Questions tagged [large-cardinals]
The large-cardinals tag has no usage guidance.
547
questions
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1answer
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Can we recover an inner model of CH after forgetting some generic information?
Suppose $\kappa$ is an inaccessible cardinal. Let $G \times H$ be $\mathrm{Col}(\omega_1,{<}\kappa) \times \mathrm{Add}(\omega,\kappa)$-generic over $V$. Let $X \subseteq \kappa$ be $\mathrm{Add}(...
0
votes
0answers
65 views
At which large cardinal property Ackermann set theory + finitization rule would stop?
By the finitization rule I mean a rule that inputs a schema in the $V$ world and outputs a single statement in the $V$ world that serves to capture that schema!
So in this sense we'll have for ...
5
votes
0answers
165 views
Inner models with all sets generic
Question: Under large cardinal axioms, what is the intersection of all inner models $M$ of ZFC such that every set in $V$ is set-generic over $M$?
Every set belongs to a generic extension of HOD, and ...
0
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0answers
70 views
Would iterating reflection and resemblance results in increment of consistency strength?
Question: Can we have a reverse hierarchy of sets resembling $V$?
That is a sequence of universes $W_\alpha$ where each is an element of the prior one, and such that each universe reflects ...
5
votes
0answers
154 views
Can a weakly inaccessible non-weakly-Mahlo cardinal carry a $\kappa$-complete, $\kappa^+$-saturated ideal?
An ideal $I$ on a regular cardinal $\kappa$ is said to be $\mu$-saturated if whenever a family $\langle S_\alpha \mid \alpha<\lambda\rangle$ of subsets of $\kappa$ is such that each $S_\alpha\notin ...
4
votes
0answers
121 views
Absoluteness and the scale property for $Π^2_2$ or $Σ^2_2$
Under the diamond principle $◊$ and large cardinal axioms, which of the two pointclasses $Π^2_2$ or $Σ^2_2$ is expected to have the scale property?
Because conditional $Σ^2_2$ absoluteness under $◊$ ...
8
votes
1answer
221 views
How to understand the interface of the consistency strength hierarchy, reverse mathematics, and proof-theoretic ordinal analysis?
I am aware of three major "hierarchies" of mathematical theories, but I don't know how to relate these hierarchies to one another. Here are the hierarchies I have in mind:
Consistency strength. My ...
0
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0answers
118 views
What's the consistency strength of resemblance + global failure of the continuum hypothesis?
Let $T$ be a theory formalized in first order logic with equality and membership and the additional primitive constant symbol $W$, with the following axioms:
Extensionality: $\forall z (z \in x \...
7
votes
0answers
151 views
$\Sigma^2_2$ absoluteness and $\diamondsuit$
This question touches on recent questions concerning the role of $\diamondsuit$ and the Continuum Hypothesis. In particular, I would like to know more information about a conjecture of Woodin on the ...
11
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0answers
293 views
$\Sigma^2_1$ and the Continuum Hypothesis
This is a follow up to Will Brian's answer to this recent question. In particular, quoting Brian:
"In fact, Paul Larson has pointed out to me that the statement "$\phi$ and $\phi^{-1}$ are conjugate"...
0
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0answers
81 views
Are there minor tweaks of hereditary replacement that can prove large cardinal properties?
Hereditary replacement: if $\phi(x,y)$ is a formula in which only symbols $x,y$ occur free, and those never occur bound, and in which symbol $B$ never occur; then: $$\big{(}\forall A [\forall x \...
14
votes
2answers
335 views
Consequences of existence of a certain function from $\omega_1$ to $\omega_1$
In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any countable elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is ...
9
votes
1answer
316 views
Is the product of commuting ultrafilters an ultrafilter?
If $U$ is a filter on $X$ and $W$ is a filter on $Y$, their product is the filter $U\times W$ on $X\times Y$ generated by rectangles $A\times B$ where $A\in U$ and $B\in W$.
In certain circumstances ...
10
votes
1answer
303 views
Does $Π^1_{2n+1} = (Σ^1_{2n+1})^{M_{2n-1}}$?
Every $Π^1_1$ formula $φ$ without free second order variables can be converted into a $Σ^1_1$ $ψ$ such that $φ ⇔ ψ^\mathrm{HYP}$, and vice versa. ($\mathrm{HYP}$ is the hyperarithmetical universe, ...
3
votes
0answers
255 views
Is there a second-order expression of '$\kappa$ is Reinhardt' in $NGB$, where $\kappa$ is a cardinal?
In their paper, "Generalizations of the Kunen inconsistency", (Annals of Pure and Applied Logic 163 (2012) 1872-1890, doi:10.1016/j.apal.2012.06.001, arXiv:1106.1951), Hamkins, Kirmayer, and ...
