Questions tagged [large-cardinals]
The large-cardinals tag has no usage guidance.
772
questions
11
votes
1
answer
437
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, ...
1
vote
0
answers
311
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 ...
6
votes
0
answers
382
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 ...
1
vote
0
answers
160
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 ...
7
votes
0
answers
307
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 ...
4
votes
0
answers
190
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
1
answer
234
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
3
answers
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 ...
14
votes
0
answers
516
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
1
answer
125
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
1
answer
366
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
1
answer
414
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
1
answer
811
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
1
answer
539
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 ...
3
votes
1
answer
443
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
votes
0
answers
116
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
1
answer
126
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
1
answer
229
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
2
answers
553
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 ...
0
votes
1
answer
281
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
votes
0
answers
174
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 ...
15
votes
1
answer
536
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 ...
5
votes
2
answers
594
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
1
answer
334
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
4
answers
1k
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
2
answers
288
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
1
answer
239
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 ...
-1
votes
1
answer
131
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
1
answer
537
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 ...
6
votes
0
answers
173
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
1
answer
409
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
1
answer
232
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 ...
12
votes
1
answer
461
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$...
13
votes
1
answer
737
views
Do we know the consistency strength of the Singular Cardinal Hypothesis failing on an uncountable cofinality?
Suppose that $\kappa$ is a strong limit cardinal. The singular cardinal hypothesis states $2^\kappa=\kappa^+$. We know that the failure of SCH requires large cardinals, and in fact is equiconsistent ...
13
votes
1
answer
561
views
End-extending cardinals
Let us say a cardinal $\kappa$ end-extending if there is a function $F : V_\kappa^{<\omega} \to V_\kappa$ such that:
(a) If $M \subseteq V_\kappa$ is closed under $F$, then $M \prec V_\kappa$.
(b) ...
3
votes
1
answer
233
views
Radin forcing preserving large cardinals
I'm wondering if there are any known result for the maximum large cardinal strength
which can be preserved by Radin forcing? For instance, with any large cardinal hypothesis in the ground model, can ...
1
vote
0
answers
90
views
Why does $p_{n}(i,1)=1$ so often where the polynomials $p_{n}$ are obtained from the classical Laver tables
So I was doing some computer calculations with the classical Laver tables and I found polynomials $p_{n}(x,y)$ such that $p_{n}(i,1)=1$ for many $n$.
The $n$-th classical Laver table is the unique ...
1
vote
0
answers
60
views
Growth rate of the critical points of the Fibonacci terms $t_{n}(x,y)$ vs $t_{n}(1,1)$ in the classical Laver tables
The classical Laver table $A_{n}$ is the unique algebra $(\{1,\dots,2^{n}\},*_{n})$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and $x*_{n}1=x+1\mod 2^{n}$ for all $x,y,z\in A_{n}$.
Define the ...
2
votes
1
answer
105
views
Attraction in Laver tables
If $X$ is a self-distributive algebra, then define $x^{[n]}$ for all $n\geq 1$ by letting $x^{[1]}=x$ and $x^{[n+1]}=x*x^{[n]}$. The motivation for this question comes from the following fact about ...
5
votes
1
answer
241
views
Amalgamation via elementary embeddings
Can there exist three transitive models of ZFC with the same ordinals, $M_0,M_1,N$, such that there are elementary embeddings $j_i : M_i \to N$ for $i<2$, but there is no elementary embedding from $...
1
vote
0
answers
75
views
Multiple roots in the classical Laver tables
The classical Laver table $A_{n}$ is the unique algebraic structure $$(\{1,\dots,2^{n}\},*_{n})$$ such that $x*_{n}1=x+1\mod 2^{n}$ and
$$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$$ for all $x,y,z\in\{1,...
1
vote
0
answers
59
views
Can we have $\sup\{\alpha\mid(x*x)^{\sharp}(\alpha)>x^{\sharp}(\alpha)\}=\infty$ in an algebra resembling the algebras of elementary embeddings?
A finite algebra $(X,*,1)$ is a reduced Laver-like algebra if it satisfies the identities $x*(y*z)=(x*y)*(x*z)$ and if there is a surjective function
$\mathrm{crit}:X\rightarrow n+1$ where
$\mathrm{...
1
vote
0
answers
60
views
In the classical Laver tables, do we have $o_{n}(1)<o_{n}(2)$ for any $n>8$?
The classical Laver table $A_{n}$ is the unique algebraic structure
$(\{1,\dots,2^{n}\},*_{n})$ where
$$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$$
and where $$x*_{n}1=x+1\mod 2^{n}$$ for $x,y,z\in\{1,\...
2
votes
0
answers
81
views
For each $n$ is it possible to have $\mathrm{crit}(x^{[n]}*y)>\mathrm{crit}(x^{[n-1]}*y)>\dots>\mathrm{crit}(x*y)$?
Suppose that $(X,*,1)$ satisfies the following identities:
$x*(y*z)=(x*y)*(x*z),1*x=x,x*1=1$. Define the Fibonacci terms $t_{n}(x,y)$ for $n\geq 1$ by letting
$$t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=...
1
vote
0
answers
44
views
Vastness of inverse systems of Laver-like algebras
Suppose that $(X,*,1)$ satisfies the identities
$x*(y*z)=(x*y)*(x*z),x*1=1,1*x=x$. Then we say that $(X,*,1)$ is a reduced Laver-like algebra if whenever $x_{n}\in X$ for all $n\in\omega$, there is ...
1
vote
0
answers
33
views
Can we always extend a finitely generated reduced Laver-like algebra to a vast inverse system of Laver-like algebras?
An $(X,*,1)$ that satisfies the identities $x*(y*z)=(x*y)*(x*z),1*x=x,x*1=1$ is said to be a reduced Laver-like algebra if whenever $x_{n}\in X$ for $n\in\omega$, there is some $N\in\omega$ where $x_{...
11
votes
3
answers
1k
views
Higher $\infty$-categories
Is there a reason we consider $\infty$-categories to be the $\omega^{th}$ step in the 2-internalization inside Cat (or enrichment over Cat if you prefer)* process made invertible above some finite ...
7
votes
1
answer
370
views
Locally presentable categories, universes, and Vopenka's principle
Some aspects of the theory of locally presentable categories depends on set-theoretic assumptions. Namely, a large cardinal axiom known as Vopenka's principle implies several nice properties of such ...
3
votes
0
answers
240
views
Ordering large cardinal axioms around the level of $n$-huge by consistency strength?
So the large cardinal axioms are for the most part considered to be linearly ordered by consistency strength. For the large cardinals between extendibility and rank-into-rank (i.e. the $n$-huge ...
2
votes
0
answers
53
views
Calibrating the strength of the quotients of subalgebras of the classical Laver tables
Define an algebraic structure $A_{n}$ by letting
$$A_{n}=(\{1,\dots,2^{n}-1,2^{n}\},*_{n})$$
where $*_{n}$ is the unique operation such that $x*_{n}1=x+1\mod 2^{n}$ for $$x\in\{1,\dots,2^{n}-1,2^{n}\}$...