Mathematicians Bridge Finite-Infinite Divide
&gt; The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption — not evident in nature — that infinite objects exist.<p>Let Y = some ridiculously huge finite number. The notion that ∞ = Y + 1 has to be one of the most insidious defects in reasoning. Even to acknowledge the defect by dismissing it is almost like saying things could have turned out that way when they _never_ could.<p>Finite means limited or bounded or quantifiable. Infinite means unlimited or unbounded or unquantifiable. They are two wholly separate classes of things. Another way to say this is discrete versus continuous (in nature).<p>Imagine I said that I could bridge the coloured&#x2F;non-coloured divide. What could I possibly mean by that. It would mean that I have found a way to talk about both coloured things and non-coloured things that applies to both class of things.<p>Put another way. Imagine I separate coloured things into green ones and not green ones. Both subclasses belong to the class of coloured things. What class then do the subclasses of finite things and non-finite things belong to? Negation constructs the distinction, negation _is_ the bridge in a way and all the could be said of both subclasses of things is that they are both things. (Which, to be clear, is not saying very much at all, is it?) If those things are numbers then all we are saying is that both things are numbers, if both things are procedures then all we are saying is that both things are procedures. And so on.<p>If I&#x27;m thinking about this properly then the paper is trying to more precisely define the term &quot;countably&quot;. That&#x27;s as most as it can do. To say otherwise is not just vastly overstating what the paper is about but outright misleading.