Shannon explicitly leaves aside much of what we mean by information when he formulates his theory. Meaning is completely ignored, but what remains is still an important element in information: novelty.
Information is what I depend on another to get. If you send me a message like 2, 4, 6, 8… I quickly hit the point where I no longer depend on you for what comes next. If you send me < >MJJJ?”?>>kgkufy6-oh… then I am very dependent on you for whatever comes next. It’s this sense of “information” that leads to the paradoxical conclusion that the meaningless and random has “more information” than a meaningful sentence fragment.
In this sense, the information content of any physical medium is limited. Even in the random message above, you still had pretty-good-odds-for-a-lottery of guessing the next character (It had to be something typeable on a standard keyboard with English-language settings, for example.)
Plato’s theory of recollection argues that all learning is from previous learning and so is infinite. Aristotelianism arrives at the same conclusion by its hylomorphism: all limitation is from physicality or matter, and so the spirituality of mind makes it an unlimited being proportionate to an infinite source of information. What does this mean?
Infinite information would be perpetually surprising or novel – the amazement of the new could not wear off. The tour guides in the Grand Canyon don’t get to have the same rush of novelty as the tourists, but this is due to the limitations in physical objects. The Romantic exultation of the sublime was really exultation of novelty or of something inseparable from the novel, and so was an argument for human spirituality.
Cognitive neuroscience has also discovered the value of the novel. The bizarre and novel has always been away of increasing memory, because the brain seems built to look for it. Novelty is an essential element in triggering dopamine, in the sense that the amount released goes down with the familiarity or predictability of the stimulus. The inevitable lassitude, burnout, and boredom that results from acting as if physical pleasures could be infinite is a result of their intrinsically limited information content. Make orgasm a god, and it will soon be no more pleasant than sneezing; make a drug a god and you will soon only do it to stave off the pain caused by its absence. Physical pleasures only link up with the infinite to the extent that they can be a part of some spiritual good – the communion of persons, the fluidity and joy of discourse, the deepening of interpersonal bonds, etc.
Any two angels are more information rich than the physical universe. To see any one part of the physical universe allows you to predict another with greater accuracy than seeing one angel would allow you to predict another. After you go to heaven and see your first angel, you will see a second one and say “what is that”? (among the damned, the novelty of the same experience will only be in horror, which has its own novelty – think jump scares.) Even this image turns on the idea that they will be in different places, when in fact this part of the metaphor falsifies it.
At the summit of novelty is the triune God, made by the continual outpouring of divine cognitive and volitional possibility. God’s power is limited in what he can create, since creation itself has limitation, but the procession if the Arche into the Logos and the Life is entirely unlimited and novel. God alone is the Heraclitean river that one cannot be twice the same. There is no re-cognition of God as he processes into his cognitive and volitional persons, only the continual amazement of seeing him for the first time. Palamas is right that the essence of Trinity is unknowable, but this is not a darkness but the perpetual impossibility of re-cognition. One cannot say of the Trinity that it is just what he saw before.
The resurrection is into spiritualized matter that overcomes the limitation of physical information. This is why the physical body of Christ was not recognizable and was trans-dimensional, appearing at will and having a true physicality even while now “in heaven”. The assumption of his Mother was not the act of levitating her into orbit but of taking her entirely into this same trans-dimensionality.