At last. If you go to my Gödel book’s website, and click on the link under “Latest Additions” then the link no longer takes you to a rambling, unfinished essay but to a much crisper extended version of the talk I gave in Oxford a week ago. I argue that ACA0 strictly speaking overgenerates, and that the official conceptual motivation for the theory in fact favours a weaker theory (a theory that doesn’t threaten to inductively inflate, yet which is as competent at generating proxies for theorems of classical analysis). And I make a similar claim about a different family of extensions to first-order PA as well, i.e. extensions by adding truth-theories. Here too I suggest that a familiar weak theory also overgenerates. (I suggest that this matters if we are concerned to evaluate and perhaps defend Dan Isaacson’s Thesis that first-order PA sets a limit to what can be established from purely arithmetical considerations plus logic alone.)
4 thoughts on “ACA0, #7: The last word”
On page 8 of your paper, you mention the following result (Theorem 3) : ACA0 exhibits exponential speed-up over PA1.
Do you know if the result also holds for T U PA over PA1 (using your notation)?
And what property should a translation have in order to respect speed-up-like properties of the translated theory over a given common base theory ? (references would also be welcome)
In fact there is an excellent argument on there today, which is:
part 1. since God knows everything, and since learning something requires knowing something one did not know before, ergo God cannot learn anything.
part 2. only stupid persons cannot learn anything, God cannot learn anything, ergo God is stupid.
This requires the assumption that God is a person, but that is pretty orthodox theology.
The link doesn’t work. The link you give on sci.logic DOES work. They are identical except the one you give here has
as a prefix, which seems to be causing the problem.
This proves that sci.logic in fact has a use.