Let ${\bf N}^\omega = \bigcup\_{m=1}^\infty {\bf N}^m$ denote the space of all finite sequences $(N\_1,\ldots,N\_m)$ of natural numbers. For want of a better name, let me call a family ${\mathcal T} \subset {\bf N}^\omega$ a *blocking set* if every infinite sequence $N\_1,N\_2,N\_3,N\_4,\ldots$ of natural numbers must necessarily contain a blocking set $(N\_1,\ldots,N\_m)$ as an initial segment. (For the application I have in mind, one might also require that no element of a blocking set is an initial segment of any other element, but this is not the most essential property of these sets.)


One can think of a blocking set as describing a machine that takes a sequence of natural number inputs, but always halts in finite time; one can also think of a blocking set as defining a subtree of the rooted tree ${\bf N}^\omega$ in which there are no infinite paths. Examples of blocking sets include


1. All sequences $N\_1,\ldots,N\_m$ of length $m=10$.
2. All sequences $N\_1,\ldots,N\_m$ in which $m = N\_1 + 1$.
3. All sequences $N\_1,\ldots,N\_m$ in which $m = N\_{N\_1+1}+1$.


The reason I happened across this concept is that such sets can be used to pseudo-finitise a certain class of infinitary statements. Indeed, given any sequence $P\_m(N\_1,\ldots,N\_m)$ of $m$-ary properties, it is easy to see that the assertion



> 
> There exists an infinite sequence $N\_1, N\_2, \ldots$ of natural numbers such that $P\_m(N\_1,\ldots,N\_m)$ is true for all $m$.
> 
> 
> 


is equivalent to



> 
> For every blocking set ${\mathcal T}$, there exists a finite sequence $(N\_1,\ldots,N\_m)$ in ${\mathcal T}$ such that $P\_m(N\_1,\ldots,N\_m)$ holds.
> 
> 
> 


(Indeed, the former statement trivially implies the latter, and if the former statement fails, then a counterexample to the latter can be constructed by setting the blocking set ${\mathcal T}$ to be those finite sequences $(N\_1,\ldots,N\_m)$ for which $P\_m(N\_1,\ldots,N\_m)$ fails.)


Anyway, this concept seems like one that must have been studied before, and with a standard name. (I only used "blocking set" because I didn't know the existing name in the literature.) So my question is: what is the correct name for this concept, and are there some references regarding the structure of such families of finite sequences? (For instance, if we replace the natural numbers ${\bf N}$ here by a finite set, then by Konig's lemma, a family is blocking if and only if there are only finitely many finite sequences that don't contain a blocking initial segment; but I was unable to find a similar characterisation in the countable case.)



Is there a name for a family of finite sequences that block all infinite sequences?
Intuitionists use the name "bar" for what you called a blocking set. The relevant context is "bar induction," the principle saying that, if (1) a property has been proved for all elements of a bar and (2) it propagates in the sense that, whenever it holds for all the 
one-term extensions of a finite sequence s then it holds for s itself, 
then this property holds of the empty sequence. (I'm omitting some technicalities here that distinguish different versions of bar induction.) 


There's also a closely related notion in infinite combinatorics, called a "barrier"; this is a collection $B$ of finite subsets of $\mathbf N$ such that no member of $B$ is included in another and every infinite subset of $\mathbf N$ has an initial segment in $B$. This is the subject of a partition theorem due to Nash-Williams: If a barrier is partitioned into two pieces, then there is an infinite $H\subseteq\mathbf N$ such that one of the pieces includes a barrier for $H$ (meaning that every infinite subset of $H$ has an initial segment in that piece). 


