# [FOM] Reverse Mathematics of Scheffer-Shnirelman paradox

José Manuel Rodriguez Caballero josephcmac at gmail.com
Tue Aug 21 06:52:56 EDT 2018

```Concerning the questions about VERY, all the answers are explicitly and
implicitly given in Jean Benabou's paper Orbits and monoids in a topos. We
also gave VERY VERY good explanations in his lecture (available on
youtube): Jean BENABOU - Very, almost, and so on, ... . So, I will focus on
questions with less clear answers, like this one:

> Sam wrote:
> Please show me your evidence/proof that real numbers do not exist in
> reality.

This question is ambiguous. So, I will interprete it in two ways:

Interpretation A: show me your evidence/proof that for each real number x,
we have that x do not exist in reality.

Answer A: Counterexamples: Zero exists in reality. Also, one, etc.

Interpretation E: show me your evidence/proof that for some real number x,
we have that x do not exist in reality.

Answer E: Let x be Chaitin's constant (it represents the probability that a
randomly constructed program will halt). Notice that x contains more
information than our cosmological horizon. So, x cannot exist in our
cosmological horizon: a pregnant woman cannot have a baby that weigh more
than herself.

I think that this argument may provide a solution to Scheffer-Shnirelman
paradox: the Euler's equations in the plane are consistent with the
spontaneous creation of energy (Ex Nihilo !). Many attempts to add physical
restrictions to Euler's equations to avoid this paradox have failed by
counterexamples. So, I guess that such the information contained in such a
spontaneous-creation-of-energy solution exceeds the information of our
cosmological horizon like in the above-mentioned argument for the Chaitin's
constant.

An open problem is to apply reverse mathematics in order avoid
Scheffer-Shnirelman paradox. I am not aware of the literature about his
approach.

By the way, Prof. Shnirelman will give a course on non-standard analysis at
Concordia during the winder 2019:

Kind Regards,
Jose M.

References:
Cosmological horizon: https://en.wikipedia.org/wiki/Cosmological_horizon
Chaitin's constant: https://en.wikipedia.org/wiki/Chaitin%27s_constant