Deep Boundary

WORK IN PROGRESS

## Uppedness

### Mundane Uppedness

Mundane one-uppedness, which is shortened to 1-up, is the most basic form of one-uppedness. It is a very weak form of one-uppedness so it is easily surpassed with more powerful methods.

Within a given field, 1-upping any given entry will create an entry that is minimally greater than that entry while still being reasonably meaningful and unarbitrary, where greater is defined as closer to the specific goal in mind, as long as the entry has not got some form of protection against being 1-upped.

Within the context of Hypergoogology, this refers to numbers of course, and on lower levels, like natural numbers, 1-upping something really is as simple as adding 1. Within a field like cooking, this would be somewhat subjective and it should be specified in which goal in mind the 1-upping is done, a reasonable goal would to have the best tastiness, and the smallest reasonable milestone could reasonably be specified to +0.05 points on a tastiness scale to 10. Let's start an analogy, this will be useful for explaining other types of one-uppedness even if it is fairly simple to understand here.

Imagine you are talking to an omnipotent, he guarantees you omnipotence as well, but you have to beat him at a game, he tells you that you must beat him in a contest to define, or name a well defined and formalised natural number. It isn't always necessary to comprehend a concept to 1-up it, as long as it can still exist uncollapsed to you, and it's not 1-up proof, you don't always need to understand a concept to 1-up it, but you might have to depending on the field. Within our example though, we are simply dealing with natural numbers and we have what's called a 1-up algorithm (which later can be extended to other kinds of one-uppedness). A 1-up algorithim in this case is quite literally just "+1", no matter what abominations of googology this omnipotent throws at us, since we're in a field where all entries can be 1-upped we can apply a 1-up algorithm to anything he throws at us (so far at least).

### Field Uppedness

#### Lower

Lower field one-uppedness, shortened to 2-up, is the 2nd level of one-uppedness, it exercises complete control over a field to assert your entry as the greatest. It is unsurpassable by using 1-upping, however it is not as simple as a 1-upping fixed point, which is far weaker than 2-upping.

Within a given field, 2-upping any given entry is to axiomatically, omnipotently or otherwise manipulate the given field within or outside that field to where your entry is seen as automatically greater than all other entries, where greater is defined as closer to the goal in mind.

Again, as said before, sometimes within more subjective fields, you may want to clarify the intended goal. Basically what 2-upping is, is like changing the judge's sense of taste at a cooking competition so that they prefer your dish over any other contestants' dish, you might do this within the field by adding some super secret chemical to your dish that changes their brains to change how they perceive taste, or outside the field by knocking them out and fiddling with their brains.

Ok, so at this stage in the analogy the omnipotent has grown tired of your constant 1-upping, so he uses his power to change the nature of natural numbers themselves so that he can place his number at the very end, here he has 2-upped you and no amount of 1-upping can beat him. Within HG, this would be directly altering HG-logic within an individual logic.

To 2-up within Hypergoogology, you need to exercise omnipotence as you cannot directly affect HG-logic regardless of any claims or cheats that are used.

#### Higher

Higher field uppedness, shortened to 3-up, is the 3rd level of uppedness. It changes field entirely to get closer to an intended goal. Again, this is not simply a 2-up fixed point.

Within a given intended goal, 3-upping any given entry is to change to another field than the one that the entry exists within to a field with greater potential to achieve the goal (eg. a field with a higher limit, or no limit), while staying as close as possible, or at least meaningfully and unarbitrarily close to the intended goal.

At this point, we surpass staying within a single field entirely, the goal with 3-upping is to move to a different field, while staying as close to the goal as possible. Imagine you were in a sandwich cooking competition, to 3-up the other contestants would be to move to making cookies, since within this hypothetical universe, cookies are always going to be way tastier than sandwiches. This is 3-upping because we are moving fields (sandwiches->cookies) while still staying as close as possible to the intended goal (tastiness).

In the analogy, you realise that you can't beat the omnipotent within natural numbers, because he has used his omnipotence to change natural numbers themselves, so instead you move to infinite ordinals, and you 3-up him by having your new number be ω. Even though you've changed the goal slightly from biggest natural number, you're still close enough because you've instead moved to biggest number. If you instead moved to best singing for example, that would no longer be meaningfully or unarbitrarily close to the original goal.

The strange thing about 3-upping is that you can beat 3-upping by 2-upping the new field, however 3-upping is more powerful because you don't even need to be omnipotent to do it in most cases. However within HG, you would have to be omnipotent to change to a new field that still has structure and objectivity that beats HG, because HG is entirely what we will create within MRL, or you could create a system in which all elements of structure and objectivity are removed (sHG).

### Transcendent Uppedness

#### Lower

Lower transcendent uppedness, shortened to 4-up, is the level of uppedness in which collapse starts becoming extremely useful. It rewrites its own definition to be stronger.

Within a loosely defined given intended goal, 4-upping any given entry or means of getting closer to the intended goal is to define 4-upping with some sense of omnipotence over the actual definition of 4-upping in our actual reality such that it can beat the entry or means of getting closer to the goal by a meaningful and unarbitrary amount.

Things are starting to get complex, if you don't understand collapse then you won't understand this. If you 4-up something, you change the literal definition that's written in turquoise to a more powerful concept of uppedness to beat something in some way. Yes, 4-upping is LITERALLY a concept that alters its own definition to be stronger, the only reason the turquoise text isn't changing is because the concept is collapsed, some black magic fuckery would have to be involved for 4-upping to actually be invoked in reality.

Within the analogy, the omnipotent actually and truly breaks out of the analogy to change how the analogy goes so that he beats you.

#### Higher

Higher transcendent uppedness, shortened to 5-up, is the level of uppedness in which things really start to ramp up, to understand this level you must have a good understanding of collapse and the previous forms of uppedness. This level of uppedness can transcend any conceptual barrier.

First we must define barriers of collapse, a barrier of collapse is a situation in which something must be collapsed to exist on a lower level. For example, 4-uppedness transcends the barrier of collapse that separates fiction/the abstract from reality, and there is a barrier of collapse between each thought level, there is a barrier of collapse between universal HG truth and individual truth, and so on.

Within a loosely defined given intended goal, 5-upping any given entry or means of getting closer to the intended goal is to transcend all barriers of collapse to get closer to a version of that intended goal that can also transcend all barriers of collapse.

Not even sure where to start with this one, first of all we notice that it surpasses the barrier of collapse between fiction/the abstract like 4-upping does, but it also would surpass the next barrier up (to rreality? thought level 4?), it also surpasses the barrier between false and absolutely true regardless of all other defintions, it would even surpass some barrier between transcendence and some other thing """"transcendent""""-ish that it would have to be collapsed to exist as transcendence, and all these properties that 5-upping has intertwine and boost eachother.

### Unbounded Uppedness

#### Lower

We are now getting close to \*-uppedness. Lower unbounded uppedness, shortened to 6-upping, is the point in which the very idea of our interpretation of 6-upping becomes a part of 6-upping itself.

First we have to talk about formality, and formality levels. We start at level 0,