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# #60 Counting Down from Infinity

June 09, 2008
Q

Dear Bill,

Me and my husband have a difficulty with a version of Kalam argument against infinite past.

You wrote in your answer to question 12 (when arguing against the thesis that the temporal series of all past events is actually infinite),

... if an actual infinite could be formed by successive addition, then various absurdities would result. ... Suppose we meet a man who claims to have been counting down from infinity and who is now finishing: . . ., -3, -2, -1, 0. We could ask, why didn't he finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished.

We are interested in what seems to us to be an implicit and crucial premise (P, for short) of the quoted argument:

(P) IF (i) the temporal series of all past events is actually infinite in its duration (as measured by equal temporal intervals), THEN (ii) there COULD be some mind/clock/counting machine/computer/angel/god which would SUCCESSIVELY pair all the past equal intervals (say, seconds) to all negative whole numbers in the corresponding order.

As we understand the argument, you assume (P), or something relevantly similar, then you argue from some other premises (including the premise "it should already have finished") to Not-(ii), and finally you derive Not-(i) from Not-(ii) and (P) by modus tollens.

Now our questions:

1. The connective "IF ..., THEN ..." in (P) suggests some kind of entailment. But entailment is a convoluted issue. Philosophers talk about different kinds of entailment (like material implication, strict implication, relevant entailment), but they seldom specify them precisely. Could you, please, explicate the entailment in (P)?

2. The word "COULD" suggests some kind of possibility. Again, scholars talk about different kinds of possibilities (like strictly logical, broadly logical, and conceptual possibility). Could you explicate the "COULD" in (P)?

3. Are there any good reasons for (P)?

For example, do they consist in a sort of imagining? If so, what is that we are imagining? Secondly, note that we can't imagine the process of counting of the given sort clearly even if we can imagine the END of such counting (like the finite sequence of "-3, -2, -1, 0, finished!"). Thus we can't argue by means of the principle "whatever is readily imaginable is possible." We admit we can imagine, e.g., a god's beginningless meter applied to a 4D model of a beginningless universe, recording how many years remain until some important event in that universe. For every second the meter would display a corresponding negative number. But then we imagine rather enumeration of a non-successive, intuitive, all-at-once apprehension by the god of each and every one-to-one correspondence between any past second the infinite temporal series with any appropriately ordered negative number. And this enumeration is different from the temporal process of SUCCESSIVE counting (pairing) referred to in (P).

Thank you very much!

Pavla

United States

## Dr. craig’s response

A

Your husband Vlastimil must figure I'm partial to the ladies, Pavla, since I hadn't yet selected his same question! O.K., I give in.

The argument I gave takes P for granted—granted, that is, by the proponents of the infinitude of the past. They assume that if the past series of events is infinite, then the order type of that series is the order type of the negative numbers, or in the usual symbolism, ω*. Logically, that isn't necessary. The series of past events could have the order type ω* + ω*, the order type exemplified by . . . , -3, -2, -1, . . . , -3, -2, -1. But it's obvious why proponents of the infinite past don't adopt this view: for then there are past events which lie at an infinite distance from the present. But then one could never traverse the infinite distance from, say, the first -3 to the second -3. That's why proponents of the infinite past always insist that the existence of an infinite past doesn't entail an infinitely distant starting point. The Tristram Shandy paradox about a man writing his autobiography so slowly that it takes him a year to record the events of a single day challenges the assumption that an infinite past would have the order type ω*. The only hope for proponents of the infinite past is to insist that the series of past events has the order type ω* so that every event lies at only a finite distance from the present. In that way, forming an infinite past by successive addition doesn't involve, they claim, traversing an infinite distance.