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comment Categorical foundations without set theory
Also I noticed the question has been edited - first order logic does not need set theory. First order logic can be given entirely syntactically
Dec
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comment Categorical foundations without set theory
@Pete: Here is an example of an alternative foundation for studying topological spaces: Abstract Stone Duality, monad.me.uk/ASD
Dec
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comment Categorical foundations without set theory
The base category is the choice of the mathematician, and many categories can take that role. So they do indeed exist. Furthermore none of this is dependent on NBG or classes. If you have them, then there is a good choice for a base category. If you don't have NBG, you can choose another base category. Finally you are seriously misguided about classes - the class of all sets is a mathematical object in NBG set theory.
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comment Encoding fuzzy logic with the topos of set-valued sheaves
The Heyting algebra for the example given consists of elements $0 < * < 1$, with obvious meet and join, and implication is mostly 1, except $1 \Rightarrow * = *$, $1 \Rightarrow 0 = 0$ and $* \Rightarrow 0 = 0$. This example can also be considered as sheaves over the topological space {0,1}, where {1} is open, but {0} is not - the Sierpinski space. Note that the partial order of open subsets agrees with the Heyting algebra above.