The story of Infinity Avenue was primarily put forth as an argument against the complete traversal of an infinite region through successive steps. That is, the argument is that such a traversal is impossible, and its impossibility is illustrated in the story. The story, so the argument goes, leads into a number of coherency problems and/or logical inconsistencies. The traversal in question is not necessarily logically impossible (in the sense of yielding a direct, formal contradiction) but the claim nonetheless is that it is metaphysically or ontologically impossible (i.e. cannot exist in actual reality under any possible circumstances).
Why would anyone care about the traversal of an infinite region? It has relevance to the idea of an infinite past. In an infinite past, the universe traversed an infinite number of years before reaching the present. If such a complete traversal is impossible, the past is finite and the universe has a beginning. If the universe necessarily had a beginning, this has important implications for a form of the cosmological argument. Basically, this argument says that the universe began to exist, something or someone (like God) had to create it. Some theists thus argue that God is the best explanation for the existence of the universe. After all, creating the entire universe is a big job. And if an infinite past is impossible and if the universe requires a cause, then the outside agency that created the universe would have to transcend space and time altogether. Once again, God fits the bill rather nicely. However, if the universe were infinitely old, no Creator of the universe is necessary. So eliminating the possibility of an infinite past would be of great help for theism.
While there are a number of alleged coherency problems generated by the story, this article will deal with only one here. In a nutshell, the argument claims that having always traversed an infinite region is not possible (any complete traversal of an infinite region—when done via successive finite steps—involves having always traversed an infinite distance). Now, on to the story.
Infinity Avenue is aptly named, for it is infinitely long and ends at Hilbert’s Hotel. With every step down the nice and perfectly flat sidewalk of Infinity Avenue, the immortal Joe Walker leaves behind an indelible footprint and walks at the speed of exactly one meter per second (1 m/s). After completely traversing Infinity Avenue, he reaches the Hotel with his infinitely long track of his indelible footprints behind him.
For Joe to have reached any given point in the Avenue, Joe must have walked an infinite number of steps to get there. In a beginningless traversal of an infinite region (again, when done via successive finite steps), one interesting thing is that the task of completely traversing an infinite distance has always been done. That is, it has always been the case that Joe has walked an infinite number of steps. However, the argument arising from the story is that if this task was always finished it was never actually done.
As an analogy, consider the scenario of Tristram Shandy having always written his autobiography. Yet an obvious necessary condition for him being finished is that he has in fact penned those pages (just as Joe Walker must actually walk the distance). But if Tristram Shandy was always finished, at no time could he have actually written it, and thus the autobiography couldn’t possibly exist to begin with. Similarly, it is argued that if Joe’s task (of walking an infinite number of steps) is always finished, he never actually did the work (of walking an infinite number of steps) necessary for the task to be complete. Because the requisite work was never done, it is not possible for Joe to have finished the task.
A crucial part of the argument is the idea that the work is in fact needed for the task to be complete (and conversely, that if Joe never did the work, the task cannot be complete). To justify this we examine two criteria that need to have been met for Joe Walker’s task to be complete.
Both of these requirements need to have been met if the task is to be complete. That the first criterion is needed is unsurprising. Joe needs to have actually walked an infinite number of steps in order to have walked an infinite number of steps. If Joe did not do the work (of walking an infinite number of steps), then it is not the case that he has walked an infinite number of steps. Denying the second criterion (the work that Joe did must be relevant, involved in the task being complete) would not make much more sense. If the second criterion is false, then the work that’s described the first criterion doesn’t need to be involved in the task being complete. This is obviously false if the first criterion is in fact needed. Thus, the work described in the criteria is necessary for the task to be complete.
A formal proof of the argument goes as follows:
Theorem: [![[All]](https://www.angelfire.com/mn2/tisthammerw/images/forall.gif)
x(Px 
Qx) · x(Px)] x(Qx)
t(Ft ~Wt)
![[Ex]](https://www.angelfire.com/mn2/tisthammerw/images/exist.gif)
t(Wt) N
~t(Ft)
t(Ft) (indirect proof assumption)
t(~Wt) 1, 4 theorem
t(Wt) 5 quantifier negation.
t(Ft) 3, 7 modus ponens
t(Ft) · ~t(Ft) 4, 8 conjunction
t(Ft) 4-9 indirect proof
The task in question is Joe walking an infinite number of steps, and that this task has always been finished (a similar thing holds true for Tristram Shandy and his autobiography). The purpose of the proof is to show that the task always being finished (i.e. t(Ft)) is impossible given the premises; in short because the task always being finished/complete means that the relevant work that is necessary for the task to be complete was never done.
The first premise says that if the task is finished and complete at time t, then it is not the case that requisite, relevant work is being done on the given task at time t. The first premise is true for both Tristram Shandy and Joe Walker, because if the task is already finished, then obviously no work is being done that is necessary for the task to be complete. And if Joe Walker has traversed an infinite number of steps, he no longer needs to continue walking in order to have walked an infinite number of steps. Joe Walker can still continue walking of course, nonetheless he is not doing any work necessary for the task (of walking an infinite number of steps) to be complete, because the task is already finished. (Even if Joe needs to walk a few more steps to get to the hotel, he has nonetheless already done the task of walking an infinite number of steps, and in fact this task has always been finished.) Any work done after the task is already finished is “too late” to be relevant work. The work is not responsible for the task being complete because the task is already finished.
