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Finite axiomatization of second-order NBG


Extending our language with a new function symbolIs there a Second-Order Axiomatization of ZF(C) which is categorical?Second order logic and quantification over formulasCan a finite axiomatization of PA be expressed in a finitely axiomatizable first order set theory?Is there a minimal axiomatization of ZFC?Infinitely many axioms of ZFC vs. finitely many axioms of NBGSecond-order logic as the basis for set theoryIs there a finite axiomatization of Tarski's geometry axioms?Alternative axioms for NBG or MKAxiom schemas vs second order axioms: which first-order predicates “exist”?













0












$begingroup$


First-order NBG set theory is finitely axiomatizable. The proof of this basically shows that the axiom schemas, in the presence of the other axioms, can be reduced to a finite set of cases, and are hence equivalent to a finite set of axioms.



Is this equivalence also valid in second-order NBG? Or does the reduction to a finite number of cases only work for definable predicates (e.g. the first order axiom schema)?










share|cite|improve this question









$endgroup$







  • 3




    $begingroup$
    What do you mean by "second-order NBG", exactly?
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:37










  • $begingroup$
    NBG with the axiom schemas replaced with a second order axiom. Is this term ambiguous?
    $endgroup$
    – Mike Battaglia
    Mar 16 at 23:42






  • 1




    $begingroup$
    Well, that is at least somewhat ambiguous (it's not obvious what second-order axiom you're using, given that an essential feature of NBG is that its comprehension schema is restricted to formulas without class quantifiers--is "second-order NBG" the same thing as "second-order MK"?). But in any case, if second-order NBG does not have any axiom schemas, it's unclear what you mean by your question.
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:46







  • 2




    $begingroup$
    I guess there is probably only one reasonable choice for the second-order axiom, and no hope of distinguishing second-order NBG from second-order MK. I still don't know what your question is--are you asking whether second-order class comprehension is equivalent to the finitely many first-order axioms used in NBG? (In that case the answer is obviously not, since the latter are only equivalent to first-order class comprehension.)
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:58






  • 3




    $begingroup$
    Both NBG and MK have a class comprehension schema; the difference is that MK allows formulas with class quantifiers in the schema. That distinction is lost if you replace the schema with a second-order axiom saying "every collection of sets is a class". (Also, NBG usually isn't stated with a replacement schema, since you can just use a class quantifier, and the necessary generality follows from the comprehension schema. This doesn't really make a difference for anything, though.)
    $endgroup$
    – Eric Wofsey
    Mar 17 at 0:13
















0












$begingroup$


First-order NBG set theory is finitely axiomatizable. The proof of this basically shows that the axiom schemas, in the presence of the other axioms, can be reduced to a finite set of cases, and are hence equivalent to a finite set of axioms.



Is this equivalence also valid in second-order NBG? Or does the reduction to a finite number of cases only work for definable predicates (e.g. the first order axiom schema)?










share|cite|improve this question









$endgroup$







  • 3




    $begingroup$
    What do you mean by "second-order NBG", exactly?
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:37










  • $begingroup$
    NBG with the axiom schemas replaced with a second order axiom. Is this term ambiguous?
    $endgroup$
    – Mike Battaglia
    Mar 16 at 23:42






  • 1




    $begingroup$
    Well, that is at least somewhat ambiguous (it's not obvious what second-order axiom you're using, given that an essential feature of NBG is that its comprehension schema is restricted to formulas without class quantifiers--is "second-order NBG" the same thing as "second-order MK"?). But in any case, if second-order NBG does not have any axiom schemas, it's unclear what you mean by your question.
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:46







  • 2




    $begingroup$
    I guess there is probably only one reasonable choice for the second-order axiom, and no hope of distinguishing second-order NBG from second-order MK. I still don't know what your question is--are you asking whether second-order class comprehension is equivalent to the finitely many first-order axioms used in NBG? (In that case the answer is obviously not, since the latter are only equivalent to first-order class comprehension.)
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:58






