0
$\begingroup$

Given Peano Arithmetic PA, is it possible to form the theory $T_1=PA$+{all true but unprovable statements of PA}?

Then define $T_{n+1}=T_n$+{all true but unprovable statements of $T_n$}

Then let $Q=T_{\infty}$. Is it possible to say anything about Q? Is it a well-defined theory? Is it consistent? Does Godel's second theorem apply to it? Is it possible to construct a true unprovable statement of Q?

Also, what is the system in which we deduce a statement to be true but unprovable in another system, what are the additional axioms?

  • 0
    @Andres: Thank you, and fair enough; but note that the OP does not seem to be aware that "true" is a relative statement, not an absolute one.2011-05-09

1 Answers 1

7

Assuming "true" refers to a particular model, any statement in $T_1$ that is true but unprovable (in $T_1$) is already unprovable in $PA$, so your recursion stops: $T_\infty = T_2 = T_1$. Any statement in $T_1$ that is true is either provable in $PA$ or is an axiom of $T_1$, so it is provable in $T_1$. But Goedel's theorems don't apply to $T_1$, because there is no formula expressing the statement "$A$ is an axiom of $T_1$".