J Williams
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 Nov 4 awarded Notable Question Feb 3 awarded Popular Question Nov 12 awarded Yearling Nov 13 awarded Yearling Jun 10 accepted Model structure on Simplicial Sets without using topological spaces Dec 5 awarded Fanatic Nov 15 awarded Nice Question Nov 14 asked Model structure on Simplicial Sets without using topological spaces Nov 13 awarded Yearling Nov 9 awarded Citizen Patrol Jun 30 awarded Popular Question Apr 10 answered If we can define a topology on the set of all the ideals of a commutative ring? Jan 11 accepted Real spectrum of ring of continuous semialgebraic functions Dec 19 asked Real spectrum of ring of continuous semialgebraic functions Dec 13 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 13 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 13 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. Dec 13 answered Categorical foundations without set theory Dec 11 awarded Enthusiast Dec 9 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.