Archives of: Kurt Godel Research Center

Chris Scambler – On Ineffable Liars

 KGRC Friday seminar on 2017‑06‑02 – 12 pm

Speaker: Chris Scambler (New York University, USA)

Abstract: The most promising non-classical approaches to the theory of truth build on that of Saul Kripke (1975) by adding a conditional satisfying reasonable laws. Among the attractive features of such approaches are their capacity to offer object-language means for classifying the defectiveness of paradoxical sentences and formulas; in (2007), Hartry Field shows his approach yields a transfinite hierarchy of determinacy operators of increasing strength that seem to play exactly this role. There are, however, difficult technical questions about the extent of the hierarchy of such operators that turn on the availability of reasonable ordinal notation systems, and these may yield philosophical issues for Field’s approach to the paradoxes. According to Field, the extent of the hierarchy is inherently ‘fuzzy’, because of indeterminacy concerning the unrestricted notion of definability. As a result, Field argues, one can’t diagonalize out of the hierarchy of determinacy operators in any meaningful sense, since the hierarchy in question is not bivalently definable. In (2014), Philip Welch has argued that on the contrary the hierarchy of determinacy operators breaks down precisely at the least $\Sigma_2$-extendible ordinal (relative to a given model M); moreover, Welch has shown how to use this result to produce “ineffable liars”, that diagonalize out of the hierarchy: these are sentences that are indeterminate on Field’s theory, but whose defectiveness is not measured by any determinacy operator in the object language.

The task of this paper is to assess the significance of Welch’s result, and to adjudicate the dispute between Field and Welch. In the opening sections, I will review the Kripke and Field constructions, focussing especially on the hierarchy of determinacy operators and their behaviour. After that, I will give an overview of Welch’s construction, culminating in the construction of an ineffable liar sentence. Finally, I will scrutinize Welch’s argument from a philosophical perspective, and suggest that Field’s project is not adversely affected by Welch’s results. Nevertheless, I will show some ways in which that the latter are still of considerable philosophical interest.

References

Saul A. Kripke: Outline of a theory of truth. Journal of Philosophy 72 (19):690-716 (1975)

Hartry Field: Solving the paradoxes, escaping revenge. In J. C. Beall (ed.), Revenge of the Liar: New Essays on the Paradox. Oxford University Press (2007)

P. D. Welch: Some observations on truth hierarchies. Review of Symbolic Logic 7 (1):1-30 (2014)

 

Vincenzo Dimonte – Rank-into-rank axioms and forcing

KGRC Research Seminar – 2017‑06‑01 at 4pm

Speaker: Vincenzo Dimonte (University of Udine, Italy)

Abstract: Rank-into-rank axioms sit on the top of the large cardinal hierarchy, and their fringe status makes them quite mysterious and evasive. In particular, research on I0 started to gain momentum just in the last few years.

In this talk we will give an overview of what is known at the moment about the interaction between such axioms and forcing, in four steps, in increasing order of complexity. The main result most of the time would be that the rank-into-rank axiom is not destroyed by the forcing, therefore providing many independence results (for example involving the behaviour of the power function, tree structures, pcf theory…). We will also note how such results pose an actual problem for the main branch of the research on I0, i.e, the quest for finding similarities between I0 and the Axiom of Determinacy.

Part of this work is joint with Sy Friedman and Liuzhen Wu.

Victoria Gitman: A model of second-order arithmetic with the choice scheme in which $\Pi^1_2$-dependent choice fails

KGRC research seminar – 2017‑05‑18 at 4pm

Speaker: Victoria Gitman (CUNY Graduate Center, New York, USA)

Abstract: 

In second-order arithmetic, the choice scheme is the scheme of assertions, for every second-order formula $\varphi(n,X,A)$, that if for every $n$ there is a set $X$ such that $\varphi(n,X,A)$ holds, then there is a single set $Y$ whose $n$-th slice $Y_n$ witnesses $\varphi(n,Y_n,A)$. While full second-order arithmetic ${\textrm Z}_2$ implies the choice scheme for $\Sigma^1_2$-assertions, the reals of the Feferman-Lévy model form a model of ${\textrm Z}_2$ in which $\Pi^1_2$-choice fails. The dependent choice scheme is the analogue ${\textrm DC}$ for second-order arithmetic and it asserts, for every second-order formula $\varphi(X,Y,A)$, that if for every set $X$ there is another set $Y$ such that $\varphi(X,Y,A)$ holds, then there is a single set $Z$, viewed as an $\omega$-sequence of sets, such that for every $n$, $\varphi(Z\upharpoonright n,Z_n,A)$ holds. The theory ${\textrm Z}_2$ implies $\Sigma^1_2$-dependent choice, and Simpson has conjectured that there is a model of ${\textrm Z}_2$ with the choice scheme in which $\Pi^1_2$-dependent choice fails. We prove Simpson’s conjecture by constructing a symmetric submodel of a forcing extension in which ${\textrm AC}_\omega$ holds, but ${\textrm DC}$ fails for a $\Pi^1_2$-definable relation on the reals.