To see this more explicitly, let’s recap the idea that for Joe Walker’s task to be complete, the following criteria need to have been met somewhere:
While the second criterion may seem redundant, it is quite crucial to the meaning of the first premise. The purpose of the second criterion is to distinguish between the kind of work that could finish the sort of task versus the work that actually does make the task complete. The latter more accurately conveys what is meant by relevant work. Any work done “too late” is not relevant.
The story of Jane the Builder nicely illustrates these points. Jane is an excellent builder of fine houses. Let T1 denote the task of building ten houses by December 6, 2004. The second criterion as applied here is “The requisite work (of building ten houses) must actually be involved in finishing the task T1.” Suppose in January of 2005, after finishing task T1, Jane goes on to build ten other houses. But the work she’s doing on those other houses was not involved in finishing T1. Even though she’s doing the same kind of work (house building) that task T1 requires, it is of no relevance in doing the work necessary for T1 to be complete. In fact, anything done after the 12/6/2004 date could not have been involved in completing the task T1 (and thus could not be involved in meeting the second criterion), because building ten houses after that date is too late. And if Jane did not meet the second criterion, then T1 was never completed even if she is building a thousand houses in January 2005. And as this example illustrates, the second criterion is essential.
As one might suspect, this idea regarding the second criterion becomes a problem if the task in question is always finished. After all, for what time t could Wt exist? Where could the requisite work be done? When could Joe do it? Any point in time is “too late,” because the task was already finished by then (hence for any time t, if Ft then ~Wt). Nowhere can Joe do any relevant work that is necessary for the task to be complete. Looking at the two criteria again:
But if the task described in #1 was always finished, at no time could he meet criterion #2 (as the formal proof confirms; see especially lines 5 and 6 of the proof). If the task was always finished it could never have been done.
The second premise is essentially a restatement of terms. If the requisite work for the task being finished never happens anywhere in time (e.g. Tristram Shandy penning those pages) then the requisite work for the task being finished was never done.
Line 4 is the indirect proof assumption, e.g., “The task of completely traversing an infinite number of steps was always done.” We see that this assumption leads to a direct contradiction given the theorem and premises.
Since the argument is valid, the only way to deny the conclusion is to reject one of the premises. But all the premises seem true. For instance, one could not reasonably deny that for the task to be finished, the requisite work that was actually involved in finishing the task had to be done (which is kind of what the third premise says).
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The weakest premise seems to be the third one. In that case a criticism could go like this: the third premise is false. The story simply doesn’t contain that allegedly crucial element: Joe doing the work. Why? At any time t, the task is already finished, so no work is ever needed. It’s impossible for relevant work to be done at any time, therefore it isn’t logically necessary.
One problem is that there seems little basis for the objection apart from wishful thinking and question-begging. “It’s impossible for relevant work to be done at any time, therefore it isn’t logically necessary.” That is not enough to prove that the crucial element (Joe did the relevant work) is not needed.
The whole point of the argument is that the element is logically necessary and the story does not (and cannot) have it, thereby making the story metaphysically impossible. Justification was given for why the work is necessary. Merely claiming it isn’t logically necessary (merely because the story doesn’t have it) isn’t enough to disprove the argument.
To disprove the third premise and disprove the idea that the work is necessary, one must attack its actual justification: the two criteria and the criteria’s supporting arguments. The two criteria are what justifies that the work is necessary for the task to be complete. Claiming that the work (described in the two criteria) is unnecessary does not seem reasonable, even to the extent of generating immediate logical inconsistencies.
For instance, take the claim that Joe needs to have actually walked an infinite number of steps in order to have walked an infinite number of steps. If Joe did not do the work (of walking an infinite number of steps), then it is not the case that Joe has walked an infinite number of steps. Claiming it can be any other way seems to generate a logical inconsistency.
Note the story of Tristram Shandy having written his autobiography. A given from the story is the requirement that, for the autobiography to exist, Tristram Shandy must have actually written it. In other words, if Tristram Shandy never wrote the autobiography it couldn't possibly exist. This is a given axiom of the story. Given this axiom, the story of the autobiography having always been finished generates into a logical inconsistency, and thus the story is not metaphysically possible. Now it is still possible under the assumptions for a book to have existed eternally, but not if the book existing requires that Tristram actually penned the pages, because if the book existed forever, then Tristram never actually wrote it (and neither did anyone else). Claiming that Tristram Shandy writing the autobiography isn’t necessary merely because he doesn’t do it in the story (“because at any time t, he’s already finished”) does nothing to make the logical inconsistency disappear.
A similar thing goes for the story of Infinity Avenue. To deny the third premise one must deny the axiom that the two criteria are necessary. But denying that the work (described in the two criteria) is necessary seems to generate logical inconsistencies. Remember Tristram Shandy’s “always finished” autobiography. A similar thing for holds true for Joe Walker in having “always” walked an infinite number of steps. The task he allegedly completed is a phantom task. He never actually did it, as the formal proof demonstrates.
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Proof of the theorem:
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