[FOM] the amazing language of set theory

[Tero TULENHEIMO]

Charlie Silver wrote:

> Envoyé: Mardi 29 Mars 2016 03:30:33
> Objet: Re: [FOM] the amazing language of set theory
> 
> For foundational studies, why not use (a sort of)  “function theory” instead
> of set theory (different from Von Neumann’s f’n theory  modified by
> Bernays,…)  There are different ways to look at f’n theory.  I’m proposing
> that the ordered pair <x,y> be the primitive sentence, saying y is a
> function of x.  

This proposal sounds pretty much like the starting point of the so-called dependence logic developed by Jouko Vaananen: one takes as primitive the statement that the value of variable x_0 is a function of values of variables x_1,...,x_n. Taken by itself, such a statement will be expressible in existential second-order logic (\Sigma^1_1). Certain extensions of dependence logic reach the expressive power of full second-order logic.

See e.g. 

http://www.math.helsinki.fi/logic/people/jouko.vaananen/Dagstuhl10.pdf

http://plato.stanford.edu/entries/logic-if/#DepLog

https://en.wikipedia.org/wiki/Dependence_logic

  Tero Tulenheimo