Questions tagged [large-cardinals]
The large-cardinals tag has no usage guidance.
773
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Why is inner model theory evidence for consistency of large cardinals?
I want to understand the viewpoint that existence of canonical inner model for a large cardinal notion is strong evidence for its consistency. For example, below is Trevor Wilson's answer to What &...
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$n$-ineffable and $n$-Ramsey hierarchies
The paper Games and Ramsey-like cardinals by Nielsen and Welch 2018 defines $n$-Ramsey cardinals as follows (this is not quite the same definition but it's equivalent): $\kappa$ is $n-1$-Ramsey if ...
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Most recent results on formulating Kunen's inconsistency theorem in ZF without choice
Kunen showed that if $j:V \rightarrow M$ is a nontrivial elementary embedding from the von Neumann universe $V$ into a transitive class $M$, then $M \neq V$, or equivalently that there are no ...
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Do precipitous ideals "always" come from collapsing?
It's well-known that if $\kappa$ is a measurable cardinal, then there is a poset $\mathbb{P}$ that forces $\kappa$ to carry a precipitous ideal.
Suppose that $\omega_1$ carries a preciptous ideal $I$.
...
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What is the strength of adding this de-schematizing inference rule to Ackermann's set theory?
Language: first order logic with equality, membership, and a constant symbol $W$.
Axioms:
Extensionality: $\forall z \, (z \in x \leftrightarrow z \in y) \to x=y$
Comprehension: $\exists x \forall y \,...
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Consistency strength of strongly compact cardinal
Where can I find a proof that strongly compact cardinal has higher consistency strength than Woodin cardinal, or even just strong? Recall that a strongly compact cardinal itself may not be strong, ...
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Is this modified H. Friedman theory bi-interpretable with ZFC + ORD is Mahlo?
The following theory is a modification of Harvey Friedman $\sf K(W)$ theory.
Language: first order logic with equality, membership, and a constant symbol $W$.
Axioms:
Extensionality: $\forall z \, (z ...
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1
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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 ...
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From HODs to corresponding models of AD
If $M$ is HOD of a model $N$ of $\text{AD}^+ + V=L(P(ℝ))$, what kind of forcing construction in $M$ gives back such an $N$?
HODs for $\text{AD}^+ + V=L(P(ℝ))$ are conjectured (and under anti-large ...
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Can proper classes have different sizes?
I'm presently working in a non-ZF set theory, where there are proper classes. (Think MK or VNBG.) And I'm interested in how to think about the possibility (or impossibility) of proper classes with ...
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Higher-order equivalence of ordinals
I wonder which ordinals are second-order equivalent, and similarly for other logical equivalences. Let the signature be fixed and include only <. For concreteness, let us first ask for the first ...
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Large cardinal axioms and Grothendieck universes
A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of ...
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Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$
I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ ...
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Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal
Is anything known about the consistency strength of the following statement?
$\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ such that $a \cap \...
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1970 question of Reinhardt - how large is this ordinal?
On page 241 of William Reinhardt's paper "Ackermann's set theory equals ZF" (Annals of Math. Logic vol. 2, 1970), question 4.15 is the following:
How large is the first ordinal $\gamma$ ...
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Can a generic ultrafilter over $\mathrm{NS}^+_{\omega_1}$ witness $\omega_1$ is Ramsey-like?
Suppose that $\kappa$ is an appropriate large cardinal (preferably a Woodin cardinal, but possibly something stronger) and let $G$ be a $\operatorname{Col}(\omega_1,<\kappa)$-generic filter over $V$...
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The strongest reflection principle that does not violate covering lemmas
#-generated reflection, or Indiscernible-generation, is considered to be the strongest reflection principle that does not violate the covering lemma in L. [1]
Is there a way to extend this success to ...
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What is the least inaccessible cardinal for Tarski-Grothendieck set theory?
Let ordinal $\alpha$ be the least ordinal such that $V_\alpha\models$ Tarski-Grothendieck set theory.
What position does $\alpha$ have in the hierarchy of inaccessible cardinals?
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Why do we need the comparison lemma?
An inner model is a standard transitive (proper class) structure which satisfies all the axioms of ZFC and contains all the ordinals. The simplest and most well-known inner model is Gödel’s $L$, which ...
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Follow up question: Shelah's "Can you take Solovay's inaccessible away?"
In this answer to the question " Shelah's "Can you take Solovay's inaccessible away?" " the following is stated:
Assume that $\aleph_1$ is not inaccessible in $L$, hence a ...
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Is $\kappa \rightarrow [\kappa]^2_3$ the same as $\kappa \rightarrow [\kappa]^2_2$ for inaccessible $\kappa$
The principle $\kappa \rightarrow [\kappa]^2_\alpha$ states that whenever we have a coloring $c:[\kappa]^2\rightarrow \alpha$ there is $H \subset \kappa$ of size $\kappa$ s.t. $|c"[H]^2|<\alpha$.
...
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Ultrafilter projections and critical points of factor maps
Suppose $j : V \to M$ is $\lambda$-supercompactness embedding derived from an $\kappa$-complete normal ultrafilter $U$ on $P_\kappa(\lambda)$, $\lambda$ regular. Suppose $\eta$ is an ordinal such ...
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What can be the measure of a Vitali set?
Suppose the continuum $\mathfrak{c}$ is real-valued measurable, i.e., there exists a countably additive probabilistic measure on $\mathfrak{c}$ that measures all subsets. Then by the construction on p....
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A Löwenheim–Skolem–Tarski-like property
I am interested in the following Löwenheim–Skolem–Tarski-like property.
Given a cardinal $\kappa$, what (if any) is a property $\phi(x)$ such that if $\phi(\kappa)$ holds, then we can prove the ...
