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Since the suggestion to close the question has not been successful, allow me to offer an answer as a way of bringing the question to a conclusion. Many mathematicians find the situation of the questi …

There are a couple of ways to interpret the non-consecutive issue in your question. One way is that you are fixing a template with infinitely many positions, and in some of those positions, a defini …

It seems to me that the various accounts and theories of formal representations of belief might be something approaching what you want. Without deductive closure, it seems to me that the probability …

I think this is a very interesting question. In response to your comment, let me argue that if 1 holds and the measure is additive, then the singleton values are all the same. This is the sense in w …

This article contains the following statements. Describing the normality property, Bailey explains that "in the familiar base 10 decimal number system, any single digit of a normal number occ …

In your second question, you are asking merely for an atomless Boolean algebra, of which there are numerous examples. One easy example related to the one given on the Wikipedia page is the collection …

You are describing what is known as a supertask, or task involving infinitely many steps, and there are numerous interesting examples. In a previous MO answer, for example, I described an entertaining …

Since the existence of non-measurable sets is often seen as undesirable, we naturally want to have as many measurable sets as possible. With Lebesgue measure on the reals, for example, if we were to s …

I am currently writing an essay on hat puzzles, and for the warm-up section I introduce some of the standard finite hat puzzles. One of these proceeds as follows: You and two friends are each given a …

The answer is no, not necessarily. It can happen that there is no such set $B$. A counterexample is provided whenever $A$ is a Borel subset of the plane, while the projection $\pi(A)$ is analytic, b …

It seems to me that you have two questions here. First, you inquire about a formal account of "usefulness". I believe that this is already provided by the formal mathematical accounts of utility in …

There are a large number of natural graph properties that are not monotone. The property of being isomorphic to a given graph is never monotone (except for the empty graph and the complete graph). …

The existence of such a measure is equiconsistent to the existence of a measurable cardinal, one of the large cardinal notions, and if ZFC is consistent, cannot be proved in ZFC. (See the notion of re …

Let me focus on the question of your title, and mention that there is another quite robust way to understand what it means to say that a random Turing machine has such-and-such property. Specifically …

Your question is really about the uniformization problem, a major focus of descriptive set theory. A set $B\subset X\times Y$ is uniformized by a set $C\subset B$ if $C$ is the graph of a function wit …