We force over $L$ with a tree iteration of Jensen’s forcing (a ccc subposet
of Sacks forcing adding a unique generic real) along the tree ${}^{\lt\omega}\omega_1$, adding a tree, isomorphic to ${}^{\lt\omega}\omega_1$, of finite sequences of reals ordered by extension, such that that the sequences on level $n$ are $L$-generic for the $n$-length iteration of Jensen’s forcing. We extend the uniqueness of generic reals properties of Jensen’s forcing (obtained earlier by Jensen and later by Lyubetsky and Kanovei) by showing that in the tree iteration extension, the only sequences of reals $L$-generic for the $n$-length iteration of Jensen’s forcing are those explicitly added on level $n$ of the generic tree. The uniqueness property implies that the generic tree is $\Pi^1_2$-definable.

The theorem arose out of our attempts to separate the analogues of the
choice scheme and the dependent choice scheme over Kelley-Morse set theory,
and we conjecture that an appropriate generalization of our arguments will
now achieve this result.

This is joint work with Sy-David Friedman.

Stefan Hoffelner: $\text{NS}_{\omega_1}$ saturated and a $\Sigma^{1}_{4}$-definable wellorder on the reals

KGRC Research Seminar – 2017‑05‑11 at 4pm.

Speaker: Stefan Hoffelner (KGRC)

Abstract: The investigation of the saturation of the nonstationary ideal $\text{NS}_{\omega_1}$ has a long tradition in set theory. In the early 1970’s K. Kunen showed that, given a huge cardinal, there is a universe in which $\text{NS}_{\omega_1}$ is $\aleph_2$-saturated. The assumption of a huge cardinal has been improved in the following decades, using very different techniques, by many set theorists until S. Shelah around 1985 realized that already a Woodin cardinal is sufficient for the consistency of the statement “$\text{NS}_{\omega_1}$ is saturated”.

Due to work of H. Woodin on the one hand and G. Hjorth on the other, there is a surprising and deep connection between definable wellorders of the reals and the saturation of $\text{NS}_{\omega_1}$: In a universe with a measurable cardinal and $\text{NS}_{\omega_1}$ saturated, it is impossible to have a $\Sigma^1_3$-wellorder. This leads naturally to the question whether there is a universe in which $\text{NS}_{\omega_1}$ is saturated and its reals have a
$\Sigma^1_{4}$-wellorder. In my talk I will outline a proof that this is indeed the case; assuming the existence of $M_1^{\#}$ there is a model with a $\Sigma^1_{4}$-definable wellorder on the reals in which  $\text{NS}_{\omega_1}$ is saturated.

This is joint work with Sy-David Friedman.

Sam Roberts: The iterative conception of properties and comprehension

KGRC Friday Seminar – 2017‑05‑05 at 12pm

Speaker: Sam Roberts (University of Oslo, Norway)

Abstract: Mathematicians appeal to proper classes: that is, collections too large to form sets. But what are classes if not sets? One response is that classes are properties. Properties are sharply distinguished from sets: they are intensional whereas sets are extensional. Fine and Linnebo have proposed theories on which properties are “built up” in a series of stages. Unfortunately, neither of these theories imply that there are very many properties. In this talk, I will propose an improvement of these theories. More precisely, by ensuring that the stages extend far enough, I will show that the they can be modified to interpret Morse-Kelly class theory, which implies the existence of a plethora of classes.

Víctor Torres-Pérez: Rado’s Conjecture, an alternative to forcing axioms?

KGRC Research Seminar – 2017‑05‑04 at 4pm

Speaker: Víctor Torres-Pérez (TU Wien)

Abstract: Rado’s Conjecture (RC) in the formulation of Todorcevic is the statement that every tree $T$ that is not decomposable into countably many antichains contains a subtree of cardinality $\aleph_1$ with the same property.
Todorcevic has shown the consistency of this statement relative to
the consistency of the existence of a strongly compact cardinal.

Todorcevic also showed that RC implies the Singular Cardinal Hypothesis,
a strong form of Chang’s Conjecture, the continuum is at most $\aleph_2$,
the negation of $\Box(\theta)$ for every regular $\theta\geq\omega_2$,
etc. These implications are very similar to the ones obtained from traditional
forcing axioms such as MM or PFA. However, RC is incompatible even with
$MA(\aleph_1)$.

