Comment by Sabotaber on 31/01/2025 at 19:44 UTC

2 upvotes, 2 direct replies (showing 2)

View submission: Logic has no foundation - except in metaphysics. Hegel explains why.

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You should take a look at Turing's Halting theorem. It's much easier to digest than Godel's work, but touches on a lot of the same ideas. An interesting exercise is to look at Machine X and figure out if it will halt, which should give you some insight into the whole metalanguage thing.

Also relevant is that information requires context to have meaning. If the universe we inhabit is informational, then it is very easy to assume(but not prove) a cosmology where it's turtles all the way down. It goes like this:

Information requires context to have meaning.

My context is made of information.

Repeat.

And this is bizarre because there are lots of reasons to think the universe might be purely informational, which also brings up the notions of philosophical ideas like Plato's world of forms and whether or not numbers exist abstractly. If they do exist abstractly and the universe is informational, then you can even argue that their existence explains our existence. But the context problem of information also lets every set argue it is uniquely represented by the number 1, which breaks any sense of mapping that you could use to describe the relationship between the evolving state of the universe and the set of all integers.

Another interesting quandary is that Graham's number is expressible in this universe. To write it out long-form would supposedly require the space and mass of a ridiculous number of universes the size of our observable universe, which means there MUST be gaps in the list of integers we can express in this universe. This is because of how the pigeon hole principle applies to the combinatorics of the arrangement of all the mass in the universe: A ridiculous number of combinations are taken up by all the possible arrangements that represent Graham's number, so there are also going to be a ridiculous number of smaller numbers that cannot have unique represensations. The big question I have is this:

What determines which integers are actually expressible in this universe?

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Comment by Nigel_Mckrachen at 01/02/2025 at 19:03 UTC

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Well, you've basically stretched my brain beyond its limits. I'll have to say that this area, wherein it has been discovered there are limits to logic and science, have reshaped my thoughts on truth in itself. Similar to what Jung said, there are scientific truths and metaphysical (spiritual?) truths. His answer to questions about God was, "I don't deal in metaphysical truths." This line of thinking has changed my idea of reality, the afterlife, and God.

Comment by CryptographerTop9202 at 02/02/2025 at 07:34 UTC

1 upvotes, 1 direct replies

An integer is expressible in our universe only if it can be described in a way that fits within the finite informational capacity available. Although mathematics provides an infinite set of integers, physical limits—implied by principles like the Bekenstein bound—restrict the amount of information that any region of space can store. This means that only those integers with sufficiently compact descriptions, whether as a numeral, formula, or algorithm, can be uniquely represented within our finite universe. For instance, while Graham’s number can be defined by a concise formula, its full expansion would far exceed any feasible physical representation. In essence, if the shortest description of an integer (its Kolmogorov complexity) is within the bounds imposed by the universe’s finite resources, then that integer is expressible; otherwise, it remains abstract despite its mathematical existence. Moreover, expressing an integer requires a shared language or system of interpretation, intertwining physical limitations with the ways we communicate and formalize information.

The short answer is that an integer is “expressible” in our universe if and only if it can be described (or encoded) in a way that fits within the universe’s finite informational capacity. In other words, the integers we can actually represent are those that have descriptions (whether as a numeral, a formula, or a computer program) whose length in bits is less than or equal to the maximum amount of information that can be physically stored.