Apparently it’s undecideable in ZFC whether ω_1→(ω_1,ω+2).

That, uh, seems like serious evidence against my claim that ordinals are on pretty solid groud, huh?

]]>In more detail, you of course need to define the oracle access mechanism in a suitable way, so that (for example) the integer k such that you want to call the level-k BB oracle gets written on an oracle tape. But if you do, then as usual, there are only finitely many n-state machines, so among those that halt, there’s one that runs for a maximal number of steps.

]]>It seems the ability to call any level-k BB (with k a positive integer) could plausibly allow us to create a family of n state halting Turing machines without an upper bound on the number of steps before halting. The finite set argument does not hold if we have infinitely many different operations (oracles) to choose from. ]]>

On a different note, when you say “That essay might still get more views than any of the research I’ve done in all the years since!”, it makes me wonder what do you think your legacy will be? I mean, what is the most important thing (by whatever measure) you have done in your career as a scientist? It doesn’t have to be a result or anything like that, I mean it more in the “something for the greater good of the humanity” sense.

]]>“If you wanted only three orders of infinity, then you’d certainly need to give up on the powerset axiom, which (together with the axiom of infinity) gives you infinitely many orders of infinity. But why 3? Why not 17?”

Scott, I will now state the axioms of my ‘Reality Theory’. My lemmas will finally provide the answer to life, the universe and everything 😉

*Axiom of Knowledge-Existence Equivalence:

Reality and knowledge are equivalent at the fundamental level! Whilst it’s *usually* true that the map and the territory are separate, this distinction breaks down at the fundamental level, and knowledge and existence become equivalent!

*Axiom of Ultra-Finite Recursion:

There is absolutely nothing to be gained from a logical language that has more than 3 levels of recursion. That is to say, full logical power is reached at the 3rd level of recursion, and additional levels are redundant because they can always be reduced to (collapse to) the 3rd level.

This is generalization of Yudkowsky’s law of ultra-finite recursion:

https://wiki.lesswrong.com/wiki/Yudkowsky%27s_Law_of_Ultrafinite_Recursion

Yudkowsky’s Law of Ultrafinite Recursion states that “in practice, infinite recursions are at most three levels deep.”

My generalization is to take out the phrase ‘in practice’ and replace with ‘in principle’.

*Axiom of Self-Similarity of Knowledge

The structure of knowledge is a fractal. That is to say, there’s a self-similarity property to the structure of knowledge such that we can find a method of knowledge representation that applies across *all* levels of abstraction

The Geddes axioms are sufficient to construct a theory of everything.

Lemma 1: Reality is a *language*

From Axiom of Existence-Knowledge Equivalence:

All of reality is a language that is reflecting upon itself.

Lemma 2: From (1) and Axiom of Self-Similarity

The language of reality is *recursive*. The universe is a fractal reflecting the structure of knowledge

Lemma 3: From (1) and (2) and Axiom of Ultra-Finite Recursion

The fundamental language of reality (call it L1) undergoes recursion and splits into 3 levels, representing 3 fundamental levels of abstraction.

But this logical system as a whole represents a new language (call it L2) and *that* undergoes recursion, again splitting into 3 levels of abstraction.

The new logical system resulting from the previous steps (L3) does not recurse.

We are left with 3^3 = 27 fundamental levels of abstraction.

Result:

There are 27 core knowledge domains that ‘cleanly’ (precisely) categorize all knowledge. These are based on 27 a-priori categories/sets/types that are necessary and sufficient for a mind to represent any aspect of reality.

The 27 core knowledge domains are shown here:

http://www.zarzuelazen.com/CoreKnowledgeDomains2.html

The domains are arranged according to level of abstraction. So as you look at the knowledge domains on the main page, you can see the whole fractal structure of knowledge across all the levels of abstraction – top-to-bottom and left-to-right. There are only 2 dimensions on the screen – to indicate the 3rd dimension of abstraction I split the domains into groups of 3 using white space.

Clicking on links takes you through to my wikibooks with A-Z lists of wikipedia articles explaining the central concepts for each domain (on average, each domain has about 100 core concepts).

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The key conjecture I want to get across here is that these 27 categories are NOT inventions. They are *fundamental* a-prior categories of existence itself! They *cleanly* (with 100% precision) carve reality at the joints. This follows from Axiom of Knowledge-Existence Equivalence. The map-territory distinction breaks!

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