[FOM] How real are real numbers?

Ben Sherman sherman at csail.mit.edu
Sat Jan 19 14:00:13 EST 2019

> From: Joe Shipman <joeshipman at aol.com>
> Date: January 17, 2019 at 11:13:30 PM EST
> I think we need a metatheorem that certain types of mathematical questions cannot be relevant to physical experimental results if the ultimate physical theories are anything like today’s, and are therefore unknowable simply by resorting to experiment, or else an extension of the Church-Turing thesis that covers phenomena not precisely formalized by the current versions.

This reminds me of Solomon Feferman’s arguments that system W, a predicative formal system that is conservative over Peano Arithmetic. In his article on Predicativity (https://math.stanford.edu/~feferman/papers/predicativity.pdf <https://math.stanford.edu/~feferman/papers/predicativity.pdf>), he says

> I had formulated the working hypothesis that all of scientifically applicable analysis can be developed in the system W, and argued that this has been verified in its core parts (cf. 1998, pp. 280-283 and 293-294). Of course, there are results of theoretical analysis which cannot be carried out predicatively, either because they are essentially impredicative in their very formulation, or because they are independent of predicative systems such as the examples given above. However, none of those affects the working hypothesis because they do not figure in the applicable mathematics.
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