Galchin Vasili wrote:
Let G be a directed graph that either has an infinite # of nodes or has edges which are loops.
Would a constructist recognize the existence of G's free category?
I'm no expert on the various schools of constructivism, but every one that I know would accept this free category (at least assuming that they already accept the infinite graph G). Note that the free category on a graph with 1 node and 1 loop is the monoid of natural numbers. Although philosophers talk about finitism (even ultra-intuitionism = ultra-finitism), everybody interested in doing mathematics itself (rather than simply the philosophy of mathematics) seems to believe in the monoid of natural numbers. Conversely, once you have the set of natural numbers, an explicit description of the free category on any graph is easy. For example, a morphism from x to y (where x and y are nodes in G) consists of a natural number n and a function from {1,...,n} to the set of edges that satisfies certain equations about endpoints. So if you believe in the set of natural numbers and you believe in G, then you ought to believe in the set of morphisms; and the rest is yet easier. Even a finitist can ~talk~ about this, even if they don't ~believe~ in it, in much the same way that a ZFC dogmatist, who believes only in small sets, can nevertheless talk about large categories with no difficulty. So only an ultra-finitist should have any problems; but ultra-finitism is a pretty ill developed approach anyway. -- Toby