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From Mathieu Marion *July 17, 2009*

*Posted by Alexandre Borovik in Uncategorized.*

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I do not know if this is of interest to you or not but here is a thought-experiment of mine, probably around the age of 7-8, for sure after 6 and before 10.

I started thinking about death and wanted to convince myself I would never die, instead of thinking about life after death… So I started thinking about an infinity in this way: first, I assumed that my entire life was only one dream in one night in another life where I am still the same person but could not fully realize that a full life goes on in each dream (an interesting point about personal identity, I guess). Now, that other life would be finite and have only a finite number of nights. So, I thought further that in each night there must be a finite number of dreams, encapsulating a finite number of lives. This was still short of infinity, so I started thinking that in each of these finitely many dreams of the finitely many nights, I would live a life that would in turn contain finitely many nights, which would contain finitely many dreams, and so on. I was not so sure that I was safe that way (i.e. that I would go on living forever), but I convinced myself that these were enough lives to live, so that even if the process would end, I would still have lived enough, and stopped thinking about it.

König’s Lemma (for trees)

This is beautiful. I had the same “grand thought” at about the same age and I remember that I had the feeling of a true realization upon fully grasping its implication. The difference being that I did not look for a solution to a problem (death) rather I had been thinking about the nature of dreams and why dreams where as realistic as my “real” life. Then this thought developed on its own.

Many years later I had one of my first lucid dreams. I was very excited and started to think about what I whished to do with the freedom I had gained knowing that I was now in dream state, only to wake up. What a disappointment. Some time later I woke up again, this time for real (?) only to find out that my prior waking up was just a change from lucid dreaming to normal dreaming. I had tricked myself out of the lucid state. Am I still dreaming?

Dream within a dream — could be this be a reflection of first attempts at reflection and introspection?

JT is right: it looks like König’s Lemma: every tree that contains infinitely many vertices, each having finite degree, has at least one infinite simple path.