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Questions tagged [large-cardinals]

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3
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1answer
66 views

Are there any I1 embeddings with interweaving critical sequences?

Can there exist non-trivial elementary embeddings $j,k:V_{\lambda+1}\rightarrow V_{\lambda+1}$ along with a strictly increasing function $r:\omega\rightarrow\omega$ such that $j^{r(2n)}(\mathrm{crit}(...
1
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0answers
34 views

A linear ordering on the quotient algebras of elementary embeddings?

We say that a finite self-distributive algebra $(A,*)$ is linear if there is some $1\in A$ where $a*1=1,1*a=a$ for all $a\in A$ and where if $\preceq$ is the relation where $x\preceq y$ if and only if ...
3
votes
0answers
51 views

What is the computational complexity of equivalence up to a critical point in the one generator free self-distributive algebra?

Suppose that there exists a rank-into-rank cardinal. Then let $R$ be the relation on the set of terms in the language of self-distributive algebras (or LD-monoids) on one generator where we set $R(s,t)...
2
votes
0answers
64 views

Arriving at the critical points in an algebra of elementary embeddings in a unique way

Let $\mathcal{E}_{\lambda}$ denote the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. Then $\mathcal{E}_{\lambda}$ can be endowed with a self-distributive operation $*$ defined ...
3
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0answers
64 views

Can algebras of elementary embeddings be sufficiently described by two element subalgebras?

Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Then $\mathcal{E}_{\lambda}$ can be endowed with a self-distributive operation $*$ where $j*...
7
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0answers
115 views

If $j_{1},…,j_{n}:V_{\lambda+1}\rightarrow V_{\lambda+1}$ are elementary embeddings, then does $j_{1}(A)=…=j_{n}(A)=A$ for some linear order $A$?

Suppose that $j_{1},\dots,j_{n}:V_{\lambda+1}\rightarrow V_{\lambda+1}$ are elementary embeddings. Then does there necessarily exist a linear ordering $A$ of $V_{\lambda}$ such that $j_{1}(A)=\dots=j_{...
1
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0answers
57 views

Consistency of reflective sequences

Is it consistent that there is a measurable cardinal $κ$, a $κ$-complete normal nonprincipal ultrafilter $U$ on $κ$, and $S∈U$ such that for every $T⊂S$ with $T∈U$ and $T$ ordinal definable from $S$ ...
-1
votes
1answer
224 views

What is the consistency strength of this kind of iterating Berkeley cardinals?

[EDIT] After suggestion from Monroe Eskew, and after having an e-mail correspondence with Prof. Joan Bagaria, I'll re-present the older question as the second in a series of 4 questions. I've had an e-...
0
votes
1answer
158 views

Why the restrictions in the definition of Berkeley cardinals?

A cardinal κ is a Berkeley cardinal, if for any transitive set $M$ with $κ∈M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(j)<\kappa$. My ...
3
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0answers
103 views

Does the Hurwitz action of the braid group on rank-into-rank embeddings tend to increase the critical points?

An algebraic structure $(X,*)$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$. Suppose that $X$ is a self-distributive algebra. Then the positive braid monoid $B_{...
0
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1answer
76 views

Strength of BTEE

What is the consistency strength of Basic Theory of Elementary Embeddings (BTEE) from The spectrum of elementrary embeddings j : V → V by Paul Corazza? BTEE uses the language of $(V,∈,j)$ and asserts:...
0
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0answers
87 views

Is there a clear inconsistency with intense reflection of top properties of the universe?

Let $V$ be the class of all sets, where sets are defined like in $MK$ as elements of classes. Properties of $V$ whose negations are unbounded (by element-hood & subset-hood) in $V$ would be ...
0
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0answers
262 views

What is the consistency status of this theory?

Let $K_2^+(W)$ be the following theory in the language $L(\in,W)$ with the constant symbol $W$. Extensionality: $\forall x (x \in a \leftrightarrow x \in b) \to a=b$ $\mathcal{Define:} \ set(x) \iff ...
8
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0answers
152 views

α-Mahlo vs weakly compact cardinals

Question: What is the consistency strength of existence of a $(κ^{++})^L$-Mahlo cardinal $κ$? I am particularly interested in how the strength compares to weakly compact cardinals (and other levels ...
2
votes
1answer
307 views

Can Ackermann set theory find a natural interpretation in light\heavy class dichotomy?

I want to coin the notion of heaviness of a class as a function from classes to ordinals such that $$heaviness(x)=|TC(x)|$$, i.e. the heaviness of a class is the cardinality of its transitive closure. ...
11
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0answers
236 views

Consistency strength of $j:L_δ→L_δ$ for some δ

What is the consistency strength of existence of a nontrivial elementary embedding $j:L_δ→L_δ$ for some ordinal $δ$? The consistency strength is strictly between totally ineffable and $ω$-Erdős ...
0
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0answers
56 views

What is the limit to iterating class comprehension, reflection and limitation of size?

