I find your work really fascinating, however, and I mean no offense by this, but you should really seek treatment for your mental health issues. You are a brilliant individual, but you really need to work on your mental illnesses. Do you take any medication? Do you have a therapist or some sort of support group? What's your current plan for treatment?
I genuinely wish you the best, and I just hope you can seek out help when you need it.
If X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality, e.g., replace X by X×{0} and Y by Y×{1}.
|X| + |Y| = | X ∪ Y|.
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Anonymous2013-08-31 20:47
\int_{a}^{b} \, f(t)\ dt \ = \infty means that f(t) does not bound a finite area from a to b
Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
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Anonymous2013-08-31 22:17
A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics.
Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.
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Anonymous2013-08-31 23:47
Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model (the constructible universe) which satisfies ZFC and thus showing that ZFC is consistent. Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF¬C (ZF with the negation of AC added as axiom) and thus showing that ZF¬C is consistent.
Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in functional programming and domain theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus. At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense).
A bifunctor (also known as a binary functor) is a functor in two arguments. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other.
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Anonymous2013-09-01 10:08
Zermelo began to axiomatize set theory in 1905; in 1908, he published his results despite his failure to prove the consistency of his axiomatic system.
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Anonymous2013-09-01 10:53
κμ + ν = κμ·κν.
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Anonymous2013-11-30 7:58
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