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This question is an extension of something I asked earlier here: Ordering of large cardinals by cardinalityOrdering of large cardinals by cardinality

I have seen large cardinals ordered by consistency strength in several places but no ordering by cardinality of the least instance, although this is probably common knowledge among experts. Based on information scattered in various articles and the answer to my prior question from Dr. Hamkins, here is what I have gathered so far (I have deliberately avoided "identity crisis" cases):-

Strongly inaccessible < Mahlo < Weakly Compact < Totally Indescribable < Measurable < Huge (1-Huge upwards) < Rank-into-Rank < Supercompact < Extendible

My questions are:

  1. Is this ordering correct ? Maybe < should be replaced by $\leq$ in some cases ?

  2. Any important types I have left out which can be placed in the ordering without ambiguity ?

  3. If a Reinhardt cardinal existed (maybe in ZF), would it be at the top of this size hierarchy ? My intuition is that it should, since a Reinhardt cardinal is an "ultimate extendible cardinal" in a sense, but I find my intuition isn't terribly good on these issues.

This question is an extension of something I asked earlier here: Ordering of large cardinals by cardinality

I have seen large cardinals ordered by consistency strength in several places but no ordering by cardinality of the least instance, although this is probably common knowledge among experts. Based on information scattered in various articles and the answer to my prior question from Dr. Hamkins, here is what I have gathered so far (I have deliberately avoided "identity crisis" cases):-

Strongly inaccessible < Mahlo < Weakly Compact < Totally Indescribable < Measurable < Huge (1-Huge upwards) < Rank-into-Rank < Supercompact < Extendible

My questions are:

  1. Is this ordering correct ? Maybe < should be replaced by $\leq$ in some cases ?

  2. Any important types I have left out which can be placed in the ordering without ambiguity ?

  3. If a Reinhardt cardinal existed (maybe in ZF), would it be at the top of this size hierarchy ? My intuition is that it should, since a Reinhardt cardinal is an "ultimate extendible cardinal" in a sense, but I find my intuition isn't terribly good on these issues.

This question is an extension of something I asked earlier here: Ordering of large cardinals by cardinality

I have seen large cardinals ordered by consistency strength in several places but no ordering by cardinality of the least instance, although this is probably common knowledge among experts. Based on information scattered in various articles and the answer to my prior question from Dr. Hamkins, here is what I have gathered so far (I have deliberately avoided "identity crisis" cases):-

Strongly inaccessible < Mahlo < Weakly Compact < Totally Indescribable < Measurable < Huge (1-Huge upwards) < Rank-into-Rank < Supercompact < Extendible

My questions are:

  1. Is this ordering correct ? Maybe < should be replaced by $\leq$ in some cases ?

  2. Any important types I have left out which can be placed in the ordering without ambiguity ?

  3. If a Reinhardt cardinal existed (maybe in ZF), would it be at the top of this size hierarchy ? My intuition is that it should, since a Reinhardt cardinal is an "ultimate extendible cardinal" in a sense, but I find my intuition isn't terribly good on these issues.

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Anindya
Anindya
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Large cardinals ordered by cardinality of least instance

This question is an extension of something I asked earlier here: Ordering of large cardinals by cardinality

I have seen large cardinals ordered by consistency strength in several places but no ordering by cardinality of the least instance, although this is probably common knowledge among experts. Based on information scattered in various articles and the answer to my prior question from Dr. Hamkins, here is what I have gathered so far (I have deliberately avoided "identity crisis" cases):-

Strongly inaccessible < Mahlo < Weakly Compact < Totally Indescribable < Measurable < Huge (1-Huge upwards) < Rank-into-Rank < Supercompact < Extendible

My questions are:

  1. Is this ordering correct ? Maybe < should be replaced by $\leq$ in some cases ?

  2. Any important types I have left out which can be placed in the ordering without ambiguity ?

  3. If a Reinhardt cardinal existed (maybe in ZF), would it be at the top of this size hierarchy ? My intuition is that it should, since a Reinhardt cardinal is an "ultimate extendible cardinal" in a sense, but I find my intuition isn't terribly good on these issues.