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Sy-David Friedman

Publications and source records attributed to Sy-David Friedman.

At least 19 recordsLinked to original sources
arXiv2026

A Foundation for the Core Mathematician

The foundations of mathematics have long been considered settled by the Zermelo-Fraenkel-Choice axioms. But set theory abounds in models with different truths and even classical questions such as the measurability of projective sets can vary between models. The core of mathematics resides in the study of structures built from the set R of real numbers. This paper proposes a foundation for core mathematics, with both a system of axioms and a definite model of those axioms, in which essentially all core mathematics is incorporated. This definite model delivers a definite truth-value, either true or false, to any core mathematical assertion.

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arXiv2021

Mutually embeddable models of ZFC

We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.

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arXiv2019

Structural Properties of the Stable Core

The stable core, an inner model of the form $\langle L[S],\in, S\rangle$ for a simply definable predicate $S$, was introduced by the first author in [Fri12], where he showed that $V$ is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname{GCH}$ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show that, if large cardinals exist in $V$, then the stable core has inner models with a proper class of measurable limits of measurables, with a proper class of measurable limits of measurable limits of measurables, and so forth. We show this by providing a characterization of natural inner models $L[C_1, \dots, C_n]$ for specially nested class clubs $C_1, \dots, C_n$, like those arising in the stable core, generalizing recent results of Welch [Wel19].

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arXiv2019

On the complexity of classes of uncountable structures: trees on $\aleph_1$

We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire-space $ω_1^{ω_1}$. First, we will show that none of these classes have the Baire property (unless they are empty). Moreover, under $(V=L)$, (a) the class of Aronszajn and Suslin trees is $Π_1^1$-complete, (b) the class of special Aronszajn trees is $Σ_1^1$-complete, and (c) the class of Kurepa trees is $Π^1_2$-complete. We achieve these results by finding nicely definable reductions that map subsets $X$ of $ω_1$ to trees $T_X$ so that $T_X$ is in a given tree-class $\mathcal T$ if and only if $X$ is stationary/non-stationary (depending on the class $\mathcal T$). Finally, we present models of CH where these classes have lower projective complexity.

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arXiv2019

Embeddings into outer models

We explore the possibilities for elementary embeddings $j : M \to N$, where $M$ and $N$ are models of ZFC with the same ordinals, $M \subseteq N$, and $N$ has access to large pieces of $j$. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line $\mathbb R$.

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arXiv2018

Generic Coding with Help and Amalgamation Failure

We show that if $M$ is a countable transitive model of ZF and if $a,b$ are reals not in $M$, then there is a $G$ generic over $M$ such that $b \in L[a,G]$. We then present several applications such as the following: if $J$ is any countable transitive model of ZFC and $M \not\subseteq J$ is another countable transitive model of ZFC of the same ordinal height $\alpha$, then there is a forcing extension $N$ of $J$ such that $M \cup N$ is not included in any transitive model of ZFC of height $\alpha$. Also, assuming $0^\#$ exists, letting $S$ be the set of reals generic over $L$, although $S$ is disjoint from the Turing cone above $0^\#$, we have that for any non-constructible real $a$, $\{ a \oplus s : s \in S \}$ is cofinal in the Turing degrees.

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arXiv2018

A model of second-order arithmetic satisfying AC but not DC

We show that there is a $β$-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a $Π^1_2$-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of ${\rm ZFC}^-$. This work is a rediscovery by the first two authors of a result obtained by the third author.

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arXiv2017

Universism and Extensions of V

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by providing a hitherto unexamined way for a Universist to interpret talk that seems to necessitate the addition of sets to $V$. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk ($V$-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist.

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arXiv2016

Cichon's Diagram for uncountable cardinals

We develop a version of Cichon's diagram for cardinal invariants on the generalized Cantor space 2^kappa or the generalized Baire space kappa^kappa where kappa is an uncountable regular cardinal. For strongly inaccessible kappa, many of the ZFC-results about the order relationship of the cardinal invariants which hold for omega generalize; for example we obtain a natural generalization of the Bartoszynski-Raisonnier-Stern Theorem. We also prove a number of independence results, both with <kappa-support iterations and kappa-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.

