Hi Dr. Dick,

i am seriously contemplating your paper.
thank you for your reply.
your analogy of how we come to understand words with respect to your discussion of embedded data has helped me understand the idea presented in your paper about which i was asking.

i've drawn up a synopsis part of which are quotes from your paper and part of it i've put in my own words. my synopsis follows below:

truth as we know it is dependent upon our definitions:
hence we must be meticulously correct in our definitions.

reality is comprehensible and communicable therefore:
reality can be represented by a set of numbers.

this set of numbers can be divided into subsets. those subsets can be construed as transformed by our senses for analysis via the fundamental transform of the model we construct.

because these subsets are examinable they must be finite.
because our purpose is to establish a mental image of the universe the number of subsets examined must be finite.
because the mental image must allow for the existence of another subset not yet examined the number of subsets that make up the universe must be infinite.

since the examined subsets (of which we shall construct our mental image) are finite they can be ordered.
they can be ordered by the continuous parameter 't' which shall be defined as time. where a specific value of t will indicate a specific subset available to be examined. time has a definite direction: the past consists of those subsets available to our senses whereas the future consists of those subsets which are not available for examination. The past is what we know and the future is what we do not know.

an observation is defined to be a member of the set of examined subsets of reality.

the task at hand is to determine from a sequence of observations, exactly what the rules of the universe are: that is, we are interested in constructing a mental model which will separate the actual observable universe from the set of all possible universes.
In order for that result to be possible, the sequence must be produced by some unknown algorithm (i.e., it is presumed that the sequence is not random or without rules)

if we knew the algorithm it would have to be independent of time since independent observers in time must be able to apply the algorithm and still be able to construct the rules of the universe or our mental model.
since the algorithm is independent of time we can not predict any particular observation unless the time of the observation is contained in some implicit manner.

If there is information implicitly embedded in the data, it must be presumed that there are patterns of data which are possible (uncontradictable) and patterns of data which are not possible(contradictable). This is the essence of what a scientist means when he says there are rules. If all things are allowed then there are no rules.

upon examining our observations (our set of numbers) we can have a collection of patterns that are unique or some patterns that may appear more than once. if we were all knowing we would know what data to add to those sets so as to make those sets unique. since we are not all knowing we will just say we will add unknownable data to those sets so as to make the collection of patterns unique.
now let us add further unknowable data such that every possible observation consists of a unique pattern even after any arbitrary element is removed from that observation. If that fact is true, then the value of the removed element may be determined via the rule and the remainder of the pattern.
this situation can be represented mathematically as below:

Xn = f(X1,X2,X3,...Xn-1)

so this is where i stop and ponder for a while....

it would be appreciated if you inform me of any errors or omissions that i may have made.

regards tim