Degrees of rigidity for Souslin trees

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=FuchsHamkins2009:DegreesOfRigidity]

We investigate various strong notions of rigidity for Souslin trees, separating them under Diamond into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under Diamond that there is a group whose automorphism tower is highly malleable by forcing.

Tall cardinals

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2009:TallCardinals]

A cardinal $𝜅$ is tall if for every ordinal $𝜃$ there is an embedding $𝑗 : 𝑉 \to 𝑀$ with critical point $𝜅$ such that $𝑗 (𝜅) > 𝜃$ and $𝑀 𝜅 \subset 𝑀$ . Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal $𝜅$ can be made indestructible by a variety of forcing notions, including forcing that pumps up the value of $2 𝜅$ as high as desired.

The proper and semi-proper forcing axioms for forcing notions that preserve $ℵ 2$ or $ℵ 3$

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=HamkinsJohnstone2009:PFA(aleph_2-preserving)]

We prove that the PFA lottery preparation of a strongly unfoldable cardinal $𝜅$ under $\neg 0 ♯$ forces $P F A (ℵ 2 - p r e s e r v i n g)$ , $P F A (ℵ 3 - p r e s e r v i n g)$ and $P F A ℵ 2$ , with $2 𝜔 = 𝜅 = ℵ 2$ . The method adapts to semi-proper forcing, giving $S P F A (ℵ 2 - p r e s e r v i n g)$ , $S P F A (ℵ 3 - p r e s e r v i n g)$ and $S P F A ℵ 2$ from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction $S P F A (ℵ 2 - p r e s e r v i n g) + S P F A (ℵ 3 - p r e s e r v i n g) + S P F A ℵ 2 + 2 𝜔 = ℵ 2$ . Since unfoldable cardinals are relatively weak as large cardinal notions, our summary conclusion is that in order to extract significant strength from PFA or SPFA, one must collapse $ℵ 3$ to $ℵ 1$ .

Changing the heights of automorphism towers by forcing with Souslin trees over $𝐿$

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=FuchsHamkins2008:ChangingHeightsOverL]

We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.

In an earlier paper with Simon Thomas, “Changing the heights of automorphism towers,” we had added such malleable groups by forcing, and the current paper addresses the question as to whether there are such groups already in L.

The ground axiom is consistent with $𝑉 \neq H O D$

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=HamkinsReitzWoodin2008:TheGroundAxiomAndVequalsHOD]

Abstract. The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, it is first-order expressible in set theory. The previously known models of the Ground Axiom all satisfy strong forms of $𝑉 = H O D$ . In this article, we show that the Ground Axiom is relatively consistent with $𝑉 \neq H O D$ . In fact, every model of ZFC has a class-forcing extension that is a model of $Z F C + G A + 𝑉 \neq H O D$ . The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a class-forcing extension with $Z F C + G A + 𝑉 \neq H O D$ in which this supercompact cardinal is preserved.

Large cardinals with few measures

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=ApterCummingsHamkins2006:LargeCardinalsWithFewMeasures]

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly $𝜅 +$ many normal measures on the least measurable cardinal $𝜅$ . This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of $𝜆$ -strong compactness or $𝜆$ -supercompactness measures on $𝑃 𝜅 (𝜆)$ can be exactly $𝜆 +$ , if $𝜆 > 𝜅$ is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.

Diamond (on the regulars) can fail at any strongly unfoldable cardinal

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=DzamonjaHamkins2006:DiamondCanFail]

If $𝜅$ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which $◊ 𝜅 (R E G)$ fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=HamkinsWoodin2005:NMPccc]

The Necessary Maximality Principle for c.c.c. forcing asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal.

The Ground Axiom

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2005:TheGroundAxiom]

This is an extended abstract for a talk I gave at the 2005 Workshop in Set Theory at the Mathematisches Forschungsinstitut Oberwolfach.

Oberwolfach Research Report 55/2005 | Ground Axiom on Wikipedia

Extensions with the approximation and cover properties have no new large cardinals

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2003:ExtensionsWithApproximationAndCoverProperties]

If an extension $¯ 𝑉$ of $𝑉$ satisfies the $𝛿$ -approximation and cover properties for classes and $𝑉$ is a class in $¯ 𝑉$ , then every suitably closed embedding $𝑗 : ¯ 𝑉 \to ¯ 𝑁$ in $¯ 𝑉$ with critical point above $𝛿$ restricts to an embedding $𝑗 ↾ 𝑉 : 𝑉 \to 𝑁$ amenable to the ground model $𝑉$ . In such extensions, therefore, there are no new large cardinals above delta. This result extends work in my article on gap forcing.

Exactly controlling the non-supercompact strongly compact cardinals

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=ApterHamkins2003:ExactlyControlling]

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify previous results of the first author.

A simple maximality principle

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2003:MaximalityPrinciple]

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence $𝜑$ holding in some forcing extension $𝑉 ℙ$ and all subsequent extensions $𝑉 ℙ * ℚ$ holds already in $𝑉$ . It follows, in fact, that such sentences must also hold in all forcing extensions of $𝑉$ . In modal terms, therefore, the Maximality Principle is expressed by the scheme $(◊ ◻ 𝜑) \to ◻ 𝜑$ , and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in $𝜑$ , is equiconsistent with the scheme asserting that $𝑉 𝛿$ is an elementary substructure of $𝑉$ for an inaccessible cardinal $𝛿$ , which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is the Necessary Maximality Principle, which asserts that the boldface MP holds in V and all forcing extensions. From this, it follows that $0 ♯$ exists, that $𝑥 ♯$ exists for every set $𝑥$ , that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.

How tall is the automorphism tower of a group?

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=Hamkins2001:HowTall?]

The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of that group, and so on, iterating transfinitely by taking the natural direct limit at limit stages. The question, known as the automorphism tower problem, is whether the tower ever terminates, whether there is eventually a fixed point, a group that is isomorphic to its automorphism group by the natural map. Wielandt (1939) proved the classical result that the automorphism tower of any finite centerless group terminates in finitely many steps. This was generalized to successively larger collections of groups until Thomas (1985) proved that every centerless group has a terminating automorphism tower. Here, it is proved that every group has a terminating automorphism tower. After this, an overview is given of the author’s (1997) result with Thomas revealing the set-theoretic essence of the automorphism tower of a group: the very same group can have wildly different towers in different models of set theory.

Joel David Hamkins

mathematics and philosophy of the infinite

Tag Archives: forcing

Degrees of rigidity for Souslin trees

Tall cardinals

The proper and semi-proper forcing axioms for forcing notions that preserve $ℵ 2$ or $ℵ 3$

Changing the heights of automorphism towers by forcing with Souslin trees over $𝐿$

The ground axiom is consistent with $𝑉 \neq H O D$

Large cardinals with few measures

Diamond (on the regulars) can fail at any strongly unfoldable cardinal

The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal

The Ground Axiom

Extensions with the approximation and cover properties have no new large cardinals

Exactly controlling the non-supercompact strongly compact cardinals

How tall is the automorphism tower of a group?

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