Questions tagged [large-cardinals]
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54 questions from the last 365 days
2
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0
answers
40
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Diamonds on supercompact $\kappa$ after a $\kappa$-c.c. forcing
Let $\kappa$ be supercompact. Then the (supercompact) Laver diamond holds at $\kappa$: There is $f:\kappa\to V_\kappa$ such that for all $\lambda\geq \kappa$ and $x\in H(\lambda^+)$ there is $j:V\to M$...
- 51
2
votes
2
answers
122
views
Existence of k-complete uniform ultrafilter over a regular cardinal, k is strongly compact
This is a question about set theory. Let $\kappa\leq \lambda$ be infinite cardinals such that $\kappa$ is strongly compact and $\lambda$ is regular. My question is: how to construct a $\kappa$-...
- 61
5
votes
1
answer
460
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NBG, ZFC+I, and Global Choice
In Shulman's 2008 paper 'Set Theory for Category Theory', he includes amongst the axioms of $\sf NBG$ the axiom of limitation of size. Being well known to imply the axiom of global choice, it seems to ...
- 53
6
votes
0
answers
188
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Is there a characterization of measurables in terms of indiscernibles?
There is a characterization of $\alpha$-Erdős cardinals in terms of sets of indiscernibles of order type $\alpha$. There is also a characterization of Ramsey cardinals in terms of sets of good ...
- 2,031
8
votes
1
answer
196
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Weakly compact cardinals in $L$: how long do branches take to appear?
Throughout, we work in $\mathsf{ZFC+V=L+}$ "There is a weakly compact cardinal," $\kappa$ is the first weakly compact cardinal and "tree" means "subtree of $2^{<\kappa}$ of ...
- 21.2k
7
votes
1
answer
307
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Proper Forcing Axiom for $|\mathbb{P}| \leq \mathfrak{c}$
Let $\mathsf{PFA}(\mathfrak{c})$ denote the Proper Forcing Axiom (PFA) restricted to posets $|\mathbb{P}| \leq \mathfrak{c}$. I think $\mathsf{PFA}(\mathfrak{c}) \implies \mathfrak{c} = \aleph_2$, but ...
- 1,372
7
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0
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259
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A version of determinacy for all sets
Under ZF + AD, some games are undetermined because of lack of choice. In fact, the axiom of choice is equivalent to determinacy of games of length 2. However, we can ask whether the lack of choice ...
- 7,545
1
vote
2
answers
228
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Can this semi-constructible structure satisfy existence of a measurable cardinal?
If we add a primitive unary function symbol $\mathfrak L$ to the first order language of set theory.
Axiom of semi-constructibility: if $\phi^\alpha (y,x_1,\ldots,x_n)$ is a formula in which all and ...
- 11.3k
7
votes
1
answer
556
views
Does this ZFC+V=L like theory, have a limit on large cardinal properties?
Let $\sf T$ be a theory that has as axioms every axiom of $\sf ZFC$, and every theorem of $\sf ZFC + [V=L]$ that is neither provable nor disprovable by $\sf ZFC$, whose addition or addition of its ...
- 11.3k
23
votes
3
answers
3k
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Why believe in the existence of large cardinals rather than just their consistency?
Large cardinal hypotheses and related hypotheses like projective determinacy are well-known to be gauges of the consistency strength of various theories. What reasons are there to believe in their ...
- 4,985
4
votes
1
answer
186
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Logical relationship between supercompact and rank-into-rank cardinals
It is well known that the large cardinals are ordered by logical consistency. In many cases, the logical consistency is, in fact, a direct implication - for example, Mahlo $\Rightarrow$ Inaccessible ...
- 463
9
votes
0
answers
274
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Has a computer search for inconsistency of large cardinals been carried out before?
In discussion about the consistency of large cardinals, a justification which occasionally appears is that despite years of work, no inconsistency has currently been discovered. For example, here are ...
- 2,031
2
votes
1
answer
150
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Weakly compact characterization
In Theorem 9.26 of Jech, it is shown that if $\kappa$ is inaccessible and has the tree property, then $\kappa \rightarrow (\kappa)^2_\lambda$ for every $\lambda<\kappa$. Jech remarks after the ...
- 41
8
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1
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280
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What is the consistency strength of "Singular worldly that is inaccessible in an inner model"?
In short, what can we say about the consistency strength of "$\kappa$ is a singular worldly and inaccessible in an inner model"?
