Questions tagged [large-cardinals]
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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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Can Ackermann's reflection be proved in this reflective theory?
If we extend the language of set theory, by adding class symbols in upper case and reserve the lower cases for sets.
Add axioms:
Bi-sorted ID theory (see end of this posting) +
The class axioms:
Sorts:...
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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 ...
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Can we interpret Reinhardt cardinals this way?
To the language of set theory add a primitive unary predicate $\operatorname {Universe}$ and a primitive total unary function $j$. Add all axioms of $\sf ZF$ in the language of this theory, i.e. the ...
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Can we have a 'universal class' for elementary embeddings $j\colon V\to V$
Work over $\mathsf{GB}$, Gödel-Bernays set theory (without choice). My question is the following:
Question. Is the following statement consistent with $\mathsf{GB}$? There is a universal class for ...
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Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?
Is it possible to iterate elementary embeddability and reflect on those stages that are elementary embeddable to themselves?
The following is a formal capture of that idea:
To the language of $\sf ZF$...
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Is there a form of choice that can elude Kunen's inconsistency theorem?
When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, ...
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If we add stratified\acyclic replacement to the wholeness axiom, would that increase its consistency strength?
If we add to the wholeness axiom, the axiom of stratified\acyclic replacement, what would be the consistency state and strength of the resulting theory?
The wholeness axiom $\sf WA$, introduced by ...
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Cat as a bicategory of monads over another category
Let's assume infinitely many Grothendieck universes exist. Let's call $\kappa$-Cat the bicategory of $\kappa$-small categories with anafunctors and anatural transformations. Now for any $\lambda$ and ...
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If a theory speaks of sets that cannot be forced to be parameter free definable, then does this entail a large cardinal property?
If we say that an effectively generated first order theory $\sf T$ extends $\sf ZF$, such that every countable model of $\sf T$ doesn't have a class forcing extension that is pointwise definable. ...
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Feferman's universes for proof assistants?
This question was prompted by a discussion from another MO question about the consistency of ZFC. There are some mathematicians who are comfortable with ZFC but uneasy with large cardinals. For them,...
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Where do the universe embedds to in Reinhardt's cardinals setting?
I just want to understand the embedding behind Reinhardt's cardinals. We have an elementary embedding $j: V \to V$. Let the background theory be $\sf MK - Choice$. We know that $V$ itself is a class ...
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Compatibility of $\mathsf{SVC}$ and Reinhardtness
Can we prove the consistency of $\mathsf{ZF+SVC}$ + "There is a Reinhardt cardinal?" (Preferably from the consistency of $\mathsf{ZF}$ with a Reinhardt cardinal, but using a stronger ...
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"Very $L$-like" models, part 2: combinatorics
Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite ...
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Some relevant questions about the consistency strength of singularity of $\omega_1$ and $\omega_2$
The following question was asked years ago on MSE, but let me recap it:
Question: Is there anything currently known about the exact consistency strength of "$\mathsf{ZF}$ + both $\omega_1$ and $\...
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First inaccessible Suslin trees in L, an interesting detail
It's known (but quite nontrivial) that $V=L$ implies that if $\kappa$ is the 1st inaccessible cardinal then there are $\kappa$-Suslin trees $T$.
Such a tree $T$ can be considered as a forcing notion ...
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How much condensed mathematics can be founded on finite order arithmetic (or ETCS) instead of ZFC?
I recently learnt from David Roberts' answer that, there is a way, due to Colin McLarty, to set up the foundations on finite order arithmetic for EGA & SGA. In particular, all usages of ...
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"Very $L$-like" models, part 1: large cardinals
(The original version of this question was much narrower and less natural; but see the edit history if interested.)
Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ ...
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Large cardinals and measurability in $L(A)$
Under suitable large-cardinal assumptions, in the inner model $L(\mathbb R)$ one can have $\omega_1$ and $\omega_2$ measurable (this follows from determinacy).
I was wondering if it is possible to ...
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Weakening of open determinacy for uncountably long games
For a cardinal $\kappa$ I'll use the phrase "$(\kappa,\kappa)$-game" to mean "two-player, perfect-information, deterministic game on $\kappa$ of length $\kappa$."
Say that a ...
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Absoluteness of the core model under a proper class of completely Jónsson cardinals
Example 2.4.2 of Larson's The stationary tower (based on Woodin's lecture) describes how we can absorb a generic filter over a set forcing in a definable inner model into a generic extension by $\...
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On the strength of higher-logic analogues of $\mathsf{ZFC}$ + Montague's Reflection Principle
Throughout, I work in $\mathsf{MK}$ in order to be able to conveniently quantify over logics; if one prefers, we can restrict attention to (say) $\Sigma_{17}$-definable logics and work in $\mathsf{ZFC}...
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Upwards-fragility of inaccessibles (again)
Although self-contained, this question is a follow-up to this earlier one. Also, thanks to Fedor Pakhomov for fixing a trivial early version of this question!
Work in $\mathsf{ZFC}$ + "There is a ...
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Canonical functions on $\kappa$ and canonical stationary sets
This is a somewhat continuation of this question. The related paper is Jech's Stationary subsets of inaccessible cardinals. See also Chapter 8 and Chapter 24 of Jech's Set Theory.
I would like to ask ...
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Fragility of large cardinals with respect to transitive end extensions
To motivate things, let me start with a special case of the question I'm interested in. Let $\mathsf{In}(x)\equiv$ "$x$ is an inaccessible cardinal."
Question 1: Is it consistent with the ...
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How strong is this "modal definability principle"?
Throughout, we work in the class theory $\mathsf{MK}$ (although I'm open to tweaking this), "logic" means "set-sized logic whose semantics is definable over $V$," and "$\...
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Why are stationary limits so ubiquitous when studying large cardinals?
While studying large cardinals, I have frequently noticed the following phenomenon: If X and Y are two different types of large cardinals, then every cardinal of type X is a stationary limit of ...
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Does $2^{\aleph_0}\rightarrow [\aleph_1]^2_3$ require that the continuum is weakly inaccessible?
A classic result of Sierpiński shows that $2^{\aleph_0}\nrightarrow [\aleph_1]^2_2$, that is, there is a coloring of pairs of real numbers using two colors such that both colors appear on any ...
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Intuition for branch uniqueness in inner model theory
In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage?
At the level ...
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Can there be no complexity bound on the definable elementary $V\rightarrow M$?
This starts with a vaguely-recalled result (which may be false!): that if $\mathcal{U}$ is a measure on the least measurable cardinal $\kappa$, then every elementary $j: L[\mathcal{U}]\rightarrow M$ ...
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Inner model theory using indiscernibles
Has an inner model theory been developed on the basis of indiscernibles rather than measures? Is there a reasonable formalization at the level of overlapping extenders?
Fine-structural models beyond $...
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How would one formulate large cardinals beyond rank into rank?
Crossposting from MSE, after deciding that this question is related to modern research in set theory: https://math.stackexchange.com/questions/4499391/how-would-one-formulate-large-cardinals-beyond-...
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