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Asaf Karagila
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There are numerous similar instances of this. Any time a large cardinal notion is witnessed by a single object or is witnessed inside some $V_\theta$ — and this would include weakly compact, Ramsey, measurable, superstrong, almost huge, huge, rank-to-rank and others — then the least instance of that cardinal will be less than the least $\Sigma_2$-reflecting cardinal and indeed less than the least $\Sigma-2$$\Sigma_2$-correct cardinal. But $\Sigma_2$ correct cardinals provably exist in ZFC, and therefore have very low consistency strength.

There are numerous similar instances of this. Any time a large cardinal notion is witnessed by a single object or is witnessed inside some $V_\theta$ — and this would include weakly compact, Ramsey, measurable, superstrong, almost huge, huge, rank-to-rank and others — then the least instance of that cardinal will be less than the least $\Sigma_2$-reflecting cardinal and indeed less than the least $\Sigma-2$-correct cardinal. But $\Sigma_2$ correct cardinals provably exist in ZFC, and therefore have very low consistency strength.

There are numerous similar instances of this. Any time a large cardinal notion is witnessed by a single object or is witnessed inside some $V_\theta$ — and this would include weakly compact, Ramsey, measurable, superstrong, almost huge, huge, rank-to-rank and others — then the least instance of that cardinal will be less than the least $\Sigma_2$-reflecting cardinal and indeed less than the least $\Sigma_2$-correct cardinal. But $\Sigma_2$ correct cardinals provably exist in ZFC, and therefore have very low consistency strength.

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Joel David Hamkins
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  • A supercompact cardinal versus a strongly compact plus an inaccessible above.

  • A supercompact cardinal versus a proper class of strongly compact cardinals.

  • A Laver-indestructible weakly compact cardinals versus a strongly compact cardinal.

  • A cardinal $\kappa$ that is $\kappa^+$-supercompact versus $\kappa$ is $\kappa++$$\kappa^{++}$-strongly compact.

  • A PFA cardinal versus a strongly compact cardinal.

  • And many others.

  • A supercompact cardinal versus a strongly compact plus an inaccessible above.

  • A supercompact cardinal versus a proper class of strongly compact cardinals.

  • A Laver-indestructible weakly compact cardinals versus a strongly compact cardinal.

  • A cardinal $\kappa$ that is $\kappa^+$-supercompact versus $\kappa$ is $\kappa++$-strongly compact.

  • A PFA cardinal versus a strongly compact cardinal.

  • And many others.

  • A supercompact cardinal versus a strongly compact plus an inaccessible above.

  • A supercompact cardinal versus a proper class of strongly compact cardinals.

  • A Laver-indestructible weakly compact cardinals versus a strongly compact cardinal.

  • A cardinal $\kappa$ that is $\kappa^+$-supercompact versus $\kappa$ is $\kappa^{++}$-strongly compact.

  • A PFA cardinal versus a strongly compact cardinal.

  • And many others.

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Joel David Hamkins
  • 229.7k
  • 44
  • 754
  • 1.3k
  • A supercompact cardinal versus a strongly compact plus an inaccessible above.

  • A supercompact cardinal versus a proper class of strongly compact cardinals.

  • A Laver-indestructible weakly compact cardinals versus a strongly compact cardinal.

  • A cardinal $\kappa$ that is $\kappa^+$-supercompact versus $\kappa$ is $\kappa++$-strongly compact.

  • A PFA cardinal versus a strongly compact cardinal.

  • And many others.

  • A supercompact cardinal versus a strongly compact plus an inaccessible above.

  • A supercompact cardinal versus a proper class of strongly compact cardinals.

  • A Laver-indestructible weakly compact cardinals versus a strongly compact cardinal.

  • A PFA cardinal versus a strongly compact cardinal.

  • And many others.

  • A supercompact cardinal versus a strongly compact plus an inaccessible above.

  • A supercompact cardinal versus a proper class of strongly compact cardinals.

  • A Laver-indestructible weakly compact cardinals versus a strongly compact cardinal.

  • A cardinal $\kappa$ that is $\kappa^+$-supercompact versus $\kappa$ is $\kappa++$-strongly compact.

  • A PFA cardinal versus a strongly compact cardinal.

  • And many others.

added 73 characters in body
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Joel David Hamkins
  • 229.7k
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  • 754
  • 1.3k
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Joel David Hamkins
  • 229.7k
  • 44
  • 754
  • 1.3k
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added 73 characters in body
Source Link
Joel David Hamkins
  • 229.7k
  • 44
  • 754
  • 1.3k
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Source Link
Joel David Hamkins
  • 229.7k
  • 44
  • 754
  • 1.3k
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