- #1
nomadreid
Gold Member
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As I understand Hilbert's omega rule for a first-order proposition P over the natural numbers,
P(0) &P(1) &P(2) &... ⇒ ∀n∈ℕ P(n)![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
which seems to be the same as ω-consistency. Is there a difference?
Further, the axiom schema of induction has each axiom for a proposition P over the natural numbers,
[P(0) & ∀n∈ℕ (P(n)⇒P(n+1))]⇒∀n∈ℕ P(n)![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
I am not sure if the following argument is correct:
Induction is a weaker principle, because
(a) in order to convert![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
to
, you would need a rule of inference allowing you to apply Modus Ponens an infinite number of times.
(b) to convert
to ![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
, you can take every pair {P(n),P(n+1)}, deduce for each one Q(n):≡ (P(n)⇒P(n+1)), and then apply
for Q to get
∀n∈ℕ (P(n)⇒P(n+1)), etc.
P(0) &P(1) &P(2) &... ⇒ ∀n∈ℕ P(n)
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
which seems to be the same as ω-consistency. Is there a difference?
Further, the axiom schema of induction has each axiom for a proposition P over the natural numbers,
[P(0) & ∀n∈ℕ (P(n)⇒P(n+1))]⇒∀n∈ℕ P(n)
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
I am not sure if the following argument is correct:
Induction is a weaker principle, because
(a) in order to convert
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
(b) to convert
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
![Star :star: :star:](/styles/physicsforums/xenforo/smilies/icons/star.gif)
∀n∈ℕ (P(n)⇒P(n+1)), etc.