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arXiv2015

Hyperclass Forcing in Morse-Kelley Class Theory

In this article we introduce and study hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK$^{**}$. We define this forcing by using a symmetry between MK$^{**}$ models and models of ZFC$^-$ plus there exists a strongly inaccessible cardinal (called SetMK$^{**}$). We develop a coding between $β$-models $\mathcal{M}$ of MK$^{**}$ and transitive models $M^+$ of SetMK$^{**}$ which will allow us to go from $\mathcal{M}$ to $M^+$ and vice versa. So instead of forcing with a hyperclass in MK$^{**}$ we can force over the corresponding SetMK$^{**}$ model with a class of conditions. For class-forcing to work in the context of ZFC$^-$ we show that the SetMK$^{**}$ model $M^+$ can be forced to look like $L_{κ^*}[X]$, where $κ^*$ is the height of $M^+$, $κ$ strongly inaccessible in $M^+$ and $X\subseteqκ$. Over such a model we can apply definable class forcing and we arrive at an extension of $M^+$ from which we can go back to the corresponding $β$-model of MK$^{**}$, which will in turn be an extension of the original $\mathcal{M}$. Our main result combines hyperclass forcing with coding methods of [BJW82] and [Fri00] to show that every $β$-model of MK$^{**}$ can be extended to a minimal such model of MK$^{**}$ with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic.

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arXiv2014

Regularity Properties on the Generalized Reals

We investigate regularity properties derived from tree-like forcing notions in the setting of "generalized descriptive set theory", i.e., descriptive set theory on $κ^κ$ and $2^κ$, for regular uncountable cardinals $κ$.

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arXiv2014

On Borel Reducibility in Generalised Baire Space

In this paper we study the Borel reducibility of Borel equivalence relations, including some orbit equivalence relations, on the generalised Baire space $κ^κ$ for an uncountable $κ$ with the property $κ^{<κ}=κ$. The theory looks quite different from its classical counterpart where $κ=ω$, although some basic theorems do generalise.

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arXiv2014

Large cardinals need not be large in HOD

We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal $\kappa$ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.

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arXiv2013

Easton functions and supercompactness

Suppose $κ$ is $λ$-supercompact witnessed by an elementary embedding $j:V\rightarrow M$ with critical point $κ$, and further suppose that $F$ is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton's theorem: (1) $\forallα$ $α<\textrm{cf}(F(α))$ and (2) $α<β$ $\Longrightarrow$ $F(α)\leq F(β)$. In this article we address the question: assuming GCH, what additional assumptions are necessary on $j$ and $F$ if one wants to be able to force the continuum function to agree with $F$ globally, while preserving the $λ$-supercompactness of $κ$? We show that, assuming GCH, if $F$ is any function as above, and in addition for some regular cardinal $λ>κ$ there is an elementary embedding $j:V\rightarrow M$ with critical point $κ$ such that $κ$ is closed under $F$, the model $M$ is closed under $λ$-sequences, $H(F(λ))\subseteq M$, and for each regular cardinal $γ\leq λ$ one has $(|j(F)(γ)|=F(γ))^V$, then there is a cardinal-preserving forcing extension in which $2^δ=F(δ)$ for every regular cardinal $δ$ and $κ$ remains $λ$-supercompact. This answers a question of B. Cody, M. Magidor, On supercompactness and the continuum function, Ann. Pure Appl. Logic, (2013).

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arXiv2012

Generalized Descriptive Set Theory and Classification Theory

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.

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arXiv2012

Subcompact cardinals, squares, and stationary reflection

We generalise Jensen's result on the incompatibility of subcompactness with square. We show that alpha^+-subcompactness of some cardinal less than or equal to alpha precludes square_alpha, but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling's hierarchy of weak squares, showing which cases are precluded by alpha^+-subcompactness, and again we demonstrate the optimality of our results by forcing the strongest possible squares under these restrictions to hold.

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arXiv2011

Analytic equivalence relations and bi-embedability

Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of L_{ω_1 ω}) is far from complete (see [5, 2]). In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of L_{ω_1 ω}, but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.

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