There's a book "Ad Infinitum" by Brian Rotman which has an appendix in which he creates an alternative number system which has "exit points", i.e. values above which numbers are undefined. If you can find a copy you might look at the appendix (which is only 13 pages). I don't recommend buying it because the rest of the book is impenetrable post-modern philosophy in the French style. Brent On 9/3/2013 2:46 PM, James Propp wrote:
Fred makes a good point (which I didn't understand till I read Andy's post). You could have a number m that's defined to equal 1 if some proposition P is true and 2 otherwise; if we define n as 3-m, then deciding which of them is as larger is just as hard as deciding whether P is true, but it doesn't have anything to do with the bigness of m and n.
(Here I'm ducking the issue of finding a specific proposition P such that the existence of a truth-value for P is provable in PA even though P and not-P are undecidable in PA, since it's not germane to my point here, which is that the mathematical question that I raised doesn't really capture my original motivating intuition.)
Maybe what I'm asking for is a "naturally occurring" example of such a pair of numbers, but I have no idea how to formalize what I mean by this.
Jim Propp
On Tuesday, September 3, 2013, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Besides, their size is surely irrelevant: only the sign of the difference is required for comparison. WFL
On 9/3/13, Charles Greathouse <charles.greathouse@case.edu> wrote:
Surely the existence of such integers cannot be proven in PA, since there are models (those without nonstandard integers) in which it is false.
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Tue, Sep 3, 2013 at 2:05 PM, James Propp <jamespropp@gmail.com> wrote:
Are there (positive) integers m,n definable in PA whose existence is provable in PA such that m \geq n and n \geq m are undecidable in PA? (I think that's the right way to ask the question I have in mind, but if my wording evinces misunderstanding of foundational issues, please enlighten me!)
What if we replace PA by a stronger theory?
The underlying intuition is that if there are incomprehensibly big numbers, at some point even comparative notions of bigness should start to fail us, so that, in a certain sense, the well-ordering of the natural numbers should become problematical.
But my intuitions may be totally off-base...
Jim Propp