In posting about a reflection principle coupled with a limitation of size axiom over Ackermann set theory, the answer is that the theory is blown up to a Mahlo cardinal. I'm here just wondering if ...
3
votes
1answer
154 views

What is the strength of adding limitation of size and a simple version of reflection to Ackermann set theory?

The following theory is formulated in first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ denoting the class ...
0
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0answers
110 views

What is the consistency strength of this kind of reflection principle?

If $\psi$ is a predicate that is definable in $FOL(\in,=)$ by a formula from parameters in $V$, then if $\psi$ hold of the class $\small ORD$ of all ordinals in $V$, then the class of all cardinals in ...
4
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0answers
295 views

The surreal numbers under a change of universe

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\...
10
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0answers
149 views

stationary reflection in $[\kappa]^\omega$

It is well-known that the following reflection principle is consistent relative to a supercompact: For all $\kappa \geq \omega_2$ and all stationary $S \subseteq [\kappa]^\omega$, there is $X \...
6
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0answers
142 views

Singular strong generator

Let $V$ be a model of $\mathrm{ZFC}$ and let $j\colon V \to M$ be an elementary embedding with a critical point $\kappa$ ($M$ is transitive). A strong generator of $j$ is an ordinal $\zeta \geq \kappa$...
11
votes
1answer
411 views

Completeness number of ultrafilters

In what I write below, by "ultrafilter" I mean a non-principal ultrafilter. Given an ultrafilter $U$ on some set $S$, let $\mu$ be the least cardinal such that $U$ is $\mu$-complete but not $\mu^+$-...
3
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0answers
117 views

If any satisfiable $\mathcal{L}_{κ,κ}(Q_{=κ})$-theory remains satisfiable when replacing $Q_{=κ}$ with $Q_{=μ}$, is $κ$ huge?

Recently, I have asked a model-theoretic question concerning a weakening of different forms of compactness. I now present another model-theoretic question as a weakening of hugeness. If any ...
0
votes
1answer
336 views

What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$

or (contrariwise) that $NGB$ + "There exists a Reinhardt cardinal" is consistent? The question is partially in the title. $NGB$ is used for the reasons stated in the Hamkins, Kirmayer, and ...
1
vote
1answer
137 views

Definition of ineffability behind reflection principles in set theory

A prominent idea found e.g. in Koellner (http://logic.harvard.edu/koellner/ORP_final.pdf) is that reflection principles in set theory are motivated by the idea that proper classes are so large as to ...
4
votes
1answer
138 views

If $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, what properties does $κ$ have?

More specifically, if $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, does $κ$ necessarily have some form of $μ$-compactness? Is it related to strong compactness in ...
2
votes
1answer
185 views

A variant of Radin forcing

Suppose $\kappa$ is a large cardinal (strong cardinal seems to be enough). Is there a forcing notion $\mathbb{R}$ with the following properties: $(1)$ Forcing with $\mathbb{R}$ adds a club $C$ into $\...
3
votes
1answer
262 views

A weakening of cardinal compactness - is it equivalent?

I was messing around with the intuition behind the size of weakly compact cardinals (in their usual characterization). I found an interesting, seemingly weaker LCA which still implies weak ...
9
votes
1answer
273 views

What is the consistency strength of weak Vopenka's principle?

Weak Vopěnka's principle says that the opposite of the category of ordinals cannot be fully embedded in any locally presentable category. Recall that one form of Vopěnka's principle says that the ...
13
votes
1answer
326 views

Does this consequence of measurability in terms of games of length $\omega+1$ imply measurability?

For any two structures $\mathcal{M}$ and $\mathcal{N}$ in the same first-order language $\mathcal{L}$ and any ordinal $\theta$, let $G_\theta(\mathcal{M},\mathcal{N})$ be the two-player game of ...
8
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0answers
197 views

Universally meager spaces and large cardinals

Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...
3
votes
1answer
276 views

What is the consistency strength of non-existence of outer automorphisms of Calkin algebra?

The Calkin algebra $C(H)$ is the quotient of $B(H)$, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space $H$, by the ideal $K(H)$ of compact operators. In 1977, ...
4
votes
1answer
467 views

What are examples of non-equivalent virtualizations of a large cardinal?