Clearly, $0^\#$ exists since we have a singular cardinal ...
- 39.7k
1
vote
1
answer
308
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Is reflection on Grothendieck universes equivalent to TG set theory?
Let take the first order set theory whose axioms are Extensionality, Separation and Universal reflection.
By $\operatorname {unv}(x)$, denoting "$x$ is a universe", we'll take it to mean ...
- 11.3k
6
votes
1
answer
199
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$L(\mathbb{R})$-absoluteness from a proper class of Woodins: source?
For a paper I'm writing I need to use (as a blackbox) the following theorem: if there is a proper class of Woodin cardinals and $G$ is set-generic, then $L(\mathbb{R})$ and $L(\mathbb{R})^{V[G]}$ are ...
- 21.2k
9
votes
0
answers
177
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Inner model of "CH + large cardinals" that satisfies MM?
I've been told that if $V$ has CH and large cardinals, then there is an inner model in which MM holds. The sketch is as follows:
Any $\Sigma^2_1$ sentence that can be forced is already true in such $V$...
- 335
5
votes
1
answer
155
views
A function $f$ such that $j_U(f)(\kappa)=[\operatorname{id}]_U$ for all ultrapower embeddings $j_U$ with critical point $\kappa$
Let $\kappa$ be a measurable cardinal. Is there a function $f\colon\kappa\to V$ such that whenever $j_U\colon V\to\operatorname{Ult}(V,U)$ is an ultrapower embedding with critical point $\kappa$, we ...
- 2,196
4
votes
1
answer
140
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Coherent sequence of ultrafilters in iterated forcing extensions
Remember that if $\kappa$ is strongly compact, then any ${<}\kappa$-complete filter extends to a ${<}\kappa$-complete ultrafilter.
Let $\Bbb P_\delta=\langle\Bbb P_\alpha,\dot{\Bbb Q}_\alpha\mid ...
- 393
4
votes
2
answers
252
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Extending normal filters
If $F$ is a $\kappa$-complete filter on some set $S$, and $F$ is generated by a basis of size $\lambda$, then $F$ extends to a $\kappa$-complete ultrafilter on $S$ when we assume that $\kappa$ is $\...
- 393
9
votes
2
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455
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Determinacy and Woodin cardinals
I am looking for a reference for the following result:
Theorem Assume $\kappa$ is a Woodin cardinal. Then after forcing with the Levy collapse $\mathbb{P}=Col(\omega, \kappa)$, the $\Sigma^1_2$-...
- 32.1k
4
votes
0
answers
177
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Recording of 2009 lecture on Harvey Friedman's work
On December 13--20 2009 at Bristol, there was a meeting devoted to thorough dissection of Harvey Friedman's work on the foundations of mathematics and his statements claimed to be equivalent to ...
- 2,031
9
votes
1
answer
252
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Copy of $P(\omega)/\mathrm{fin}$ on $\omega_1$
$\newcommand{\fin}{\mathrm{fin}}$Under what hypotheses does there exist a uniform ideal $I$ on $\omega_1$ such that $P(\omega_1)/I \cong P(\omega)/\fin$? What is the consistency strength?
It follows ...
- 18.6k
1
vote
1
answer
112
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Constructible cardinality downslides and their consistency strengths?
Posting "Large cardinals and constructible universe" mentions that $\omega_1^L < \omega_1$ if we assume Ramsey cardinal.
My question can we have more downslides like for example $\omega_2^...
- 11.3k
1
vote
0
answers
62
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Can this theory interpret TG? Would its Reinhardt's extension be equivalent to the usual one?
Language: FOL
Primitives: $=, \in$
Axioms:
Extensionality: as in Z
Define: $\operatorname {set}(y) \iff \exists x: y \in x$
Comprehension: $$n=0,1,2, \ldots \\ \forall \operatorname {set} x_1, \cdots, ...
- 11.3k
12
votes
1
answer
322
views
Which $L$-like principles are known to be relatively consistent with large cardinals?
For which of the standard large cardinal axioms $\varphi_{LC}$ and which $L$-like principles $\psi$ (e.g. GCH, $\mathrm{V}=\mathrm{HOD},$ the ground axiom, and various diamond and square principles) ...
- 4,316
6
votes
1
answer
227
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Forcing $\neg\square_{\omega_1}$ from a Mahlo cardinal, reference
In Jensen's The fine structure of the constructible hierarchy, it is stated that Solovay proved the consistency of $\neg\square_{\omega_1}$ by collapsing a Mahlo cardinal to $\omega_2$. I was ...
