Yes, pending a slight change to the question. (see A → B, below)

I suggest we utilize Solomonoff induction (LW's narrative intro for those who don't know some basic theory of computation) in order to examine the issue. I will need one other tool: the concept of a non-recursively enumerable set of axioms. I claim that there are two kinds of infinite being:

1. infinite in number
2. infinite in description

Many people seem to think in terms of #1, but I want to argue that this is not a helpful way to think. Instead, I say we should think of an infinite being as not being describable by a [finite] computer program. That is, no longer how many lines of code you right in your simulator of the infinite being, you'll always be in error. You could get closer and closer to describing that infinite being, but you'd never 'finish', except as t → ∞ (which may be what John 17:3 claims).

Solomonoff induction plays on the idea of simplicity, which is notoriously hard to define. We may, however, be able to use Kolmogorov complexity, as argued in The Computational Theory of the Laws of Nature (I suggest reading the section "God's Problem" and "God's language"). When we want to explain a sequence of observations, per Ockham's razor, we want to pick the simplist explanation. That would be the algorithm with lowest Kolmogorov complexity. So, I can rephrase the question:

Could a sequence of observations ever be best-explainable via a type-2 infinite being, where "best-explainable" means having lower Kolmogorov complexity than any description?

I claim that this, however, is not quite the right question. It asks us to explain what already is. What if we wish, instead, to predict the next observation? This is a subtle shift, from:

     A. What best explains a given sequence of observations?
     ↓
     B. Where does a given sequence of observations most likely point?

If I am allowed to tweak your question from A → B, then we can talk about two different options for B:

     I. The next observation will be like previous, and require no increase in Kolmogorov complexity.
    II. The next observation will be different, and require an increase in Kolmogorov complexity.

If we run into enough II-type observations, I claim we are justified in inferring that our universe is infinite in description, via induction, despite the problem of induction. Furthermore, via Fitch's Paradox, the potential for infinite knowledge requires an extant omniscient being. So either everything that is knowable is already known, which would lock us into an epistemically finite universe, or there is an extant omniscient being, which means that being is infinite.

Yes, pending a slight change to the question. (see A → B, below)

I suggest we utilize Solomonoff induction (LW's narrative intro for those who don't know some basic theory of computation) in order to examine the issue. I will need one other tool: the concept of a non-recursively enumerable set of axioms. I claim that there are two kinds of infinite being:

1. infinite in number
2. infinite in description

Many people seem to think in terms of #1, but I want to argue that this is not a helpful way to think. Instead, I say we should think of an infinite being as not being describable by a [finite] computer program. That is, no longer how many lines of code you right in your simulator of the infinite being, you'll always be in error. You could get closer and closer to describing that infinite being, but you'd never 'finish', except as t → ∞ (which may be what John 17:3 claims).

Solomonoff induction plays on the idea of simplicity, which is notoriously hard to define. We may, however, be able to use Kolmogorov complexity, as argued in The Computational Theory of the Laws of Nature (I suggest reading the section "God's Problem" and "God's language"). When we want to explain a sequence of observations, per Ockham's razor, we want to pick the simplest explanation. That would be the algorithm with lowest Kolmogorov complexity. So, I can rephrase the question:

Could a sequence of observations ever be best-explainable via a type-2 infinite being, where "best-explainable" means having lower Kolmogorov complexity than any description?

I claim that this, however, is not quite the right question. It asks us to explain what already is. What if we wish, instead, to predict the next observation? This is a subtle shift, from:

     A. What best explains a given sequence of observations?
     ↓
     B. Where does a given sequence of observations most likely point?

If I am allowed to tweak your question from A → B, then we can talk about two different options for B:

     I. The next observation will be like previous, and require no increase in Kolmogorov complexity.
    II. The next observation will be different, and require an increase in Kolmogorov complexity.

If we run into enough II-type observations, I claim we are justified in inferring that our universe is infinite in description, via induction, despite the problem of induction. Furthermore, via Fitch's Paradox, the potential for infinite knowledge requires an extant omniscient being. So either everything that is knowable is already known, which would lock us into an epistemically finite universe, or there is an extant omniscient being, which means that being is infinite.