollection of U of size less than the size of X, the
intersection of the sets in U0 belongs to U. We say that U is nonprincipal
if U consists only of infinite sets and we say that X is measurable if X is
uncountable and there is a nonprincipal measure on X. A cardinal number1
κ is measurable if it is the cardinality of a measurable set, which can be taken
to be κ itself. Scott’s Theorem states that if there is a measurable cardinal,
then V is larger than L.

If U is a nonprincipal measure on the uncountable cardinal κ then we
can form the universe of sets constructible from U, denoted by L[U]. Silver
[20] showed that L[U] is a very nice “L-like” m...

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