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Comment by packetlost | Hacker News Reader

Maybe you can theoretically encode arbitrary (but not infinite) information, but there are still discrete underlying states.

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> encode arbitrary (but not infinite) information

What leads you to claim that you can't encode infinite information given an infinite set to work with?

reality does not have an infinite set though.

Infinity is an abstraction that simplifies a lot of maths, it isn't real though.

So you have proof spacetime is discrete?

I suspect a looseness with language; this often happens with maths done in natural language, and I may be guilty of it even when I try not to be.

If spacetime is infinite in extent yet discrete at the Plank scale, it would be ℵ₀; the Hilbert space is real-valued and therefore (a vector of) ℵ₁.

The Bekenstein bound, in practice, would do limit how much information can be actually encoded, even if all quantum states have an underlying real value and therefore even just one quantum value of one particle is ℵ₁ all by itself.

> the Hilbert space is real-valued and therefore (a vector of) ℵ₁.

But cardinality of real numbers might not be ℵ₁. It's literally unprovable in ZFC.

I had to look this up, and if I understand right: yes, all that's certain is |ℝ| = ℵₖ, k≥1? If so, same point applies.

No instrument gives you an infinite precision in which to encode information with or decode it back out (though in quantum systems the result quantizes to one of N possible states depending on the system, so this is moot). The other option is an infinite amount of time to continuously encode information, but practically decoherence becomes the limiting factor.

I don't have a formal education in QP so this comes from my engineering perspective of working on quantum computers.

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