Questions tagged [large-cardinals]
The large-cardinals tag has no usage guidance.
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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What is the evidence for and against the HOD conjecture?
I'm aware that the HOD conjecture is implied by the Ultimate-L conjecture, but I don't know what the evidence is for the Ultimate-L conjecture. On the other hand, I'm aware the evidence against the ...
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Large cardinal near inconsistencies
I am looking for examples of results about large cardinals, large cardinal axioms, or other objects of high (or seemingly high) consistency strength that are almost inconsistencies. I am looking for ...
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stating large cardinal axioms in ZF
Can I ask whether there is a good reference for how to state the standard large cardinal axioms in the context of $ZF$? My concern is that it seems that the usual proof that embeddings defined from ...
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Cardinality of maximal elements in terms of set-theoretic inclusion in the space $c_0(\mathbb{N})$
Let $c_0(\mathbb{N})$ be the space of real-valued sequences converging to zero.
Here, each element $\{\alpha_n\} \in c_0(\mathbb{N})$ itself is a "set", so that we can think about set-...
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Universe V = Ultimate L inside set theoretic multiverse
Good day to you all,
I would like to ask a question about relation between Prof. H. Woodin V = Ultimate L and a concept of set theoretical multiverse as proposed by Prof. Hamkins.
If V = Ultimate L ...
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Omega logic, V = Ultimate L and hierarchy of laws collapse
I would like to ask a question about the omega conjecture and its relationship with the V = Ultimate L axiom.
In his lecture Prof. Hugh Woodin have stated that "Assuming the omega conjecture, ...
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Is determinacy of (some) very long open games consistent?
For $\varphi$ a first-order sentence in the language of set theory and $\kappa$ an ordinal, let $G_\varphi^{\kappa}$ be the game of length $\kappa$ in which players $1$ and $2$ alternately play ...
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Is this determinacy principle consistent?
Let $\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$ be the following principle ("determinacy for simple open length-$\omega_1$ games"):
If $\kappa$ is any ordinal and $X\subseteq \kappa^{<\...
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The consistency of $\Sigma_1$-elementary embeddings $j\colon V_{\lambda+2}\to V_{\lambda+2}$ over $\mathsf{ZFC}$
One way to stratify the large cardinal hierarchy between I3 and I1 is by using second-order elementary embeddings. We may view $j\colon V_{\lambda+1}\to V_{\lambda+1}$ as a second-order embedding $j\...
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Some questions about the Hyperuniverse Program
The Hyperuniverse Program, founded by Sy D. Friedman, intends to produce new second-order axioms of set theory which appropriately formalize "the universe is maximal" in one of a few ways. A ...
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Cofinal inconsistency
Apologies in advance if this question is obvious/not research level.
Let $\preceq$ be the consistency strength relationship on theories. Working over $ZF$ or $ZFC$, is there some large cardinal ...
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Large cardinals in ZF + DC + AD
The Axiom of Dependent Choice (DC) is often considered to be an "intuitive and non-controversial" version of choice used in the proofs of many theorems in Analysis. Similarly, the Axiom of ...
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$0^{\#}$ and self embeddings of $L_\gamma$
Let us assume that there is a non-trivial elementary embedding $j \colon L_\gamma \to L_\gamma$ and $\gamma \geq \omega_1^V$. Can we conclude that $0^{\#}$ exists?
In general, it is known that if ...
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Stationary vs measurable limits for large cardinals
This is a follow-up to a question I asked earlier over here.
Why are stationary limits so ubiquitous when studying large cardinals?
I have noticed that there appears to be a stronger limit notion that ...
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Proof (or reference) about the cc-ness of termspace forcing
Recall that for $P$ a forcing order and $\dot{Q}$ a $P$-name for a forcing order, the termspace forcing $T(P,\dot{Q})$ consists of minimal-rank names for elements of $\dot{Q}$, ordered by $\dot{q}'\...
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Can one extend higher randomness theory to the entire analytical hierarchy under certain large cardinal assumptions?
In the "Recursion Theory" book by C.T Chong, Liang Yu, towards the end of the book they list a few "open" research areas connected to higher computability theory. One such ...
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Systems of elementary embeddings
Recently I've been thinking about elementary embeddings, partition cardinals, etc. as part of my never-ending quest for understanding of consistency strength :p
I came up with this idea, called I* ...
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What large cardinals are needed to imply projective sets have the perfect set property?
If there are infinitely many Woodins, then every projective set is determined, whence every projective set has the perfect set property (PSP). Since determinacy is such a stronger property than the ...
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How to express Kunen's inconsistency, Reinhardt and Wholeness axioms, by single sentences?
Working in $\sf NBG, $ can we express the property of a class being set theoretically definable, by a single sentence? Like for example, the following way:
$$\operatorname {std}(X) \iff \exists x_1 \...
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Ramsey-like property with order condition
I wonder if there are regular cardinals $\lambda$ and $\kappa$ such that $\kappa < \lambda \leq 2^\kappa$ and such that, consistently, the following holds:
Let $c: [\lambda]^2 \to \kappa$ be such ...
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Can we have full choice prior to Reinhardt cardinals?
Working in $\sf ZF + Reinhardt \ cardinal$, can we have full choice over all stages $V_{\alpha < \kappa}$ where $\kappa$ is the Reinhardt cardinal, i.e., the critical point of the elementary ...
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Can this method let choiceless large cardinals be smaller than cardinals compatible with choice?
Recall question "Can we have this sequence where choice fails and returns?"
Can that theory be extended with requiring the $\mathcal V_n$'s to fulfill a choiceless large cardinal extension ...
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Can we have this sequence where choice fails and returns?
Can we have a sequence of transitive sets $\langle\mathcal V_0, \mathcal V_1, \mathcal V_2,...\rangle$, all modeling $\sf ZF$, such that $\mathcal P(V_n) \subset \mathcal V_{n+1}$, and the cardinality ...
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Harvey Friedman: The expanding mind
In reference 1, Friedman writes:
I discuss my efforts concerning 3 crucial issues in the foundations of mathematics that are deeply connected with the great work of Kurt Gödel.
[...]
B. Are there ...