5
votes
0answers
213 views
General theory of the reals in Solovay-like models
Solovay's model is a famous model of $\sf ZF$ where we start in $L$ with $\kappa$ inaccessible, and we collapse all the ordinals below $\kappa$ to be countable, without collapsing $\kappa$ itself, and ...
0
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0answers
75 views
What's the strength of capturing set theory in labeled Mereology?
To Atomic General Extensional Mereology + Bottom, add a primitive one place partial function symbol $L$, signifying "the label of", an add axioms:
Distinctiveness: $Lx=Ly \to x=y$
Labels: $\forall x (...
1
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0answers
108 views
A parsimonious large cardinal axiom
The ordering of large cardinals by consistency strength is well known.
I was wondering what one can say regarding an ordering by direct implication.
In particular, I am looking for is a parsimonious ...
6
votes
0answers
208 views
$0^\#$ in weak theories vs large cardinals in $L$
To better understand the transition from large cardinal axioms consistent with the constructible universe $L$ to large cardinal axioms transcending $L$, I am looking for natural equiconsistencies ...
3
votes
0answers
149 views
Elementary self-embeddings conservative over ZFC
Question: Is the following theory conservative over ZFC? And if not, what is its strength?
Language: $∈$, $j$ (unary function symbol)
Axioms:
1. ZFC (without separation and replacement for formulas ...
3
votes
1answer
166 views
The intersection of all normal ultrafilters on a measurable cardinal
Suppose $\kappa$ is a measurable cardinal. Let $W$ be the intersection of all normal ultrafilters on $\kappa$.
I am interested in a precise characterization of the filter $W$.
One sure way to ...
12
votes
3answers
1k views
Necessary use of large cardinals in mathematics [duplicate]
There are some statements, whose consistency (or the consistency of their negation) require the existence of large cardinals (in the sense that if the statement (or its negation) is consistent, then ...
10
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0answers
253 views
Seeing what gets Harvey Friedman's “tangible incompleteness” principles into large cardinal territory
I'm trying to wrap my head around some of Harvey Friedman's recent, unpublished work on his tangible incompleteness project, and I'm trying to see the link between his "tangible statements" (...
1
vote
1answer
86 views
Complexity of a proper class of extendibles
If consistent, is existence of a proper class of extendible cardinals provably equivalent to a $Σ^V_5$ statement?
Recall that in ZFC, a cardinal $κ$ is extendible iff for every $λ>κ$ there is an ...
-1
votes
1answer
284 views
What is the consistency strength of adding this ordinal reflection scheme on top of Ackermann set theory?
Axiom scheme of Ordinal Reflection: if $\phi$ is a formula that doesn't use the symbol $V$, whose parameters are among $x_1,..,x_n$; then: $$\forall x_1 \in V,\dotsc,\forall x_n \in V: \phi(On) \to \\\...
6
votes
1answer
380 views
When does “sufficient genericity” actually suffice?
Fix a forcing notion $\mathbb{P}$. Say that a formula $\varphi(x)$ with parameters is $\mathbb{P}$-enforceable if there is some countable set $\mathcal{D}$ of dense sets in $\mathbb{P}$ such that for ...
6
votes
1answer
513 views
Very large axiom of choice
let me say that I am not a set theorist, but I have to settle up some things in category theory and I need your help.
What I'd like to do is, in some way, use axiom of choice for proper classes.
I ...
5
votes
1answer
394 views
Can the category of partial orders be fully embedded in the category of linear orders?
Q(1): Can the category of partial orders be fully embedded in the category of linear orders?
Vopěnka's principle, or VP, is a very intriguing axiom with many equivalent forms and consequences ...
2
votes
1answer
197 views
(ZC + $\Sigma_2$ replacement + inaccessible cardinal) equiconsistent with (ZFC + inaccessible cardinal)?
Randall Holmes has made a quite convincing argument against the fact that the full axiom schema of replacement should be considered as “intuitively obvious”—even though he does believe ZFC to be ...
0
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0answers
67 views
Is having multiple embedding proper class hierarchies consistent?
Question 1: Is it consistent to extend "MK - Extensionality + weak Extensionality (as in NFU) - Infinity - limitation of size + set existence+ axiom of subsets (that is: a subclass of a set is a set)" ...
0
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0answers
99 views
What's the consistency strength of this theory of Stretchable Hierarchies?
Working in Morse-Kelley set theory:
A hierarchy is defined as a class that is the union of sets uniquely indexed by ordinals, called as stages, such that each stage is the power set of the ...
-2
votes
1answer
117 views
Is the principle of indifference of hierarchical construction consistent? What's its consistency strength?
Sometimes when one tries to capture the abstract aspect of some notion that is intuitively considered as being truly of abstract nature, in set theoretic terms, then this can extend the theory in a ...
