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      Questions tagged [large-cardinals]

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      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 \...
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      161 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 ...
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      2answers
      612 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 ...
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      2answers
      113 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
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      1answer
      163 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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      1answer
      102 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
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      1answer
      221 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 ...
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      84 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? $...
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      0answers
      125 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 $S$ ...
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      1answer
      131 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
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      1answer
      194 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 ...
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      1answer
      229 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
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      1answer
      582 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 ...
      12
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      1answer
      417 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
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      1answer
      146 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 ...

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