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Comment by scythe

4 years ago

People complain about infinite information, but the theory of real fields is complete and consistent, while arithmetic is not:

https://en.wikipedia.org/wiki/Decidability_of_first-order_th...

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scythe

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j-pb  4 years ago

I often wonder if we simply constructed math the wrong way around.

People tend to mentally construct the natural numbers from set theory, wholes from naturals, rationals from wholes, reals from rationals and so on.

But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.

The very act of formalising mathematical concepts into words and paper is a quantisation step after all, because both are symbols. Maybe there are proofs that can be inuitioned about (assuming brains are continuuous in some sense) but neither verbalised nor formalised.

  • burrows  4 years ago

    > But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.

    Presumably this hypothetical universe implements arithmetic, so it’s not complete.

    • j-pb  4 years ago

      You missed my point.

      Incompleteness means that there are true statments for which there are no proofs.

      But that doesn't preclude proofs that are beyond the proof system that you proofed incompleteness for.

      A non discete/symbolic proof might exist after all.

      It would therefore not be the existence of the natural number subset that causes undecidability, but the missing parts of the non-natural superset required to talk about the proofs that cause undecidability.

      4 replies →

syki  4 years ago

How do you know arithmetic is not complete or consistent? In ZFC arithmetic is complete and arithmetic is consistent. This assumes that ZFC is consistent. The second order Peano axioms are categorical so I assume you mean only the first order theory.

At any rate, what does any of this have to do with information capacity in the universe? Is the information capacity of the universe related to the consistency/completeness of arithmetic?

  • burrows  4 years ago

    > How do you know arithmetic is not complete or consistent? In ZFC arithmetic is complete and arithmetic is consistent. This assumes that ZFC is consistent.

    Aren’t you just begging the question by assuming ZFC is consistent to demonstrate that arithmetic is consistent?

    • syki  4 years ago

      I don’t think so. The point is, saying “arithmetic is inconsistent” doesn’t mean anything without talking about where this theory resides. The larger point is that this has no relationship to whether or not the universe is infinite so it shouldn’t be talked about at all within that context.

      Suppose for a moment that it makes sense to say arithmetic is part of the fabric of the universe (whatever that is supposed to mean). How would one know if arithmetic is consistent or complete within the context of being part of the universe? Suppose ZFC is part of the universe. Then arithmetic (the model of it as being part of the universe) is complete and consistent. Now what? I claim nothing of consequence follows from this in relation to whether or not space is continuous.

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