Transfinite iteration of consistency [closed]

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Consider the operation X, which takes any theory T and returns T + “T is consistent”.



Now transfinitely iterate X on (say) ZFC. Call this new theory Y. What useful things can Y prove that ZFC cannot?



More specifically:



Consider any theory $T$. We can construct a function $F: T rightarrow T + textT is consistent$.



Now apply $F$ to $T$ $n$ times, for some infinite ordinal $n$






set-theory







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closed as off-topic by Andrés E. Caicedo, Jendrik Stelzner, José Carlos Santos, amWhy, Adrian Keister Sep 11 at 0:10


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, Jendrik Stelzner, José Carlos Santos, amWhy, Adrian Keister
If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    The consistency of ZFC.
    – Andrés E. Caicedo
    Sep 8 at 12:35






  • 1




    Anyway, "transfinitely iterating consistency" is not a well-defined operation. It depends on the ordinal notation you choose to represent the length of the iteration, and on the way you code the theories themselves. Different codings lead to vastly different results.
    – Andrés E. Caicedo
    Sep 8 at 12:37










  • @AndrésE.Caicedo True, but that is trivial.
    – Demi
    Sep 8 at 16:26










  • @AndrésE.Caicedo Would you mind going into detail and making that an answer?
    – Demi
    Sep 8 at 16:26














up vote
-1
down vote

favorite












Consider the operation X, which takes any theory T and returns T + “T is consistent”.



Now transfinitely iterate X on (say) ZFC. Call this new theory Y. What useful things can Y prove that ZFC cannot?



More specifically:



Consider any theory $T$. We can construct a function $F: T rightarrow T + textT is consistent$.



Now apply $F$ to $T$ $n$ times, for some infinite ordinal $n$






set-theory







share|cite|improve this question















closed as off-topic by Andrés E. Caicedo, Jendrik Stelzner, José Carlos Santos, amWhy, Adrian Keister Sep 11 at 0:10


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, Jendrik Stelzner, José Carlos Santos, amWhy, Adrian Keister
If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    The consistency of ZFC.
    – Andrés E. Caicedo
    Sep 8 at 12:35






  • 1




    Anyway, "transfinitely iterating consistency" is not a well-defined operation. It depends on the ordinal notation you choose to represent the length of the iteration, and on the way you code the theories themselves. Different codings lead to vastly different results.
    – Andrés E. Caicedo
    Sep 8 at 12:37










  • @AndrésE.Caicedo True, but that is trivial.
    – Demi
    Sep 8 at 16:26










  • @AndrésE.Caicedo Would you mind going into detail and making that an answer?
    – Demi
    Sep 8 at 16:26












up vote
-1
down vote

favorite









up vote
-1
down vote

favorite











Consider the operation X, which takes any theory T and returns T + “T is consistent”.



Now transfinitely iterate X on (say) ZFC. Call this new theory Y. What useful things can Y prove that ZFC cannot?



More specifically:



Consider any theory $T$. We can construct a function $F: T rightarrow T + textT is consistent$.



Now apply $F$ to $T$ $n$ times, for some infinite ordinal $n$






set-theory







share|cite|improve this question















Consider the operation X, which takes any theory T and returns T + “T is consistent”.



Now transfinitely iterate X on (say) ZFC. Call this new theory Y. What useful things can Y prove that ZFC cannot?



More specifically:



Consider any theory $T$. We can construct a function $F: T rightarrow T + textT is consistent$.



Now apply $F$ to $T$ $n$ times, for some infinite ordinal $n$






set-theory




set-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Sep 14 at 0:41

























asked Sep 8 at 4:49









Demi

19916




19916




closed as off-topic by Andrés E. Caicedo, Jendrik Stelzner, José Carlos Santos, amWhy, Adrian Keister Sep 11 at 0:10


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, Jendrik Stelzner, José Carlos Santos, amWhy, Adrian Keister
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Andrés E. Caicedo, Jendrik Stelzner, José Carlos Santos, amWhy, Adrian Keister Sep 11 at 0:10


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, Jendrik Stelzner, José Carlos Santos, amWhy, Adrian Keister
If this question can be reworded to fit the rules in the help center, please edit the question.







  • 1




    The consistency of ZFC.
    – Andrés E. Caicedo
    Sep 8 at 12:35






  • 1




    Anyway, "transfinitely iterating consistency" is not a well-defined operation. It depends on the ordinal notation you choose to represent the length of the iteration, and on the way you code the theories themselves. Different codings lead to vastly different results.
    – Andrés E. Caicedo
    Sep 8 at 12:37










  • @AndrésE.Caicedo True, but that is trivial.
    – Demi
    Sep 8 at 16:26










  • @AndrésE.Caicedo Would you mind going into detail and making that an answer?
    – Demi
    Sep 8 at 16:26












  • 1




    The consistency of ZFC.
    – Andrés E. Caicedo
    Sep 8 at 12:35






  • 1




    Anyway, "transfinitely iterating consistency" is not a well-defined operation. It depends on the ordinal notation you choose to represent the length of the iteration, and on the way you code the theories themselves. Different codings lead to vastly different results.
    – Andrés E. Caicedo
    Sep 8 at 12:37










  • @AndrésE.Caicedo True, but that is trivial.
    – Demi
    Sep 8 at 16:26










  • @AndrésE.Caicedo Would you mind going into detail and making that an answer?
    – Demi
    Sep 8 at 16:26







1




1




The consistency of ZFC.
– Andrés E. Caicedo
Sep 8 at 12:35




The consistency of ZFC.
– Andrés E. Caicedo
Sep 8 at 12:35




1




1




Anyway, "transfinitely iterating consistency" is not a well-defined operation. It depends on the ordinal notation you choose to represent the length of the iteration, and on the way you code the theories themselves. Different codings lead to vastly different results.
– Andrés E. Caicedo
Sep 8 at 12:37




Anyway, "transfinitely iterating consistency" is not a well-defined operation. It depends on the ordinal notation you choose to represent the length of the iteration, and on the way you code the theories themselves. Different codings lead to vastly different results.
– Andrés E. Caicedo
Sep 8 at 12:37












@AndrésE.Caicedo True, but that is trivial.
– Demi
Sep 8 at 16:26




@AndrésE.Caicedo True, but that is trivial.
– Demi
Sep 8 at 16:26












@AndrésE.Caicedo Would you mind going into detail and making that an answer?
– Demi
Sep 8 at 16:26




@AndrésE.Caicedo Would you mind going into detail and making that an answer?
– Demi
Sep 8 at 16:26















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