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...
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