Skip to main content
MathOverflow

Return to Question

added 6 characters in body
Source Link
Cosmonut
Cosmonut
  • 781
  • 5
  • 13

Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (http://cantorsattic.info/Upper_attic)

Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type B cardinals can coexist), I define:

*Type A > Type B if smallest Type A cardinal has higher cardinality than smallest Type B

*Type A = Type B if smallest Type A and Type B have same cardinalities

*Type A ~$\perp$ Type B if the ordering in the sense above is undecidable

So, for instance, Inaccesssible < Hyper Inaccessible < Mahlo

What is known about the ordering of large cardinals from (http://cantorsattic.info/Upper_attic) in this sense ?

I am particularly interested in the "large" large cardinals from measurable upwards. For eg: How would one order measurable, extendible, huge and rank-into-rank ?

Motivation: I understand researchers are mostly focused on consistency strength, but I am interested in the intuitive notion of getting "much bigger infinities" from each successive large cardinal axiom.

Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (http://cantorsattic.info/Upper_attic)

Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type B cardinals can coexist), I define:

*Type A > Type B if smallest Type A cardinal has higher cardinality than smallest Type B

*Type A = Type B if smallest Type A and Type B have same cardinalities

*Type A ~ Type B if the ordering in the sense above is undecidable

So, for instance, Inaccesssible < Hyper Inaccessible < Mahlo

What is known about the ordering of large cardinals from (http://cantorsattic.info/Upper_attic) in this sense ?

I am particularly interested in the "large" large cardinals from measurable upwards. For eg: How would one order measurable, extendible, huge and rank-into-rank ?

Motivation: I understand researchers are mostly focused on consistency strength, but I am interested in the intuitive notion of getting "much bigger infinities" from each successive large cardinal axiom.

Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (http://cantorsattic.info/Upper_attic)

Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type B cardinals can coexist), I define:

*Type A > Type B if smallest Type A cardinal has higher cardinality than smallest Type B

*Type A = Type B if smallest Type A and Type B have same cardinalities

*Type A $\perp$ Type B if the ordering in the sense above is undecidable

So, for instance, Inaccesssible < Hyper Inaccessible < Mahlo

What is known about the ordering of large cardinals from (http://cantorsattic.info/Upper_attic) in this sense ?

I am particularly interested in the "large" large cardinals from measurable upwards. For eg: How would one order measurable, extendible, huge and rank-into-rank ?

Motivation: I understand researchers are mostly focused on consistency strength, but I am interested in the intuitive notion of getting "much bigger infinities" from each successive large cardinal axiom.

Source Link
Cosmonut
Cosmonut
  • 781
  • 5
  • 13

Ordering of large cardinals by cardinality

Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (http://cantorsattic.info/Upper_attic)

Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type B cardinals can coexist), I define:

*Type A > Type B if smallest Type A cardinal has higher cardinality than smallest Type B

*Type A = Type B if smallest Type A and Type B have same cardinalities

*Type A ~ Type B if the ordering in the sense above is undecidable

So, for instance, Inaccesssible < Hyper Inaccessible < Mahlo

What is known about the ordering of large cardinals from (http://cantorsattic.info/Upper_attic) in this sense ?

I am particularly interested in the "large" large cardinals from measurable upwards. For eg: How would one order measurable, extendible, huge and rank-into-rank ?

Motivation: I understand researchers are mostly focused on consistency strength, but I am interested in the intuitive notion of getting "much bigger infinities" from each successive large cardinal axiom.