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Incompleteness I 7^{th} February 2014
The
importance of the role of actual measurements in providing a complete
and consistent description of a physical system can not be
underestimated. The completeness requires a measurement to be a
zero
entropy measurement. The consistency requirement for a
measurement results in a measurement
algorithm which can be used to verify the completeness of the
measurements being performed.
Consider for example a circle, being measured by two observers of different capabilities. The human observer will look at it and determine it to be a circle which is essentially a single measurement, whereas the bug crawling on the perimeter does not know what a circle is and may not even complete the measurements before its lifecycle is ended. Therefore for a human observer entropy for the problem of determination of the circle is k_{B}log_{e}1 = 0, whereas for the bug the entropy may be infinite as the number of measurements is very large. (k_{B} is the Boltzmann constant.) Herein lies the importance of the entropy in characterizing a system. For same result, different observers will have different entropies depending upon the system they have designed to make measurements based on their capabilities. The result or the event being measured is independent of the entropy and is considered absolute or a PE1 event. When it comes to the universe, it is an ongoing measurement with human observer in the position of the bug. We have already identified the problem of the determination of the origin is the critical one. Therefore we can state the following: "A system as designed by humans to measure the universe can never have the complete information about the origin and therefore can not be absolute." Also it is important to note that,
It also implies that any event which is determined to be measured complete can not represent a infinite lifetime event or a stable structure. Therefore the lifetimes for the particles such as protons and electrons may be very long, but they can not be infinite. Similarly with the given capabilities a human observer can not completely measure an event of infinite life time. Quite a dilemma we are in, what we measure does not matter eventually and what really matters we can not measure. To be continued...
Thanks go to JB for suggesting that we should look into Gödel's work. As crazy as it may sound we have a feeling that the metric suggested by Gödel for GTR may be in the right direction. Information on
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