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Thank you so very much for your post and the additional explanation!

I agree that we most likely do not need randomness to do probability, statistics or physics!

The reason I’m circling this like a buzzard is that it is important in my hypothesis that algorithm at inception is proof of intelligent design. In other words, should we discover information content at inception (either big bang or abiogenesis) – then I must have some way of knowing it is algorithmically irreducible or my hypothesis is just so much hot air (not that anyone else cares, but I do.)

Chaitin refers to Champernowne in his article, Randomness & Complexity in Pure Mathematics. In speaking of algorithmic randomness, Chaitin says that Champernowne’s is the first example of a normal real number and “it follows from the fact that the halting probability Omega is algorithmically irreducible information, that this

0 < Omega = Sum_{p halts} 2^-|p| < 1

is normal in any base.” He explains this later in the section as follows:

“Let me try to explain better what this means. It means that you can only get out as much as you put in. If you want to prove whether an individual bit in a specific place in the binary expansion of the real number Omega is a 0 or a 1, essentially the only way to prove that is to take it as a hypothesis, as an axiom, as a postulate. It's irreducible mathematical information. That's the key phrase that really gives the whole idea.

Irreducible Mathematical Information
0 < Omega = Sum_{p halts} 2^-|p| < 1
Émile Borel --- normal reals
Champernowne
.01234567891011121314...99100101...

Okay, so what have we got? We have a rather simple mathematical object that completely escapes us. Omega's bits have no structure. There is no pattern, there is no structure that we as mathematicians can comprehend. If you're interested in proving what individual bits of this number at specific places are, whether they're 0 or 1, reasoning is completely useless. Here mathematical reasoning is irrelevant and can get nowhere. As I said before, the only way a formal axiomatic system can get out these results is essentially just to put them in as assumptions, which means you're not using reasoning. After all, anything can be demonstrated by taking it as a postulate that you add to your set of axioms. So this is a worst possible case---this is irreducible mathematical information. Here is a case where there is no structure, there are no correlations, there is no pattern that we can perceive.

Unless you see a flaw in my reasoning, I’ll use Chaitin for a definition of algorithmic irreducible information.

Sigh... I need another cup of coffee. That sentence,

Unless you see a flaw in my reasoning, I’ll use Chaitin for a definition of algorithmic irreducible information.

should have read

Unless you see a flaw in my reasoning, I’ll use Chaitin for a definition of algorithmically irreducible information.

Actually we do get the first few bit of Chaitin's omega: http://apru.nus.edu.sg/pdf%20files/student/Shu%20abstract.pdf.