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I want to know if there are any papers or research along the following lines. From a philosophical point of view there is no reason to think size has a limit, if we can describe it it exists, provided it does not contradict the axioms of our favourite set theory.

I think this can be expressed as a meta-axiom schema as follows. Let $AX$ be the set of axioms you like in the language $<\in >$ of set theory.

Axiom X - Let $\varphi $ be a statement with one free variable $\kappa$. If $AX\cup \{Cardinal[\kappa ], \varphi\}$ is consistent then $\exists\kappa\varphi$ is an axiom.$\square$

I would like to know if this could be converted to an internal axiom, or axiom schema, of set theory using extenders or such like. This would result in a non-conservative but consistent set theory in which all large cardinals exist.

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  • $\begingroup$ The problem is how do you know that they are consistent ? You cant even prove that an inaccessible cardinal is consistent. So the axiom may have no provable consequences. $\endgroup$ – Rene Schipperus Oct 8 '17 at 22:32
  • $\begingroup$ @ReneSchipperus you can't prove consistency within, but I think you can prove consistency outside? $\endgroup$ – Kenny Lau Oct 8 '17 at 22:33
  • $\begingroup$ @KennyLau Outside set theory ? Assuming what ? $\endgroup$ – Rene Schipperus Oct 8 '17 at 22:35
  • $\begingroup$ @ReneSchipperus I mean, in the meta-theory. OP did say that the schema is meta. $\endgroup$ – Kenny Lau Oct 8 '17 at 22:37
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    $\begingroup$ You need some sort of restriction on what $\varphi$ can be (and that seems very difficult to formalize). Otherwise this theory is obviously inconsistent (take $\varphi$ to be any statement independent of $AX$ and its negation). $\endgroup$ – Eric Wofsey Oct 8 '17 at 22:47
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This is very clearly inconsistent, e.g. "There exists an inaccessible cardinal greater than all other inaccessible cardinals that is less than the least 1-inaccessible cardinal," "There exists a Σ2-reflecting cardinal."

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