A proof of God's existence based on information theory
In our physical reality, there's never an instance where a quantity is infinite. That statement, in and of itself, is profound. Black holes, mislabeled singularities, have finite mass. The mass contained in our entire universe is ultimately finite. On the other end of the scale spectrum, the smallest detectable particles still have physical extent; there's nothing 'infinitely small'. Finally, energy/mass density of infinite magnitude can never be directly observed (again, the so-called black-holes) and so cannot be cited as examples of 'infinity in nature'. Infinity in nature has never been observed and never will be directly observed. There is nothing infinite in our physical reality.
Information theory is most conveniently expressed in terms of binary logic. Boolean values take on only two values: 0 or 1. A binary number is a sequence of 0s and 1s. A binary fraction uses the decimal point just as we do for decimal numbers: to divide the whole-number part from the fractional part. An example of this is 10.01. Binary numbers are, in a very practical sense, the simplest representation of numbers in general. You cannot find a more simple representation of any number than binary. This is the main reason why computers store information in binary format. At the same time, any number can be represented in binary form. This is not so profound as the statement above, but still is. Now information can be identified as a unique sequence of 0s and 1s, a binary number, with some arbitrary assigned meaning. For instance, the signal '0101' might mean: “kill your target now” for some assassin hit-squad. Or it might be the balance in your bank account (let's hope not .. unless it's in millions). It's clear that any assigned meaning is arbitrary yet it should also be clear that number holds information content – if only for its value.
There is a branch of mathematics called real analysis. One of the most profound results of that branch is the fact: if you pick a number between 0 and 1 at random, chances are – that number will be irrational. What are the chances? 100%. This is mind-boggling. What's the reasoning? It has to do with the density, or Lebesgue measure, of the irrationals which are a proper subset of the reals. It's almost as if the rationals and integers don't matter. Those numbers are what we use in every-day life! Every single number stored in every computer everywhere in the universe is a binary number. In other words, every piece of information stored anywhere in our universe is part of an irrational number! (Part here means subsequence.) Finally, we only need to consider the interval between 0 and 1 inclusive. From that single interval, we can reconstruct all reals. Put another way, that interval represents all reals.
This second-to-last section is 'somewhat speculative' in the sense it has not been proven or established as scientific fact. The entire chain of logic depends on this crucial link. If we prove it wrong, we prove the argument false – not necessarily the non-existence of God .. It has been proposed there is a one-to-one relationship between spacetime and R^4 (a four-dimensional representation of the reals). What this means is: if we take any point in spacetime, there's a corresponding point in R^4. Sounds simple enough but some sophisticated individuals question the assumption. For instance, time/space may be discrete or like the rationals. Time or space may be 'clumpy'. Time may be a random boolean value with positive tendency. We simply don't know. However, if true, spacetime has density properties akin to the reals. (As an 'aside', we need to make this question the primary one in physics (not the Higgs).) What this means is: if spacetime is anything like the reals, there's infinite information density in any arbitrary region!
Finally, there's only one proposed entity, fictional or otherwise, which has any infinite attribute: God. If there is any instance of infinite information density observable in our universe, the only possible source is an entity with some infinite attribute. In other words, only God could create infinite information density; if spacetime is anything like the reals, God exists.
PS1: if there are any readers who are deeply interested in mathematics and are excited about properties of real numbers, please write to me off-forum about pursing the following with me: as we know, the density of the irrationals determine the density of the reals, but there is a subset of irrationals called the transcendentals which may perform a similar role for irrationals. The question is: what is the density of the transcendentals? If that density is zero, what subset (if any) determines irrational density?
PS2: if the argument and associated notions above are correct, this may give some evidence for 'the soul'. One problem in 'metaphysical circles' is how the soul stores information. Some say it does not. Some say it doesn't exist. If it does and there is some form of information storage, that information must be stored somehow. The argument above allows information to be stored in the fabric of spacetime. In this view, the body/brain would be a parallel 'device' which allows dual storage in the physical plane as well as allow the soul to interact physically.