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I am not sure what you imagine, but once one makes the move to games with infinite play, then various set-theoretic issues come to light, and the subject becomes more set-theoretic and less like combi …

The suggestion in the comments that a linear order embeds into $\mathbb{R}$ just in case it has a countable dense set is not quite true. For example, let $2\times\mathbb{R}$ be the doubled real line ( …

I have engaged with the Lewis-style set-theoretic mereology in a few papers, undertaken jointly with Makoto Kikuchi. My interest in this topic was inspired originally by a MathOverflow question, Why h …

I claim that a set is primitive recursive enumerable if and only if it is computably enumerable. So the answer to your question is affirmative. Clearly any p-r.e. set is c.e., since primitive recurs …

You need a uniformity requirement in your limit characterization of $0'$, that is, the functions $f_k(n)$ need to be uniformly computable, in that there is a single computable function $f(k,n)$, such …

Another criterion may be to make HOD far from $V$ with respect to forcing. For example, in our paper G. Fuchs, J.D. Hamkins, J. Reitz, Set-theoretic geology, Annals of Pure and Applied Logic, vol. …

This is an immediate consequence of the fact that every $\Pi^1_1$ statement is equivalent to the assertion that a certain relation is well-founded, and well-foundedness is absolute between transitive …

Let me sketch the argument. Philip Welch is also on MO, and I would encourage him to post further explanation and details. The main question left open in the original ITTM paper Joel David Hamkins …

The answer is yes. Consider any limit ordinal $\gamma$ and let $X$ be the space of all binary $\gamma$-sequences that are not eventually ones, ordered under the lexical ordering. We may put the order …

Here is a way to get the tree property directly from the partition property, and this gets you to Mahloness from the references in the links provided in the comments. Assume that $\kappa$ is an uncou …

I don't agree that the "usual" use of forcing uses only countable transitive models. Perhaps this used to be true, years ago, and forcing is sometimes still taught this way now, because it is somewhat …

Perhaps it is helpful to mention that an equivalent formulation of your question concerns partial orders rather than graphs. Namely, if $(V,E)$ is a directed acylic graph, then your reachability rela …

In the case of a countable group, this kind of thing often arises in the subject of Borel equivalence relation theory, which has been considered in a few MO questions (see also links in this answer). …

It seems that there is no such space. Indeed, I claim that every non-isolated point in a Boolean space is the limit of a long sequence in that space. We may assume that the space $X$ is the Stone sp …

If the machine transition induces a partial order, then the machine cannot find itself in the same local configuration again after leaving it, since a partial order has no loops. It follows that the l …