Gödel’s Infinite Candy Store
Posted: Mon Aug 16, 2021 6:21 pm
As Donald said, the formal is the "bones" that together with the "meat" of non-formal constitute the "body" of the totality of the experiential reality, so the totality is definitely not reducible to formal, yet there is a certain connection, or we can say, "inclusion".Shajan624 wrote: ↑Tue Aug 17, 2021 10:43 am Consciousness having a hidden mathematical structure is an intriguing idea but I don’t see how Gödel’s theorem and Idealism can be connected. Incompleteness theorems applies to formal systems where the primitive symbols have no connection with the world of experience - the symbols are not ‘about anything’. Consciousness has to be about something so it is difficult to see how theorems of formal systems apply.
Eugene I wrote: ↑Tue Aug 17, 2021 4:27 pm Cleric, we should extrapolate the Godel's theorem to the whole reality only metaphorically and as a simplified model/example. But the core idea is that the variety of ideal forms is inexhaustible and there is no end to their exploration and evolution, both into horizontal and vertical/hierarchical dimensions.
I'm still skeptical about the actual existence of eternity/infinity though, for me this is an undecidable question. I can have an idea of infinity/eternity, but I have no actual conscious experience of it, so for me it remains an abstraction. If you do have an actual experience of infinity then good for you
Well, that's the point of the Godel "candy shop": it is impossible to ever experience every possible ideal form from either individual or overall/collective perspectives. The candy shop is in principle inexhaustible.
May be Santeri will explain better The idea of the actual infinity is an abstraction, what we know from experience, as well as from computational mathematics, is only endless sequences of phenomena or events. To my knowledge nobody has ever experienced the whole infinity of events. For example, the sequence of the events of iterating addition x(n+1) = x(n) + 1 has no end, it never stops, however, the computational process will never reach the whole infinity of all x(n) numbers. So, every specific number x(n) actually exists because there is an algorithm (path) to reach to it (experience it). However, there is no algorithm/path to reach to the infinite wholeness of all numbers. In mathematical terms, the infinity of all numbers is incomputable, and similarly, the set of all ideal forms is un-experienceable. So, the set of all potential ideations is simply an abstraction, an idea, the actual set of all ideations it is not possible for us to experience, and therefore there is no experiential evidence that it actually exists.Also. how can there be "no end", "inexhaustible", etc. yet also no eternal-infinite aspect to ideation? How can you have an idea of something which is not within the set of all potential inexhaustible ideation?
Eugene wrote:May be Santeri will explain better The idea of the actual infinity is an abstraction, what we know from experience, as well as from computational mathematics, is only endless sequences of phenomena or events. To my knowledge nobody has ever experienced the whole infinity of events. For example, the sequence of the events of iterating addition x(n+1) = x(n) + 1 has no end, it never stops, however, the computational process will never reach the whole infinity of all x(n) numbers. So, every specific number x(n) actually exists because there is an algorithm (path) to reach to it (experience it). However, there is no algorithm/path to reach to the infinite wholeness of all numbers. In mathematical terms, the infinity of all numbers is incomputable, and similarly, the set of all ideal forms is un-experienceable. So, the set of all potential ideations is simply an abstraction, an idea, the actual set of all ideations it is not possible for us to experience, and therefore there is no experiential evidence that it actually exists.Ashvin wrote: Also. how can there be "no end", "inexhaustible", etc. yet also no eternal-infinite aspect to ideation? How can you have an idea of something which is not within the set of all potential inexhaustible ideation?
Now, if you assume a hypothesis that the set of all ideation still actually exists, you will run into the Russel self-referencing paradox. The set of all ideations is obviously itself an ideation, so it must contain itself. But that is not possible and leads to contradiction, here is a simple math proof that goes like this (but you need to know the set theory to decode it):
Suppose there were a set U of all sets (for example, an ideation of all ideations). Let A be defined by A={S∈U|S∉S} Since A is a set, it's an element of U , and you can ask if A∈A. Following Russel's argument, you can show that A∈A if and only if A∉A. Since the assumption of the existence of a set U of all sets leads to a contradiction, therefore such a set doesn't exist.
I gave you a simple illustration from mathematics: a never-ending iterative algorithm of adding +1 actually exists and you can compute and experience every possible iteration of it. According to this algorithm, the sequence of natural numbers is inexhaustible and the execution of the iterative algorithm is never ending (because you cannot reach a number x(n) to which you can not add 1 and compute a larger number x(n)+1). But the wholeness of infinity of all natural numbers in incomputable and un-experienceable and therefore there is no experiential evidence that it actually exists, because "actual existence" means something that can be experienced (or computed in mathematics). This is not to say that the infinity can not exist as an abstraction (idea) - you can think of an infinity, but you cannot actually experience or imagine the whole actual infinity of all natural numbers at once. And I remember you saying elsewhere "if something can not be experienced, then it does not exist".AshvinP wrote: ↑Tue Aug 17, 2021 10:22 pm I am just referring to an apparent internal contradiction in your logic. You conclude that Gödel's candy shop theorem means Reality will unfold into inexhaustible, never-ending ideal forms. How can that conclusion be true if it is also true that "infinity" has no "actual existence"? I don't see how those can be logically reconciled.
Eugene I wrote: ↑Tue Aug 17, 2021 11:15 pmI gave you a simple illustration from mathematics: a never-ending iterative algorithm of adding +1 actually exists and you can compute and experience every possible iteration of it. According to this algorithm, the sequence of natural numbers is inexhaustible and the execution of the iterative algorithm is never ending (because you cannot reach a number x(n) to which you can not add 1 and compute a larger number x(n)+1). But the wholeness of infinity of all natural numbers in incomputable and un-experienceable and therefore there is no experiential evidence that it actually exists, because "actual existence" means something that can be experienced (or computed in mathematics). This is not to say that the infinity can not exist as an abstraction (idea) - you can think of an infinity, but you cannot actually experience or imagine the whole actual infinity of all natural numbers at once. And I remember you saying elsewhere "if something can not be experienced, then it does not exist".AshvinP wrote: ↑Tue Aug 17, 2021 10:22 pm I am just referring to an apparent internal contradiction in your logic. You conclude that Gödel's candy shop theorem means Reality will unfold into inexhaustible, never-ending ideal forms. How can that conclusion be true if it is also true that "infinity" has no "actual existence"? I don't see how those can be logically reconciled.
The idea of actual infinity is akin to the idea of matter: you can have an idea of matter, but you can never actually experience it.
Ashvin wrote:Right, let me rephrase the question this way - will you or I, under the "candy shop" view, eventually get to experience all ideal forms that other perspectives have experienced?