Skip to main content
added 2 characters in body
Source Link
user36136
user36136

First note to the following well known theorems:‎‎

Theorem (1): ‎The ‎notion ‎of ‎"‎$‎‎x$ ‎is a strongly inaccessible cardinal‎" ‎is ‎first ‎order ‎expressible ‎and ‎‎$‎‎\Pi_{1}$‎.

**Theorem (2):**‎‎ ‎The ‎notion ‎of ‎"‎$‎‎x$ is a measurable cardinal‎" ‎is ‎first ‎order ‎expressible ‎but ‎not ‎‎$‎‎‎\Pi‎_{1}$‎.‎

Theorem (3): ‎The ‎notion ‎of ‎"‎$‎‎x$ is a Reinhardt cardinal‎" ‎is ‎not ‎first order expressible.‎ ‎ Now

Now there are some questions here:‎ ‎

Question (1): ‎Are ‎larger ‎large ‎cardinals ‎more ‎complicated ‎in ‎first ‎order ‎expressibilty? ‎Is ‎there ‎any ‎exception?‎ ‎

Question (2): ‎Is ‎there a‎ ‎non ‎first ‎orde‎r expressible large cardinal weaker than Reindhardt cardinal?‎ ‎

Question (3): ‎What ‎is ‎the ‎largest $‎‎\Pi_{1}$ - ‎expressible ‎large ‎cardinal? ‎For ‎example the notions of being a ‎Mahlo ‎or ‎weakly ‎compact ‎cardinal ‎are ‎first ‎order ‎expressible ‎and‎ $‎‎\Pi_{1}$.

First note to the following well known theorems:‎‎

Theorem (1): ‎The ‎notion ‎of ‎"‎$‎‎x$ ‎is a strongly inaccessible cardinal‎" ‎is ‎first ‎order ‎expressible ‎and ‎‎$‎‎\Pi_{1}$‎.

**Theorem (2):**‎‎ ‎The ‎notion ‎of ‎"‎$‎‎x$ is a measurable cardinal‎" ‎is ‎first ‎order ‎expressible ‎but ‎not ‎‎$‎‎‎\Pi‎_{1}$‎.‎

Theorem (3): ‎The ‎notion ‎of ‎"‎$‎‎x$ is a Reinhardt cardinal‎" ‎is ‎not ‎first order expressible.‎ ‎ Now there are some questions here:‎ ‎

Question (1): ‎Are ‎larger ‎large ‎cardinals ‎more ‎complicated ‎in ‎first ‎order ‎expressibilty? ‎Is ‎there ‎any ‎exception?‎ ‎

Question (2): ‎Is ‎there a‎ ‎non ‎first ‎orde‎r expressible large cardinal weaker than Reindhardt cardinal?‎ ‎

Question (3): ‎What ‎is ‎the ‎largest $‎‎\Pi_{1}$ - ‎expressible ‎large ‎cardinal? ‎For ‎example the notions of being a ‎Mahlo ‎or ‎weakly ‎compact ‎cardinal ‎are ‎first ‎order ‎expressible ‎and‎ $‎‎\Pi_{1}$.

First note to the following well known theorems:‎‎

Theorem (1): ‎The ‎notion ‎of ‎"‎$‎‎x$ ‎is a strongly inaccessible cardinal‎" ‎is ‎first ‎order ‎expressible ‎and ‎‎$‎‎\Pi_{1}$‎.

**Theorem (2):**‎‎ ‎The ‎notion ‎of ‎"‎$‎‎x$ is a measurable cardinal‎" ‎is ‎first ‎order ‎expressible ‎but ‎not ‎‎$‎‎‎\Pi‎_{1}$‎.‎

Theorem (3): ‎The ‎notion ‎of ‎"‎$‎‎x$ is a Reinhardt cardinal‎" ‎is ‎not ‎first order expressible.‎ ‎

Now there are some questions here:‎ ‎

Question (1): ‎Are ‎larger ‎large ‎cardinals ‎more ‎complicated ‎in ‎first ‎order ‎expressibilty? ‎Is ‎there ‎any ‎exception?‎ ‎

Question (2): ‎Is ‎there a‎ ‎non ‎first ‎orde‎r expressible large cardinal weaker than Reindhardt cardinal?‎ ‎

Question (3): ‎What ‎is ‎the ‎largest $‎‎\Pi_{1}$ - ‎expressible ‎large ‎cardinal? ‎For ‎example the notions of being a ‎Mahlo ‎or ‎weakly ‎compact ‎cardinal ‎are ‎first ‎order ‎expressible ‎and‎ $‎‎\Pi_{1}$.

Source Link
user36136
user36136

Are larger large cardinals less expressible?

First note to the following well known theorems:‎‎

Theorem (1): ‎The ‎notion ‎of ‎"‎$‎‎x$ ‎is a strongly inaccessible cardinal‎" ‎is ‎first ‎order ‎expressible ‎and ‎‎$‎‎\Pi_{1}$‎.

**Theorem (2):**‎‎ ‎The ‎notion ‎of ‎"‎$‎‎x$ is a measurable cardinal‎" ‎is ‎first ‎order ‎expressible ‎but ‎not ‎‎$‎‎‎\Pi‎_{1}$‎.‎

Theorem (3): ‎The ‎notion ‎of ‎"‎$‎‎x$ is a Reinhardt cardinal‎" ‎is ‎not ‎first order expressible.‎ ‎ Now there are some questions here:‎ ‎

Question (1): ‎Are ‎larger ‎large ‎cardinals ‎more ‎complicated ‎in ‎first ‎order ‎expressibilty? ‎Is ‎there ‎any ‎exception?‎ ‎

Question (2): ‎Is ‎there a‎ ‎non ‎first ‎orde‎r expressible large cardinal weaker than Reindhardt cardinal?‎ ‎

Question (3): ‎What ‎is ‎the ‎largest $‎‎\Pi_{1}$ - ‎expressible ‎large ‎cardinal? ‎For ‎example the notions of being a ‎Mahlo ‎or ‎weakly ‎compact ‎cardinal ‎are ‎first ‎order ‎expressible ‎and‎ $‎‎\Pi_{1}$.