  • 3




    $begingroup$
    Both NBG and MK have a class comprehension schema; the difference is that MK allows formulas with class quantifiers in the schema. That distinction is lost if you replace the schema with a second-order axiom saying "every collection of sets is a class". (Also, NBG usually isn't stated with a replacement schema, since you can just use a class quantifier, and the necessary generality follows from the comprehension schema. This doesn't really make a difference for anything, though.)
    $endgroup$
    – Eric Wofsey
    Mar 17 at 0:13














0












0








0





$begingroup$


First-order NBG set theory is finitely axiomatizable. The proof of this basically shows that the axiom schemas, in the presence of the other axioms, can be reduced to a finite set of cases, and are hence equivalent to a finite set of axioms.



Is this equivalence also valid in second-order NBG? Or does the reduction to a finite number of cases only work for definable predicates (e.g. the first order axiom schema)?










share|cite|improve this question









$endgroup$




First-order NBG set theory is finitely axiomatizable. The proof of this basically shows that the axiom schemas, in the presence of the other axioms, can be reduced to a finite set of cases, and are hence equivalent to a finite set of axioms.



Is this equivalence also valid in second-order NBG? Or does the reduction to a finite number of cases only work for definable predicates (e.g. the first order axiom schema)?







set-theory axioms foundations






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Mar 16 at 23:18









Mike BattagliaMike Battaglia

1,5181127




1,5181127







  • 3




    $begingroup$
    What do you mean by "second-order NBG", exactly?
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:37










  • $begingroup$
    NBG with the axiom schemas replaced with a second order axiom. Is this term ambiguous?
    $endgroup$
    – Mike Battaglia
    Mar 16 at 23:42






  • 1




    $begingroup$
    Well, that is at least somewhat ambiguous (it's not obvious what second-order axiom you're using, given that an essential feature of NBG is that its comprehension schema is restricted to formulas without class quantifiers--is "second-order NBG" the same thing as "second-order MK"?). But in any case, if second-order NBG does not have any axiom schemas, it's unclear what you mean by your question.
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:46







  • 2




    $begingroup$
    I guess there is probably only one reasonable choice for the second-order axiom, and no hope of distinguishing second-order NBG from second-order MK. I still don't know what your question is--are you asking whether second-order class comprehension is equivalent to the finitely many first-order axioms used in NBG? (In that case the answer is obviously not, since the latter are only equivalent to first-order class comprehension.)
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:58






  • 3




    $begingroup$
    Both NBG and MK have a class comprehension schema; the difference is that MK allows formulas with class quantifiers in the schema. That distinction is lost if you replace the schema with a second-order axiom saying "every collection of sets is a class". (Also, NBG usually isn't stated with a replacement schema, since you can just use a class quantifier, and the necessary generality follows from the comprehension schema. This doesn't really make a difference for anything, though.)
    $endgroup$
    – Eric Wofsey
    Mar 17 at 0:13













  • 3




    $begingroup$
    What do you mean by "second-order NBG", exactly?
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:37










  • $begingroup$
    NBG with the axiom schemas replaced with a second order axiom. Is this term ambiguous?
    $endgroup$
    – Mike Battaglia
    Mar 16 at 23:42






  • 1




    $begingroup$
    Well, that is at least somewhat ambiguous (it's not obvious what second-order axiom you're using, given that an essential feature of NBG is that its comprehension schema is restricted to formulas without class quantifiers--is "second-order NBG" the same thing as "second-order MK"?). But in any case, if second-order NBG does not have any axiom schemas, it's unclear what you mean by your question.
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:46







  • 2




    $begingroup$
    I guess there is probably only one reasonable choice for the second-order axiom, and no hope of distinguishing second-order NBG from second-order MK. I still don't know what your question is--are you asking whether second-order class comprehension is equivalent to the finitely many first-order axioms used in NBG? (In that case the answer is obviously not, since the latter are only equivalent to first-order class comprehension.)
    $endgroup$
    – Eric Wofsey
    Mar 16 at 23:58






