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I found the article https://ncatlab.org/nlab/show/field+with+one+element that explains how important the field with one element is. I am more interested in logic and type theory (and cognition, consciousness that can be modelled/implemented by them, knowledge representation and automation) and just wanted to ask this - are there some connections, analogies, applications etc. between field with one element and the (Higher order) logic, set theory, type theory (lambda calculus) or arithmetic hierarchy of (in)computable functions? Is there are such connections then they would strongly motivate me to study this field furhter. Of course, I see a lot of applications of algebraic geometry in coding and cryptography, but they are not about field with one element. Thanks.
I appreciate your time, that is why only some links, key terms, list of notions suffice, all the remaining I can find myself, I just need some started, some hole through which to lean further.
I am starting to see - HoTT is the or The language for homotopy theories and maybe it is already expressive enough to reason about F1, or maybe HoTT should be enhanced as well?
I guess there could be a range of $\mathbb{F}_1$ entities as in some sense most basic, along the lines of
With the identification $\mathbb{F}_1 Mod \simeq FinSet^{\ast/}$ from above it follows that the algebraic K-theory over $\mathbb{F}_1$ is stable cohomotopy
Is there anything to fill the gap $\mathbb{F}_1 ??? \simeq Fin \infty Grpd^{\ast/}$?
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