Skip to main content
added 27 characters in body
Source Link
Vanessa
  • 1.4k
  • 7
  • 20

Consider $\phi(A)$ a formula of second-order arithmetic with one free variable $A$ of type "set". Suppose $\exists A : \phi(A)$ is a true sentence. Does it follow (not in second order arithmetic itself, but in a stronger theory of your choice, e.g. ZFC) that there is a formula of second-order arithmetic $\psi(n)$ with one free variable $n$ of type "number", such that $\phi(\lbrace n | \psi(n)\rbrace)$ is a true sentence?

Consider $\phi(A)$ a formula of second-order arithmetic with one free variable $A$ of type "set". Suppose $\exists A : \phi(A)$ is a true sentence. Does it follow (not in second order arithmetic itself, but in a stronger theory of your choice, e.g. ZFC) that there is a formula $\psi(n)$ with one free variable $n$ of type "number", such that $\phi(\lbrace n | \psi(n)\rbrace)$ is a true sentence?

Consider $\phi(A)$ a formula of second-order arithmetic with one free variable $A$ of type "set". Suppose $\exists A : \phi(A)$ is a true sentence. Does it follow (not in second order arithmetic itself, but in a stronger theory of your choice, e.g. ZFC) that there is a formula of second-order arithmetic $\psi(n)$ with one free variable $n$ of type "number", such that $\phi(\lbrace n | \psi(n)\rbrace)$ is a true sentence?

Source Link
Vanessa
  • 1.4k
  • 7
  • 20

Are there "non-constructive" sets in second-order arithmetic?

Consider $\phi(A)$ a formula of second-order arithmetic with one free variable $A$ of type "set". Suppose $\exists A : \phi(A)$ is a true sentence. Does it follow (not in second order arithmetic itself, but in a stronger theory of your choice, e.g. ZFC) that there is a formula $\psi(n)$ with one free variable $n$ of type "number", such that $\phi(\lbrace n | \psi(n)\rbrace)$ is a true sentence?