This is a follow up to my previous question concerning virtual large cardinals, that are generally weaker axioms of infinity obtained from ordinary large cardinals through the so-called virtualization ...
3
votes
1answer
306 views

On the Actual Potential of Virtual Large Cardinals

Virtual large cardinals belong to a relatively new breed of strong axioms of infinity. They often appear as statements of the following form: Definition. Suppose $A$ is a large cardinal property ...
9
votes
1answer
402 views

On the Large Cardinal Strength of Normal Moore Space Conjecture

In his seminal 1937 paper, Jones [1] proved the following result about Moore spaces: Theorem. (Jones) If $2^{\aleph_0}<2^{\aleph_1}$ then all separable normal Moore spaces are metrizable. Then ...
6
votes
2answers
333 views

Paul Mahlo's Original Large Cardinal Paper

It is well known that Paul Mahlo (1883-1971) developed a systematic hierarchy of inaccessible cardinals of the type $\pi_{a,b}$ where $\pi_{1,b}$ enumerates the strongly innacessible cardinals, $\pi_{...
-4
votes
1answer
157 views

Can there be such an elementary embedding?

EDIT: it appears that my original question has some confusion between auto-morphisms and elementary embeddings as it is obvious from the answer below, therefore I'll clarify here what I exactly want. ...
11
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2answers
298 views

Does $0^{\#}$ imply the failure of upward directedness in the set generic universe over $L$?

Question 1. Suppose that $0^{\#}$ exists. Are there set generic filters $g,h$ over $L$ ($g,h \in V$) such that for any $f$ ($\in V$) which is generic over $L$ we have that $L[g] \cup L[h] \not \...
3
votes
1answer
308 views

Strong Total Failures vs. Weak Instances of the Generalized Continuum Hypothesis

The exponentiation operator inflicts a subtle information loss on the transfinite numerical equations, pretty similar to the case of $a^2=b^2 \nRightarrow a=b$ in real numbers. In fact, for the ...
9
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0answers
253 views

ZF + “every Suslin set of reals is ${\bf \Sigma}^1_2$”

What is known about the theory ($\ast$) ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"? By "reals" I mean elements of the Baire space $\omega^\omega$. For a cardinal $\kappa$, a set of ...
9
votes
2answers
287 views

Does Easton forcing preserve measurable cardinals?

The question is in the title. For Easton's theorem see Wikipedia. Loosely speaking we can use forcing to manipulate the powerset function on regular cardinals as much as we like given we satisfy the ...
4
votes
1answer
129 views

Can an algebraic variety over a field $k$ be the union of proper closed subsets $(S_i)_{i\in I}$ with $I < k$

Let $k$ be an algebraically closed field (of characteristic zero, if it helps). Let $X$ be an algebraic variety over $k$. Let $I$ be an index set such that the cardinality of $I$ is smaller than the ...
11
votes
3answers
926 views

Cops, Robbers and Cardinals: The Infinite Manhunt

Cops & Robbers is a certain pursuit-evasion game between two players, Alice and Bob. Alice is in charge of the Justice Bureau, which controls one or more law enforcement officers, the cops. Bob ...
7
votes
2answers
446 views

Which large cardinals have a Matryoshka characterization?

What on Earth do Russian Matryoshka dolls have in common with large cardinal axioms?! Well, the answer lies in Jónsson algebras! Here is how: As illustrated in the pictures, a Matryoshka set is a ...
2
votes
1answer
186 views

Concrete mathematical statements in relation to Choice versus Reinhardt cardinals?

Harvey Friedman is well known for investigating concrete mathematical statements that requires strong assumptions, i.e. those that can only be interpreted in a strong extension of $\text{ZF(C)}$. My ...
4
votes
1answer
209 views

Consistency of the size of the least real-valued measurable cardinal, vis-a-vis the continuum

A well-known result of Solovay states that $ZFC$ + "the continuum is real-valued measurable" is equiconsistent with $ZFC$ + "there is a measurable cardinal", over $ZFC$. That the continuum is ...
6
votes
1answer
304 views

Regarding extenders

I asked this at math.stackexchange, but got no answers: Let $j\colon M\rightarrow N$ be an elementary embedding (between inner models) with $\operatorname{crit}(j)=\kappa$. Let $\kappa<\lambda$. ...
6
votes
1answer
398 views

fake and weak cardinals

Suppose $\lambda$ is a successor of a singular cardinal. We will say $\lambda$ fake if there is a transitive set $M$ such that $\lambda \subseteq M$ satisfying $\mathrm{ZFC}^-$ (ZFC without powerset) ...
3
votes
1answer
140 views

A version of the Martin–Solovay tree for $H_\kappa$

Consider a fixed $\Pi^1_2$ property of reals, $A(x)$. Is it true that relative to a regular cardinal $\kappa$, one can define a version of the Martin–Solovay tree $T_2$ for $A$ with the ...