In the previous post, we have had a first look at the connections between incompleteness, or logical independence -- roughly, the fact that for any mathematical system, there exist propositions that that system can neither prove false nor true -- and quantumness. In particular, we saw how quantum mechanics emerges if we consider a quantum system as a system only able to answer finitely many questions about its own state; i.e., as a system that contains a finite amount of information. The state of such a system can be mapped to a special, random number, an Ω-number or

*halting probability*, which has the property that any formal system can only derive finitely many bits of its binary expansion; this is a statement of incompleteness, known as

*Chaitin's incompleteness theorem*, equivalent to the more familiar Gödelian version.

In this post, we will exhibit this analogy between incompleteness and quantumness in a more concrete way, explicitly showcasing two remarkable results connecting both notions.

The first example is taken from the paper '

*Logical Independence and Quantum Randomness*' by Tomasz Paterek et al. Discussing the results obtained therein will comprise the greater part of this post.

The second example can be found in the paper '

*Measurement-Based Quantum Computation and Undecidable Logic*' by M. Van den Nest and H. J. Briegel; the paper is very interesting and deep, but unfortunately, somewhat more abstract, so I will content myself with just presenting the result, without attempting to explain it very much in-depth.