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    If Cons(F) is expressed via a specific arithmetical encod... — Carmelics
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    Challenges→The consistency statement Cons(F) cannot be provable in F

    If Cons(F) is expressed via a specific arithmetical encoding, different encodings can yield extensionally equivalent but proof-theoretically distinct statements, as Feferman demonstrated in 1960.

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    Key Terms

    Arithmetical encoding(in logic and computer science)
    A method of translating logical or mathematical statements into sequences of numbers so that a computer or formal system can process them.
    Cons(F)(Formal provability logic)
    The consistency statement for F, expressed as ¬Prov_F(⌜⊥⌝), meaning F does not prove falsum (bottom)
    Feferman(as a historical figure in logic and philosophy of mathematics)
    Solomon Feferman (1928–2016), a logician and philosopher who studied the foundations of mathematics and how different formal systems relate to each other.
    Proof-theoretically distinct(in mathematical logic)
    Statements that may mean the same thing in practice but require different logical steps or arguments to prove them—like two different recipes that produce the same cake.

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    extensionally equivalent(Applied to Church's thesis and Turing's thesis regarding functions of positive integers)
    Two theses are extensionally equivalent when they are about one and the same class of functions

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    The consistency statement Cons(F) cannot be provable in F

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