In this talk we will take the opportunity to give an overview of our
results with different coauthors obtained in the last few years together
with recent ones, involving RC, certain weak square principles and
instances of tree properties. These new implications seem to continue
suggesting that RC is a good alternative to forcing axioms. We will discuss
to which extent this may hold true and where we can find some limitations.
We will end the talk with some open problems and possible new directions.

Alberto Marcone: Some results about the higher levels of the Weihrauch lattice

KGRC Research Seminar – 2017‑04‑27 at 4pm

Speaker: Alberto Marcone (Università di Udine, Italy)

Abstract: In the last few years Weihrauch reducibility and the ensuing Weihrauch lattice have emerged as a useful tool for studying the complexity of mathematical statements viewed as “problems” or multi-valued functions. This approach complements nicely the reverse mathematics approach, and has been very successful for statements which are provable in ${\mathsf{ACA}_0}$. The study the Weihrauch lattice for functions arising from statements laying at higher levels, such as ${\mathsf{ATR}_0}$, of the reverse mathematics spectrum is instead in its infancy. We will present some results (work in
progress with my graduate student Andrea Cettolo).

In some cases we obtain the expected finer classification, but in other we observe a collapse of statements that are not equivalent with respect to provability in subsystems of second order arithmetic. This is in part due to the increased syntactic complexity of the statements. Our preliminary results deal with comparability of well-orderings, $\Sigma^1_1$-separation, and
$\Delta^1_1$-comprehension.

Matteo Viale: Useful axioms

KGRC Research Seminar – 2017‑04‑26 at 4pm

Speaker: Matteo Viale (Università di Torino, Italy)

Abstract: I overview several aspects of forcing axioms which (in my eyes) give solid mathematical arguments explaining why these axioms are so useful in establishing new (consistency) results and/or theorems.

  • The first aspect outlines that forcing axioms are natural strengthenings not only of Baire’s category theorem, but also of the axiom of choice (these are two of the most useful non-constructive principles in mathematics), and also strengthenings of most large cardinal axioms (at least for cofinally many of them).
  • The second aspect outlines that Shoenfield’s absoluteness, Cohen’s forcing theorem, and Los theorem for standard ultrapowers of a first order structure by a non principal ultrafilter are all specific instances of a more general form of Los theorem which can be declined for what I call boolean ultrapowers.
  • The third aspect outlines how strong forcing axioms and Woodin’s generic absoluteness results are two sides of the same coin and will try to explain how stronger and stronger forms of generic absoluteness can be obtained by asserting stronger and stronger forcing axioms. In this context category theoretic ideas start to play a role and we are led to analyze forcings whose conditions are (certain classes of) forcing notions and whose order relation is given by (certain classes of) complete embeddings.

There is a surprising analogy between the theory of these class forcings, the theory of towers of normal ideals, and many of the classical arguments yielding generic absoluteness results. For what concerns the first two aspects of my talk, I do not claim authorship of essentially none of the result I will be talking about, nonetheless it is hard to attribute correctly the relevant results.

Peter Holy: Small embedding characterizations for large cardinals, and internal large cardinals

KGRC Research Seminar – 2017‑04‑06 at 4pm

Speaker: Peter Holy (University of Bonn, Germany)

Abstract:

Many notions of large cardinals are characterized in terms of the existence of certain elementary embeddings with the large cardinal in question as their critical point. A small embedding characterization of a large cardinal notion is one that requires the existence of certain elementary embeddings that map their critical point to the relevant large cardinal. One classic example of such a small embedding characterization is Magidor’s small embedding characterization of supercompactness. We show that many other large cardinal notions have small embedding characterizations, in particular also large cardinal notions for which no embedding characterizations have been known to exist at all.

In the second part of this talk, I will then sketch an application of small embedding characterizations, that yields what we call internal large cardinals, which essentially describe what is left of large cardinals after they have been destroyed or collapsed by sufficiently nice forcing. The basic idea is to lift the small embeddings that characterize the initial large cardinals.

This is joint work with Philipp Lücke.

Luca Incurvati: Metalogic and the Overgeneration Argument

KGRC Friday seminar – 2017‑03‑31 at 12pm

Speaker:  Luca Incurvati (University of Amsterdam, North Holland, Netherlands)

Abstract:

A prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. We prove a number of results establishing that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that, by moving from the standard set-theoretic semantics for second-order logic to a semantics which makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic.

This is joint work with Salvatore Florio.