  • 3




    $begingroup$
    Both NBG and MK have a class comprehension schema; the difference is that MK allows formulas with class quantifiers in the schema. That distinction is lost if you replace the schema with a second-order axiom saying "every collection of sets is a class". (Also, NBG usually isn't stated with a replacement schema, since you can just use a class quantifier, and the necessary generality follows from the comprehension schema. This doesn't really make a difference for anything, though.)
    $endgroup$
    – Eric Wofsey
    Mar 17 at 0:13








3




3




$begingroup$
What do you mean by "second-order NBG", exactly?
$endgroup$
– Eric Wofsey
Mar 16 at 23:37




$begingroup$
What do you mean by "second-order NBG", exactly?
$endgroup$
– Eric Wofsey
Mar 16 at 23:37












$begingroup$
NBG with the axiom schemas replaced with a second order axiom. Is this term ambiguous?
$endgroup$
– Mike Battaglia
Mar 16 at 23:42




$begingroup$
NBG with the axiom schemas replaced with a second order axiom. Is this term ambiguous?
$endgroup$
– Mike Battaglia
Mar 16 at 23:42




1




1




$begingroup$
Well, that is at least somewhat ambiguous (it's not obvious what second-order axiom you're using, given that an essential feature of NBG is that its comprehension schema is restricted to formulas without class quantifiers--is "second-order NBG" the same thing as "second-order MK"?). But in any case, if second-order NBG does not have any axiom schemas, it's unclear what you mean by your question.
$endgroup$
– Eric Wofsey
Mar 16 at 23:46





$begingroup$
Well, that is at least somewhat ambiguous (it's not obvious what second-order axiom you're using, given that an essential feature of NBG is that its comprehension schema is restricted to formulas without class quantifiers--is "second-order NBG" the same thing as "second-order MK"?). But in any case, if second-order NBG does not have any axiom schemas, it's unclear what you mean by your question.
$endgroup$
– Eric Wofsey
Mar 16 at 23:46





2




2




$begingroup$
I guess there is probably only one reasonable choice for the second-order axiom, and no hope of distinguishing second-order NBG from second-order MK. I still don't know what your question is--are you asking whether second-order class comprehension is equivalent to the finitely many first-order axioms used in NBG? (In that case the answer is obviously not, since the latter are only equivalent to first-order class comprehension.)
$endgroup$
– Eric Wofsey
Mar 16 at 23:58




$begingroup$
I guess there is probably only one reasonable choice for the second-order axiom, and no hope of distinguishing second-order NBG from second-order MK. I still don't know what your question is--are you asking whether second-order class comprehension is equivalent to the finitely many first-order axioms used in NBG? (In that case the answer is obviously not, since the latter are only equivalent to first-order class comprehension.)
$endgroup$
– Eric Wofsey
Mar 16 at 23:58




3




3




$begingroup$
Both NBG and MK have a class comprehension schema; the difference is that MK allows formulas with class quantifiers in the schema. That distinction is lost if you replace the schema with a second-order axiom saying "every collection of sets is a class". (Also, NBG usually isn't stated with a replacement schema, since you can just use a class quantifier, and the necessary generality follows from the comprehension schema. This doesn't really make a difference for anything, though.)
$endgroup$
– Eric Wofsey
Mar 17 at 0:13





$begingroup$
Both NBG and MK have a class comprehension schema; the difference is that MK allows formulas with class quantifiers in the schema. That distinction is lost if you replace the schema with a second-order axiom saying "every collection of sets is a class". (Also, NBG usually isn't stated with a replacement schema, since you can just use a class quantifier, and the necessary generality follows from the comprehension schema. This doesn't really make a difference for anything, though.)
$endgroup$
– Eric Wofsey
Mar 17 at 0:13











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