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Applications of infinite Ramsey's Theorem (on N)?
Finite Ramsey's theorem is a very important combinatorial tool that is often used in mathematics. The infinite version of Ramsey's theorem (Ramsey's theorem for colorings of tuples of natural numbers) ...
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Measuring big stuff
Often during informal discussion with colleagues, the following pattern emerges when we are stuck trying to prove a theorem about $x \in X$.
A: "let's assume this hypothesis $H$ on $x$"
B: "most ...
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Can we have more malleable proper classes without sacrificing conservativity?
NBG is a conservative extension of ZFC that includes a concept of "proper class." Now I like the conservativity, since it means anytime I want to prove something in ZFC, I am free to work in NBG. ...
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Forcing in Ackermann's Set Theory
How would one do forcing in Ackermann's set theory? C. Alkor published an article entitled "Forcing in Ackermanns Mengenlehre" (Zeitshcr. f. math. Logik und Grundlagen d. Math., Bd. 25,8.265-286 (...
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Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics
To be clear, I am not a mathematics educated student and I can not follow the details of the technicality of the forcing extension, but I feel that I have a good understanding of the big picture (of ...
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Are Berkeley cardinals easier to refute in ZFC than Reinhardt cardinals?
Kunen showed that Reinhardt cardinals are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. Berkeley cardinals are stronger than Reinhardt cardinals. You can ...
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Can Friedman's property fail at or above a supercompact cardinal?
If $\kappa>\omega_1$ is a regular cardinal, $FP_\kappa$ is the assertion that every stationary subset of $\kappa$ consisting of ordinals of countable cofinality has a closed subset of order type $\...
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How can we control the cardinality of $j(\kappa)$ for $\kappa$ an $\aleph_1$-strongly compact cardinal?
The following question was posted to MathStackExchange (original here). As there were no comments/answers on the original, I have ported it unedited.
I am interested in determining the cardinality of $...
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A weak (?) form of Shelah cardinals
The following definition of a large cardinal property combines parts of the definitions of "Shelah cardinal" and "Woodin cardinal":
A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \...
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Is this strengthening of the definition of weakly Shelah cardinals equivalent to being weakly Shelah?
The MathOverflow question A weak (?) form of Shelah cardinals by Trevor Wilson defines weakly Shelah cardinals as follows:
A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \kappa$ ...
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Operations on the set of large cardinal axioms
Here's a question from a non-set-theorist, but a sometime-user of large cardinals.
The name Cantor's attic is pretty evocative for the collection of large cardinal axioms: looking through the pages ...
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Collapsing every cardinal outside the Prikry sequence
All variants of Prikry forcing with collapses that i have been able to find preserve some points outside of the generic sequence (at least the successors). This is done for two reasons, (1) to obtain ...
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Should there be a true model of set theory?
As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
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Is Vopěnka's principle inherited by Grothendieck topoi?
I call the Vopěnka's principle:
Every subfunctor of an accessible functor is accessible
but other formulations (which may lose equivalence in weak contexts?) are also interesting to me.
If this is ...
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Does the consistency of a large cardinal axiom imply the $\omega$-consistency of that axiom?
Let $P$ be some large cardinal property (or indeed any first-order formula in the language of set theory, but lets focus on large cardinals for now). Does the $\omega$-consistency of $\mathsf{ZFC}+P$ ...
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Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?
Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
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Has there been any progress on this open problem about co-well-poweredness of accessible categories?
On the relations between accessible categories and large cardinal axioms, one big example is the following:
Assume the existence of a proper class of strongly compact cardinals. Then every accessible ...
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Model of ZF + $\neg$C in which Solovay's Theorem on stationary sets fails?
It is a theorem of Solovay that any stationary subset of a regular cardinal, $\kappa$ can be decomposed into a disjoint union of $\kappa$ many disjoint stationary sets. As far as I know, the proof ...
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Embedding large countable ordinals into the complex plane
Consider large countable ordinals (e.g. $\epsilon_0$ which is not "large", but still interesting).
These are countable sets, so they inject into the complex plane ( or even the real line).
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"$\kappa$ strongly inaccessible" = "every function $f:V_\kappa\to V_\kappa$ can be self-applied"?
Strongly inaccessible cardinals are usually introduced either as (a) cardinalities of models of ZFC or (b) cardinals which are not the power set of a smaller cardinal nor the supremum of a set with ...
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How much information do we need to guess a large cardinal?
Suppose $\kappa$ is a cardinal and we want to guess if $\kappa$ is a large cardinal, and if so what kind, by looking at the large cardinal status of a selection of cardinals below $\kappa$.
The ...
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Can MK+"Ord is almost-huge"+MM$^{++}$ be new standard foundations instead of ZFC?
I'll try to explain what this looks like to a non-expert in set theory. First, $MK$ is just a second-order $ZFC$, and there are moments when we would like to use second-order statements, for example, ...
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Logical endofunctors of Set?
What set-theoretic assumptions are necessary and sufficient to ensure the existence of a nontrivial (i.e. not isomorphic to the identity) endofunctor of the category Set which is logical (i.e. ...
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On statements independent of ZFC + V=L
Let $V=L$ denote the axiom of constructibility. Are there any interesting examples of set theoretic statements which are independent of $ZFC + V=L$? And how do we construct such independence proofs? ...
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Does ZF+AD settle the original Suslin hypothesis?
Everyone knows that the real line $\langle\mathbb{R},<\rangle$ is
the unique endless complete dense linear order with a countable
dense set. Suslin's
hypothesis is
the question whether we can ...
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Why isn't there more interest in "large powerset axioms"?
By a large powerset axiom, let us mean informally an axiom that says that for some cardinal numbers $\kappa$, we have that $2^\kappa$ is somehow "large" or "difficult to access from below." For ...
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