Alex proposes to discuss the following argument from Craig:
1. A collection formed by successive addition cannot be actually infinite.
2. The temporal series of past events is a collection formed by successive addition.
3. Therefore, the temporal series of past events cannot be actually infinite.
Alex will dispute 1. For Alex, 1 is true if you assume 3. Before I get to the body of the blog entry, I don't see how this follows. If the argument transitioned from 1 to 3 without the assistance of 2, I can see why someone might infer this. But with the inclusion of 2, it's not the case that 1 is true only if you assume 3. But perhaps I'm missing something and Alex perhaps answers this later.
2. The Counterexample - Alex points out our associations of "sequences of whole numbers with durations of time", and formalizes our intuitions regarding the association in terms of 'a counting function' of inputs and outputs. I have some trouble with the language Alex uses to describe the function. Alex argues that input 'x' (which 'returns an output 'y') represents 'some amount of time that has passed' (e.g. 1 minute) and 'y' represents 'whatever number has been counted'. Let's break this down. I'll spell it out in chronological (or, more precisely, logical/explanatory order).
a. Assume input/output 'counting function'.
b. Plug into the function: 1 minute.
c. Out comes a 'counted number': say, the counting of #1.
For b, Alex wants to plug in, not 1 minute (or any 'finite amount of time'), but an 'infinite amount of time' (where the amount of elapsed time is denominated by some 'transfinite number'). The immediate issue with this move (it seems to me) is that if you start out with a transfinite number, such a number hasn't been formed via successive addition, which contradicts premise 1 without really arguing against it. I'm not sure why Alex jumps to talking about ordinals (I thought we were remaining within the domain of cardinals), but I'll go along with it. Alex makes a curious distinction between a number that's greater than any finite number, but less than any transfinite number. By any definition that I'm acquainted with, a transfinite number just is a number that's greater than any finite number. But that's neither here nor there. The issue is that of course, the function will yield (as an output) an actual infinite if we input an actual infinite. The problem is that the initial input hasn't been demonstrated to have been accumulated via successive addition. Yet Alex segues from such an initial input to seemingly conflating such an input with someone who 'has been counting' (a temporal process?) from eternity past. It's metaphysically impossible (the question at issue) to have your initial input in the first place if this person had been counting from eternity: the entertainment of such a scenario constitutes the antecedent in a counterpossible proposed as a thought experiment. Alex will object that he hasn't 'specified that' he 'has not' been 'using successive addition', that he has 'explicitly said that this is what' he 'is doing'. I will not deny that Alex 'says' that is what he is doing (for an infinite time). What I question is how he can justify the initial move to make the initial input an actual infinite number of past-eternal 'countings' without begging the question against premise 1. Premise 1 doesn't seem to me to have anything to do with counting for a finite time and seeing what would follow from this. What would follow from this is already evident in affirming that you've counted for a finite time! What premise 1 seems to be drawing my attention to is the idea that if you've been counting in such a way that your countings have followed a pattern analoguous to successive addition 'over time' (the second premise!), then the collection formed can never be an actual infinite. To have Alex's initial, functional input of an actual infinite, we'd already need good reason to think that such an input was metaphysically possible (even if mathematically legitimate) and we'd just have to assume with Alex that we already have this successively formed collection that somehow became actually infinite to have our initial, functional input in the first place. There doesn't seem to be any kind of dialectical ingenuity here. It just denies the first premise for the sake of adopting its contrary, and appropriating the contrary for the sake of a mathematical function that doesn't seem to have any of the requisite metaphysical implications anyway.
There aren't any restrictions on the amount of time spent counting. Craig is drawing out a corollary (via his thought experiments) of the very nature of counting (a process in time) and the impossibility of attaining a transfinite number of countings using this process.
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