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To repeat Emil and Andreas's comments, it can be found in Stephen G. Simpson, "Systems of Second-Order Arithmetic", the first chapter of which is available here: http://www.personal.psu.edu/t20/sosoa …

For a paper I am writing, I need these two facts. The proofs are fairly short, but I would rather just cite them. This is for martingales index by natural numbers. Also, I call a martingale which co …

As for your question on prerequisites, the more logic you know the better. Of course the basic concepts---proofs, models, Peano arithmetic, incompleteness, compactness, nonstandard models, primitive …

Recall that for sets $A, B \in 2^\omega$ that we say $A \leq_{tt} B$ if there is a total Turing functional $F \colon 2^\omega \to 2^\omega$ such that $F(B)=A$. This is called truth-table reducibility …

I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write …

I'm worried that I'm misunderstanding your question, but I think one could argue there is no satisfactory answer here except to say that this result likely predates "computability" itself. From the wi …

As the reader, you can also ask mathoverflow for help finding the paper. There have been some very interesting MO questions along this line. In particular, this record helps others to also track dow …

If your interest is type theoretic foundations, you might want to look into modern (type-theory based) theorem provers. This is how I learned both type theory and dependent type theory. This has the …

There is a theorem as follows: Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \m …

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is $$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$ if $X$ and $Y$ take values in the a me …

(At François's request, my comment in now an answer.) Yes, it is still an active research area. It however is spread out throughout a number of camps (traditions): The Weihrauch camp, the reverse mat …