Different infinities?

[Ripheus23]

I was puzzled to read in different Cosmere texts that the god-level beings could each show people an "infinite" amount of power, yet some of them were apparently stronger than others AND all were under Adonalsium, in principle. Yet I wonder...

... I wonder about my memory

So, for years and years now I've been obsessed with "Cantor's paradise." It's the belief, among many (most?) mathematicians, that there are actually an infinite number of different kinds of infinities. For example, the rational numbers are called "aleph-zero," the irrationals/reals are (believed to be...) "aleph-one," and so on. However, the number-suffix can be an entire aleph-number, too, so there's also "aleph-aleph-zero," "aleph-aleph-one," "aleph-aleph-aleph-zero," "aleph-aleph-aleph one," and so on and on, forever and ever, amen. (They're usually written out with subscripts, so they look more... manageable, haha!) So, for example, there's an aleph-aleph-aleph-aleph-aleph-aleph-42.

Also there are other dimensions of these numbers, that I have next to no idea how to write out. I think they're called "inaccessible cardinals," among other things. I'm guessing (purely guessing) that they're called "inaccessible" because they can't be expressed in notation that involves only changing the number-suffix/subscript for that original aleph-series members. But I honestly don't know.

Anyway, let's suppose the power-level per Shard, so to speak, can be measured in aleph-numbers. Since the first such number is indexed zero, let's say that Adonalsium was equal to aleph-16 and the Shards each match up to one of the 16 aleph-numbers from zero to fifteen. (So interestingly enough, Adonalsium would itself be the Seventeenth Shard, in a way?)

OR Adonalsium was aleph-aleph-...-aleph [16 aleph-steps] -something (10? there's got to be a reason people keep messing up, like on Scadrial or Roshard, and thinking there are only 10 of the things that there should be more of, like the metals?). Let's say you could hold multiple Shards and "sort of" aggregate their individual power levels, but if you go the Harmony route (total speculation!!!) you compound their aleph-dimensionality, so Harmony is not just the aleph-number for Ruin + the aleph-number for Preservation, but is, for lack of a better way to put it, (R+P)^2.

QED 

[Weltall]

1 hour ago, Ripheus23 said:

I was puzzled to read in different Cosmere texts that the god-level beings could each show people an "infinite" amount of power, yet some of them were apparently stronger than others AND all were under Adonalsium, in principle. Yet I wonder...

If ir helps un-break your brain trying to contemplate infinities, here's a handy Word of Brandon on the subject.

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Questioner [PENDING REVIEW]

For Adonalsium to create the universe, therefore he must have infinite power to create an infinitely sized universe. Therefore, infinity divided by sixteen is equal to infinity. Therefore, why don't the Shards have infinite power, which they clearly don't, because they can be killed?

Brandon Sanderson [PENDING REVIEW]

The power can't be killed. The entity controlling the power can. Infinite power existing and being able to access the infinite power are different things, and a finite mind, even added to a very powerful sense of power, isn't necessarily able to tap all of that.

Questioner [PENDING REVIEW]

What about Ruin and Preservation in Well of Ascension? We hear about Ruin using some of its power. Therefore, it must not have infinite power, because if you minus something from infinity, it's still infinity...

Brandon Sanderson [PENDING REVIEW]

So, infinite power is changing forms. It's not going anywhere, right? So, the Investiture, the power, is becoming energy, which is doing work, which is being released back into the system. Nothing's growing or shrinking. It is simply changing forms, and potential energy is becoming kinetic.

source

So, the Shards may have effectively infinite power but the Vessels that hold that power were originally finite and they can't actually comprehend and use all of their power at once. Think of it as each Shard having a 'tank' of power that the Vessel is able to recognize and access. Some of that power can be cycled into the world and thus 'lost' to the Shard at least for a time until they retrieve it or it returns to them naturally. Preservation and Ruin for example put some of their power into the world to create Scadrial but Preservation put a little more in than Ruin, thus becoming weaker in a general sense. Similarly, on a day to day basis Preservation loses some (really an infinitesimally small amount) of his power whenever someone uses allomancy, as that draws directly on Preservation's Investiture. However, once done the Investiture cycles back to Preservation so there's no net loss of power.

Shards can also increase their power (and the size of their 'tank') by essentially exploring and finding Investiture in the Cosmere that is 'theirs'. Brandon has a very long WoB discussing this, particularly in the context of what Autonomy is doing. In other words, Bavadin is expanding her awareness of her infinite power and thus is able to use more of it than she started with.

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Anyway, let's suppose the power-level per Shard, so to speak, can be measured in aleph-numbers. Since the first such number is indexed zero, let's say that Adonalsium was equal to aleph-16 and the Shards each match up to one of the 16 aleph-numbers from zero to fifteen. (So interestingly enough, Adonalsium would itself be the Seventeenth Shard, in a way?)

Not every shard is associated with a number.

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The Only Joe (paraphrased)

Do all shards have a number they're associated with?

Brandon Sanderson (paraphrased)

Some do, (most/some) don't.

source

 

[Ripheus23]

Aleph-numbers aren't like ordinary numbers, in many ways. That is, aleph-16 isn't some aleph-ness multiplied by 16. The context of Sanderson's remark is numerological, in that some Shards are numerologically associated with specific numbers. But all the Shards are associated with infinity, which is a number, or type of number (in the aleph-context).

OTOH your explanation/citation is easier to follow than the kind of algebra I was thinking of...

[ZenBossanova]

If the central mystery of Adonalsium, Hoid or the Cosmere hinges on the Continuum Hypothesis, that is going to be the ultimate RAFO to end all RAFOs. 

[The One Who Connects]

5 hours ago, Ripheus23 said:

I was puzzled to read in different Cosmere texts that the god-level beings could each show people an "infinite" amount of power, yet some of them were apparently stronger than others AND all were under Adonalsium, in principle.

In Allomancy, metals burn at a set rate. You could have infinite metal, but you can't burn all of it at once, nor can your stomach hold all of that metal at once. That's what's hitting the Shards. To continue the Allomantic Example:

---

Power tied up in something(invested in the people/planet/shardpool/etc..) functionally reduces your burn rate, because it is currently being "used." That's why Shards become "weaker" when they invest into a planet. They can't use as much of their power at once, regardless of how much they have.

Picking up another Shard seems to increases the size of your stomach, which is one reason that Harmony is more powerful than Odium. In a straight shoving match, the one with more power wins because their opponent runs out of power first(much like two people burning Atium). We've seen this before, ending with Ruin and Preservation both dead because they were too equal in power. Odium doesn't fight on equal footing though, which is part of why he's still around.

Disclaimer: This is all speculation based on what we know, and what we think we know.