2
votes
1answer
196 views
Possible inconsistency related to embeddings $j: M\prec V$
In the paper
Vickers, J.; Welch, P. D., On elementary embeddings from an inner model to the universe, J. Symb. Log. 66, No. 3, 1090-1116 (2001). ZBL1025.03049.
it is stated to that if $Ord$ is ...
0
votes
2answers
294 views
Why do ordinal collapsing functions use regular cardinals?
Inaccessible cardinals are defined as regular strong limit cardinal, and weakly inaccessible cardinals as regular weak limit cardinal. These cardinals are used by some ordinal collapsing functions. My ...
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0answers
36 views
Why do ordinal collapsing functions use regular cardinals? [duplicate]
Inaccessible cardinals are defined as regular strong limit cardinal, and weakly inaccessible cardinals as regular weak limit cardinal. These cardinals are used by some ordinal collapsing functions. My ...
0
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0answers
144 views
Is there no anti-foundational theory exists Reinhardt and hold Global Choice?
J. D. Hamkins proved in "The foundation axiom and elementary self-embeddings of the universe" that, working in $ZFGC^− +BAFA$, there are nontrivial automorphisms and elementary embeddings of the ...
0
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0answers
140 views
Can second order ordinal arithmetic be extended to the same extent as ZFC?
In a prior posting, I've posed the idea of reducing set theory to an extended kind of second order ordinal arithmetic $\small \sf`` 2 oO A"$. The idea was to have a domain of ordinals and sets of ...
11
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0answers
262 views
What would cohomological localization be good for?
An open problem in algebraic topology is whether arbitrary cohomological localizations of simplicial sets (or, equivalently, topological spaces) can be proven to exist in ZFC. It's provable in ZFC ...
4
votes
1answer
286 views
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 \...
1
vote
1answer
281 views
Proving independence with large cardinals?
Suppose I want to prove some statement S is independent of ZFC.
Now instead of the usual approach of making models, I do the following:
- Take two large cardinal axioms L1 and L2
- Prove that ZFC + L1 ...
8
votes
4answers
914 views
“Bootstrapping” an unbounded class of inaccessible cardinals
The "richness principle" of set theory asserts roughly that "everything that happens once should happen an unbounded number of times".
An example would be the existence of an unbounded class of ...
5
votes
2answers
156 views
Stationary sets and $\kappa$-complete normal ultrafilters
Let $\kappa$ be a measurable cardinal, and let $u$ be a normal $\kappa$-complete ultrafilter over $\kappa$. It is a standard easy fact that every closed unbounded set must belong to $u$ (notice that ...
5
votes
1answer
196 views
Uniqueness of countable version of $L[U]$?
Suppose $\alpha$ is a countable ordinal and $U_0,U_1,\kappa$ are such that $L_\alpha[U_i] \models \mathrm{ZFC} + U_i$ is a normal ultrafilter on $\kappa$. Does $U_0 = U_1$?
The argument for ...
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votes
1answer
110 views
What is the strength of claiming that the class of all $V_\kappa$ stages that are $H_\kappa$ when $\kappa$ is regular, is inaccessible?
[EDIT] This posting had been edited to assert that we are speaking about regular mutual stages.
Let $H_{\kappa}$ be the set of all sets that are hereditarily strictly smaller in cardinality than ...
4
votes
1answer
286 views
Upward reflection of rank-into-rank cardinals
Rank-into-rank cardinals have the rather intriguing property that they reflect upwards. I would be interested to know how far the upward reflection goes:
1) Does "There exists a rank-into-rank ...
0
votes
0answers
90 views
Can the cardinality of the set of all intervening cardinals between sets and their power sets be always singular?
This is a question that I've posted to Mathematics Stack Exchange, that was un-answered.
To re-iterate it here:
Is the following known to be consistent relative to some large cardinal assumption?
$...
6
votes
0answers
136 views
Generalized graph-minor theorem?
Consider the following generalized graph-minor theorem:
GM($κ,λ$): Given any collection $S$ of $κ$ simple undirected graphs each with less than $λ$ vertices, there are distinct graphs $G,H$ in $S$ ...
0
votes
1answer
157 views
Proving that being an inaccessible cardinal is absolute, for $V_\kappa$, where $\kappa$ is inaccessible?
I'm going through the proof that if $\kappa$ is inaccessible then $V_\kappa \vDash \mathrm{ZFC}$ and how thus we have $\mathrm{ZF} \nvdash \text{"There exist inaccessible cardinals"}$.
So the last ...
6
votes
1answer
203 views
Generic saturation of inner models
Say that an inner model $M$ of $V$ is generically saturated if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses ...
10
votes
1answer
259 views
Does the statement 'there exists a first-order theory $T$ with no saturated models' have any set theoretic strength?
Exercises 6, 7, and 8 in section 10.4 of Hodges' big model theory textbook contain an outline of a proof of the consistency of the following statement
There exists a countable first-order theory $T$...
