Questions tagged [large-cardinals]
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869 questions
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Theories satisfying Vopenka's Principle for class-sized logics
Given a logic $\mathcal{L}$ and a class of structures $\mathbb{K}$ all in the same signature, say that $\mathbb{K}$ is $\mathcal{L}$-Vopenka iff every proper class $\subseteq\mathbb{K}$ contains a ...
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Is there a generically-undetermined analytic game weaker than $0^\sharp$?
Throughout, I assume large cardinals in $V$ - at least a measurable, and I'm happy to assume more.
There are (lightface) analytic games which are determined iff $0^\sharp$ exists - e.g. $\{a: a$ ...
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Definition of n-huge cardinals in Harvey Friedman's works
In his works like here https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/OrdInvInc090214a-11dg5ov.pdf Harvey Friedman uses notion of $n$-huge cardinals different from the usual one. ...
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How "natural" is Paris-Harrington?
In his The Future of Set Theory, Shelah observes the following:
A very interesting phenomenon, attesting to the naturalness of these [large cardinal] axioms, is their being linearly ordered (i.e., of ...
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Recursion theory of $0^\sharp$?
As the notation suggests, one can foolishly view $0^\sharp$ as an inner-model theoretic $0'$.
Has any existing research addressed any problems of the following forms, possibly weakened/strengthened ...
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Designing turing machines that are nonhalting unprovably from strong large cardinals
The busy beaver function BB(n) is typically defined as the maximum amount of steps any of the n-state turing machines (TMs) will take before halting. BB(n) not only grows faster than any computable ...
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How strong can certain analogues of ZFC be?
This requires a fair amount of definitional overhead; I'm first going to simply state the question with a rough explanation of what it means, and then below the fold give the precise definitions along ...
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What is the consistency strength of SRP (Stationary Ramsey Property)?
SRP, or "Stationary Ramsey Property", is a formal theory mentioned by Harvey Friedman in some of his papers. It came of interest to me when I saw the paper on the 7,910 state turing machine ...
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If there is a $\kappa$-additive atomless probability measure, can arbitrarily large measure algebras be realized by $\kappa$-additive measures?
Suppose there is an atomless probability measure $\mu$ on some measurable space and that $\mu$ is $\kappa$-additive for some regular cardinal $\kappa>\omega_1$. Now, consider an infinite cardinal $\...
- 5,459
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Measure algebra of a real-valued measurable cardinal
Suppose there is a real-valued measurable cardinal $\kappa \leq \mathfrak{c}$. Then there is an atomless $\kappa$-additive probability measure $\mu$ on $(\kappa, \mathcal{P}(\kappa))$. Is there a ...
- 5,459
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Is it consistent every $\mathfrak{c}$-complete ideal on $\mathfrak{c}$ is contained in the null ideal of an everywhere-defined probability measure?
Let $\mathfrak{c}$ denote the continuum. Is the following statement consistent with ZFC (relative to some large cardinal assumptions)?
$\mathfrak{c}$ is a regular cardinal. Furthermore, for every ...
- 5,459
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How class-Berkeley cardinal compares to SuperReinhardt cardinal?
From Cantor's Attic:
A cardinal κ is a Berkeley cardinal, if for any transitive set $M$ with $κ∈M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(j)&...
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Are there large cardinal axioms compatible with choice yet not with class well ordering principle?
Add a new primitive binary relation $ \prec$ to the language of Morse-Kelley class theory "$\sf MK$", which is meant to be a strict well order on all classes. So, it obeys the following ...
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Collapsing cardinals to extend the ranges of parameters of large cardinal properties
There are quite a lot large cardinal properties that have an ordinal parameter with bounded value. I'm curious that can we extend their range by collapsing cardinals.
For example, consider the $\alpha$...
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Understanding tree normalization and inflations
I am trying to get myself familiar with normalization of iteration trees, and one technical concept which I find especially hard to parse is inflation. I am following the notation of Farmer ...
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