[FOM] Improving Set Theory

[José Manuel Rodríguez Caballero]

Harvey Friedman wrote:

> I am particularly interested in what people think is lacking or is
> flawed about ZFC.


Maybe ZFC is not the most natural framework in order to express emergent
properties related to the so-called Sorites Paradox [3]. For example, one
neuron is not an intelligent brain. If we have a system consisting of n
neurons that is not an intelligent brain, to add one neuron more to the
system will not transform it into an intelligent brain. The conclusion
seems to be that there are not intelligent brains in nature, which is
contradicted by experience. Therefore, sciences depending on emergent
properties (neurology, biology, etc) may need mathematical machinery which
is not founded on ZFC.

The theory of elementary topos, which is not part of ZFC, can be used in
order to formalize some emergent properties, e.g., the notion of very
large, which was formalized by Jean Benabou [1, 2] in a way that it cannot
be done in ZFC, because the Boolean topos, which is associated to ZFC,
contains what category-theoreticians call a natural number object [4].
Benabou's construction [1] cannot be defined in structures having a natural
number object.

Kind Regards,
José M.

[1] Benabou, Jean, and Bruno Loiseau. "Orbits and monoids in a topos."
Journal of Pure and applied algebra 92.1 (1994): 29-54.
URL =  https://www.sciencedirect.com/science/article/pii/0022404994900450

[2] Benabou, Jean - Lecture: Very, almost, and so on, ...
URL =  https://youtu.be/_7uONqXQvp8

[3] Hyde, Dominic and Raffman, Diana, "Sorites Paradox", The Stanford
Encyclopedia of Philosophy (Summer 2018 Edition), Edward N. Zalta (ed.),
URL = https://plato.stanford.edu/entries/sorites-paradox/

[4] Natura Number Object in nLab
URL =  https://ncatlab.org/nlab/show/natural+numbers+object


--

José Manuel Rodríguez Caballero

arvutiteaduse instituut / Institute of Computer Science
Tartu Ülikool / University of Tartu

Personal Research Page: https://josephcmac.github.io/
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