I share most of Paddy Leahy's concerns and areas of confusion with regard to the "Why Occam" discussion so far. I really don't understand what it means to explain appearances rather than reality. It's hard to get my mind around this kind of explanation and what to expect from it. Also the way the Anthropic Principle applies to infinite strings seems extremely vague until we have a clearer picture of how those strings relate to reality.
One area I differ: Paddy Leahy writes, quoting Russell: > > However, as the cardinality of "my" ensemble is actually "c" > > (cardinality of the real numbers), it is quite probably a completely > > different beast. > > There you go again with your radical compression. Without the reading I've > been doing in the last two weeks, I wouldn't have been able to decode this > statement as meaning: > > 2^\aleph_0 = \aleph_1 (by definition) > > To assume c = \aleph_1 is the Continuum Hypothesis, which is unprovable > (within standard arithmetic). Actually Russell did not bring aleph_1 into the picture at all. All that he referred to was aleph_0 and c which by definition is 2^aleph_0. c is the cardinality of the reals and of infinite bit strings. This is all just definitional. Whether c is the "next" infinite cardinal after aleph_0 is the Continuum Hypothesis, but that is not relevant here. Another area I had trouble with in Russell's answer was the concept of a prefix map. I understand that a prefix map is defined as a mapping whose domain is finite bit strings such that none of them are a prefix of any other. But I'm not sure how to relate this to the infinite bit strings that are "descriptions". In particular, if "an observer attaches sequences of meanings to sequences of prefixes of one of these strings", then it seems that he must have a domain which does allow some inputs to be prefixes of others. Isn't that what "sequences of prefixes" would mean? That is, if the infinite string is 01011011100101110111..., then a sequence of prefixes might be 0, 01, 010, 0101, 01011, .... Does O() apply to this sequence of prefixes? If so then I don't think it is a prefix map. I want to make it clear by the way that my somewhat pedantic and labored examination of this page is not an attempt to be difficult or stubborn. Rather, I find that by the third page, I don't understand what is going on at all! Even the very first sentence, "In the previous sections, I demonstrate that formal mathematical systems are the most compressible, and have highest measure amongst all members of the Schmidhuber ensemble," has me looking to see if I skipped a page! I don't see where this is discussed in any way. So I hope that by pinning down and crystalizing exactly what the first page is claiming, it will help me to see what the more interesting third page is actually able to establish. I think Paddy is in much the same situation. Hal Finney