- 2,286
5
votes
1
answer
484
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Large cardinals approached through $\infty$-categories
I am an undergraduate student (rising junior) majoring in philosophy and mathematics. For some time, I have been interested in homotopy type theory and so-called "univalent foundations". On ...
- 225
5
votes
1
answer
139
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On a question about ordinals $\xi$ satisfying $j_0(\xi)=j_1(\xi)$ for an $I_3$-embedding $j$
Let $j\colon V_\lambda\to V_\lambda$ be an $I_3$ embedding with the critical sequence $\kappa_n$. Define $j_0=j$, $j_1 = j[j]=\bigcup_{\alpha<\lambda} j(j\upharpoonright V_\alpha)$. My question is ...
- 3,042
5
votes
0
answers
150
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Consistency upper bounds for $\neg\square_{\aleph_\omega}$
In the introduction of Cummings and Friedman's $\square$ on the singular cardinals the following is written:
Failure of $\square_\lambda$ for $\lambda$ singular is stronger and rather more ...
- 2,286
4
votes
1
answer
247
views
Jensen's proof that $\diamondsuit$ holds at subtle cardinals
At the end of these notes by Ronald Jensen (which I found from this question) there is a proof that $\diamondsuit_\kappa$ (diamond principle) holds if $\kappa$ is a subtle cardinal.
By induction on $\...
- 1,097
4
votes
1
answer
158
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Impact of coining $L$ in $\mathcal L_{\omega_1, \omega}$ on which large cardinals it can satisfy?
What happens to the constructible universe $L$ if we build it in the first of infinitary languages $\mathcal L_{\omega_1, \omega}$?
Would the usual limitation of $L$ not satisfying existence of a ...
- 11.3k
4
votes
1
answer
148
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Reference request: $\kappa^+$-saturated ideal from iterated Cohen forcing on measurable $\kappa$
In Kunen [1] the author makes the following note: Let $\kappa$ be measurable with normal measure $\mathscr{U}$ in a model of $\mathsf{GCH}$. Let $\mathbb{P}$ be an iteration of $\operatorname{Add}(\...
- 2,196
7
votes
1
answer
793
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Must there be a proper class of Reinhardt cardinals if there is a Reinhardt cardinal?
A cardinal is Reinhardt if $\kappa$ is the critical point of a nontrivial elementary embedding of $V$ to itself, where $V$ is the class of all sets. As Reinhardt cardinals are inconsistent with $\...
- 2,031
8
votes
0
answers
246
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Large cardinals beyond choice and HOD(Ord^ω)
Are Reinhardt and Berkeley cardinals (and even a stationary class of club Berkeley cardinals) consistent with $V=\mathrm{HOD}(\mathrm{Ord}^ω)$ ?
It seems natural to expect no, but I do not see a proof....
- 7,545
6
votes
2
answers
308
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Reference: If $X$ is metrizable, then $X$ is realcompact iff $|X|$ is non-measurable
Note: What I call a measurable cardinal seems to be non-standard among set theorists, and should be called a $\sigma$-measurable cardinal.
I know that a discrete space is realcompact iff its non-...
- 1,201
3
votes
0
answers
200
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Weak extender models for supercompactness without choice
Assume ZFC and a supercompact cardinal $κ$. Is it consistent that there is a weak extender model $N⊨\text{ZF}$ for supercompactness of $κ$ such that the axiom of choice and well-ordering of $P_κ(λ)^N$...
- 7,545
15
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2
answers
1k
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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 &...
- 667
9
votes
1
answer
313
views
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 $I$.
...
- 1,142
10
votes
1
answer
416
views
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, ...
- 333
1
vote
0
answers
150
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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 \,...
- 11.3k
1
vote
0
answers
121
views
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 ...
- 11.3k
6
votes
0
answers
125
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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 ...
- 7,545
12
votes
1
answer
845
views
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 ...
5
votes
0
answers
190
views
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 ...
9
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0
answers
258
views
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 $\mathsf{SCH}$ ...
- 877
5
votes
1
answer
196
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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 $V$...
- 3,042
11
votes
1
answer
441
views
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 $\gamma$ ...
- 2,031
2
votes
0
answers
144
views
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 ...
- 695
9
votes
1
answer
844
views
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?
- 3,574