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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 ...
Lorenzo's user avatar
  • 2,216
3 votes
1 answer
305 views

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 ...
safsom's user avatar
  • 33
4 votes
1 answer
95 views

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 ...
Hanul Jeon's user avatar
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4 votes
0 answers
113 views

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 ...
Lorenzo's user avatar
  • 2,216
4 votes
1 answer
203 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 $\...
Arvid Samuelsson's user avatar
4 votes
1 answer
142 views

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 ...
Zuhair Al-Johar's user avatar
4 votes
1 answer
138 views

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}(\...
Calliope Ryan-Smith's user avatar
7 votes
1 answer
684 views

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 $\...
C7X's user avatar
  • 1,400
7 votes
0 answers
152 views

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....
Dmytro Taranovsky's user avatar
6 votes
2 answers
248 views

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-...
Jakobian's user avatar
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3 votes
0 answers
151 views

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 $P_κ(λ)^N$...
Dmytro Taranovsky's user avatar
14 votes
2 answers
991 views

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 &...
n901's user avatar
  • 555
9 votes
1 answer
289 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 $I$. ...
Toby Meadows's user avatar
  • 1,132
9 votes
1 answer
353 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, ...
Lxm's user avatar
  • 323
1 vote
0 answers
147 views

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 \,...
Zuhair Al-Johar's user avatar
1 vote
0 answers
112 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 ...
Zuhair Al-Johar's user avatar
6 votes
0 answers
117 views

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 ...
Dmytro Taranovsky's user avatar
12 votes
1 answer
730 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 ...
Anonymous grad student's user avatar
5 votes
0 answers
164 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 ...
Alexey Slizkov's user avatar
9 votes
0 answers
228 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 $\mathsf{SCH}$ ...
new account's user avatar
5 votes
1 answer
176 views

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 $V$...
Hanul Jeon's user avatar
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10 votes
1 answer
375 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 $\gamma$ ...
C7X's user avatar
  • 1,400
2 votes
0 answers
131 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 ...
Ember Edison's user avatar
8 votes
1 answer
822 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?
Frode Alfson Bjørdal's user avatar
12 votes
1 answer
458 views

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 ...
Binary198's user avatar
  • 704
8 votes
2 answers
1k views

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 ...
C_M's user avatar
  • 83
10 votes
0 answers
209 views

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. $|c"[H]^2|<\alpha$. ...
Jiachen Yuan's user avatar
3 votes
2 answers
321 views

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 ...
Monroe Eskew's user avatar
  • 17.8k
7 votes
1 answer
564 views

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....
new account's user avatar
8 votes
1 answer
601 views

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 ...
Arian's user avatar
  • 183
23 votes
4 answers
3k views

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 ...
Nai-Chung Hou's user avatar
14 votes
1 answer
1k views

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 ...
Tim Campion's user avatar
  • 61.9k
12 votes
1 answer
424 views

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 $\...
Ben Goodman's user avatar
6 votes
2 answers
277 views

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 $...
Calliope Ryan-Smith's user avatar
5 votes
1 answer
242 views

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 ...
Hannes Jakob's user avatar
  • 1,622
6 votes
0 answers
182 views

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 ...
Arshak Aivazian's user avatar
0 votes
0 answers
139 views

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 $\mathsf{ZFC}+P$ ...
Calliope Ryan-Smith's user avatar
5 votes
0 answers
125 views

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 $...
Tim Campion's user avatar
  • 61.9k
16 votes
2 answers
730 views

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 ...
Tim Campion's user avatar
  • 61.9k
8 votes
0 answers
334 views

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 ...
interregno's user avatar
5 votes
2 answers
539 views

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). ...
0x11111's user avatar
  • 493
1 vote
0 answers
183 views

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 ...
Erin Carmody's user avatar
-1 votes
1 answer
318 views

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, ...
PaleChaos's user avatar
8 votes
0 answers
402 views

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 ...
Someone211's user avatar
8 votes
2 answers
914 views

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 ...
Joseph Van Name's user avatar
4 votes
0 answers
219 views

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 ...
Rupert's user avatar
  • 2,055
1 vote
0 answers
55 views

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-...
Isaac's user avatar
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3 votes
0 answers
207 views

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 ...
Pan Mrož's user avatar
  • 171
4 votes
0 answers
588 views

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, ...
Pan Mrož's user avatar
  • 171
2 votes
0 answers
259 views

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 ...
Noah Schweber's user avatar

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