Sunday, October 14, 2012

Infinite Sets an Impossibility?

If any readers are familiar with Thomas Aquinas, or simply arguments pertaining to the existence of God having to do with the impossibility of an infinite set, then hopefully this post will be enjoyable.

Here is an example of an impossible infinite set:

Premise 1: The universe has existed forever
Premise 2: Time flows forward at a finite rate

Conclusion: It would be impossible for us to ever reach the 'present' time, because there will always be non-zero amounts of time that have to pass before we will reach the present.

Another way of saying this is as follows: If time goes backwards into the past, and there was never a beginning of time, we would never reach the present. There would quite simply be 10 more years before 10 more years before 10 more years, etc. before we ever reached the present. In fact, you would never reach any point in time, because there would always be another 10 years (actually an infinite number of years) before you ever reached any given point.

Tonight I was also considering the 'traveling half the distance' paradigm. This argument is as follows:
If you continue traveling half the distance from point A to point B, you will never reach point B. You will get pretty darn close, but there will always be space between the two points.

I have to go study for a test now, but I wanted to jot down my ideas and throw them out there for consideration and comment! I acknowledge there are key differences between the two concepts I have introduced. Also, there are many issues to bring up, especially concerning the first two premises of argument 1.


  1. If you choose to only travel half distances, you will never reach your goal. One of those darn Greeks got there before you - Zeno was his name. If you look at his paradoxes, I think you'll find that the Achilles and the tortoise one is basically what you are getting at time. (Link:'s_paradoxes)

    The thing is ... the logic is the same, but Achilles will in fact overtake the tortoise (or perhaps we should say a hare will overtake a tortoise, or a McLaren supercar will overtake a Vespa scooter). Similarly, the logic seems to tell you that if the universe were eternal in the past (deliberately avoiding the word "forever"), then we'd never reach "now". However, just as the Achilles (or the hare) reaches the tortoise, we can reach the present - even if the universe is eternal in the past.

    Any chance of you reporting on this little event at your campus - ?

  2. Time is measured by cause and effect. As long as there is no motion, there can be no time. So in one sense, “time” cannot go back any further than the universe.

    Mathematically, though, it is perfectly possible for something to have infinite backwards time. The set of real numbers is fully infinite, yet the number 0 is still at a known location. (When modeling using differential equations, it is often necessary to assume that the system has been at its initial state for an “infinite” amount of time.) Not only can we have infinite sets, but we can have “countably” and “uncountably” infinite sets; continuums which can be bisected infinitely without loss of continuity.

    Which segues to the next part:

    I enjoy talking about Zeno’s paradox, because it provides a very good illustrative example of the derivative. The problem with the paradox is that each progression is considered to be taken at an equal time interval. For instance, you make it ½ of the way in 1 second, then ¾ of the way in another second, then 7/8 of the way in another second, and so on. If you were to do this, you would approach your goal asymptotically, but only would arrive there at infinite time.

    However, this is not how we usually move: instead, we move such that for each unit of distance traveled takes a proportional amount of time. No matter how small the unit is, the ratio of distance traveled to time elapsed will be finite (i.e. the speed.) This is how calculus works: your speed is equal to dx/dt –i.e. the ratio of an infinitesimally small bit of distance to the infinitesimally small amount of time required to travel that distance. Despite the fact that they are infinitesimally small, their ratio is finite.

    Therefore, what Zeno failed to realize is that as you bisect the distance between you and your goal, you also bisect the remaining transit time!