-2
$\begingroup$

Gödel's second theorem asserts that the consistency of a formal theory containing arithmetic cannot be proved by the tools of the relevant theory itself (provided that the theory is in fact consistent).

Because we are inside and bound by the tools of our universe, this seems to imply that either we can never develop a formal arithmetical theory that fully explains the workings of our universe or that our universe is inconsistent.

Is my logic correct?

Gödel's incompleteness theorems

https://www.encyclopediaofmath.org/index.php/Consistency

$\endgroup$

closed as off-topic by Renan, Frostfyre, Mołot, Ash, kingledion Nov 21 '18 at 13:58

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about worldbuilding, within the scope defined in the help center." – Renan, Frostfyre, Mołot, Ash, kingledion
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ Maybe this should be adked in Math.SE $\endgroup$ – Renan Nov 21 '18 at 12:40
  • $\begingroup$ But there is a Mathematics tag for this SE so I thought it would be appropriate. $\endgroup$ – chasly from UK Nov 21 '18 at 12:55
  • 3
    $\begingroup$ The [tag:mathematics} states that it is for question about mathematics "focusing on their effects on societies and civilizations." The question as stated is asking for an existential proof of our real, existing universe. The two are not related. $\endgroup$ – Frostfyre Nov 21 '18 at 13:15
  • 3
    $\begingroup$ But your question isn't asking about the impacts of mathematics on society, and that is what the tag is for. This question is asking for clarification about an existing theory. That's not worldbuilding. $\endgroup$ – Frostfyre Nov 21 '18 at 13:27
  • 2
    $\begingroup$ I've got to agree, this seems more like a philosophical discussion than a worldbuilding question. Mike Scott does hit the nail on his head with the answer, and you can see that this answer has little to do with worldbuilding. $\endgroup$ – kingledion Nov 21 '18 at 13:58
8
$\begingroup$

Your logic is incorrect, because our universe is not a formal system of logic, and so the Incompleteness Theorem does not apply to it. Furthermore, the Incompleteness Theorem requires an infinite domain, but the observable universe is finite, so there are only a finite number of statements you can make about it. A system which admits only a finite number of statements can be consistent and provable.

$\endgroup$
  • 1
    $\begingroup$ So the latter half of your reply implies that there is a 'bottom' to the granularity of the universe. The overall size of the universe may be finite but the depth we could go in discovering ever smaller and smaller fundamental 'particles' is at least conceivably infinite. $\endgroup$ – chasly from UK Nov 21 '18 at 12:47
  • 2
    $\begingroup$ @chaslyfromUK Unbounded is not the same thing as infinite. There may be no lower bound on the size of the smallest particle we can observe, and thus no upper bound on the number of particles in the universe, but that doesn't mean that it can be infinite. Think of the biggest whole number you can -- it's unbounded, and can be as high as you can imagine, but it can't be infinite. $\endgroup$ – Mike Scott Nov 21 '18 at 13:18
  • $\begingroup$ @chaslyfromUK why is it conceivable for there to be an infinite number of smaller particles? You shouldn't think about it from a human perspective, looking at a large atom and finding out it's made up of smaller and smaller elements. You should think about it from a logical perspective, which is to start at the beginning. If it's infinite, there is no beginning, and your building blocks all depend on a smaller element without there ever being a fundamental, smallest one. Now whether or not we will ever be able to observe that is another story. $\endgroup$ – Thymine Nov 21 '18 at 14:08
  • $\begingroup$ @chaslyfromUK No. The latter half of the answer doesn't imply granularity 'bottoms out". Instead it states a finite universe can be described consistently and provably by a finite number of statements. $\endgroup$ – a4android Nov 22 '18 at 4:22

Not the answer you're looking for? Browse other questions tagged or ask your own question.