Cardinality issues block and confuse discussions of NKS and the significance of universality.

Some QM fans stump for more than Turing computation as physically real based on the notion that

access to continuum cardinalities can break the limited fetters of countable computables.

Others stick to Turing computables but make more than practical use of its own countable

infinity and span more than anyone will ever span, and pretend things are the same or have

been reduced to one another, sub species infinitum. We even see men educated in computational

complexity theory speak as though anything exponential corresponds to uncountables while

anything polynomial corresponds to countably infinities. Against all these loose associations,

it is necessary to insist forcefully that finite does not mean small. Even finite computation

exceeds all realizable grasp. Against cosmologists dreaming of towers of continuum

infinities and microscopists strident for infinitessimal distinction, both as the supposed

origin of limitations on knowledge or uncertainty in the external world, I must insist that

even denying all such claims a purely finite, discrete, and computable universe has nothing

simple about it. The operative cause of a limit on exhaustive knowledge is not the

hypothetical presence of infinite cardinals of any description, but follows simply and directly

from the term “universe”, and our existing minimal knowledge of its scope and complexity, and

is there already even if every such infinity is denied. People need to give up the equation

between “finite” and “simple”; it is a mere mistake. And if this is established it is

already enough to show that no appeal to experienced limits of knowledge can count as evidence

of any kind, for a real existence for any such hypothetical infinities.

I present a mesh to cover the practical universe and allow for its possible laws and regularities. The

smallest spatial distinction we think can matter for physical behaviors is the planck length.

But I allow for underlying generators below that scale, 100 orders of magnitude smaller. We

experience 3 large scale spatial dimensions, but some theories employ more; I’ll allow for

10,000. We differentiate a zoo of a few score particles, themselves understood as quanta of

fields, on that space. I’ll allow a million, no need to skimp. The smallest unit of time

we think likely to have physical meaning is the planck time, but that’s too long. Slice the

time domain in a manner unique at every single location as specified above, into time units

100 orders of magnitude shorter than the planck time. We can see tens of billions of light

years in any direction, but extend this outward 1000 fold, and allow for every location from

which a light signal might enter a forward position on our light cone 800,000 billion years

in the future, as the spatial extent we care about. Now let us consider every possible field

value for every possible hypothetical field quanta at each such location – quanta to the power

of the number of locations. Those are states. Now let us consider their transitions one

after another, not as compressed by some definite law, but the pure power set, any state can

go to any other state, as a purely formal and one-off transition rule. Allow this transition

to be multivalued and indeterministic, such that the same exact prior state can go to

literally any other distinct subsequent, or otherwise put, multiple by the number of time

instances at each location as though they are all independent. Any regularity actually seen

is a strict compression on this possibility space. Now add elaborate running commentary on

all events as they happen, in 3000 billion billion languages, surrounding the physical text.

All independent and voluminous, billions of times larger than all human thought to date, about

each infinitessimal instance. Don’t worry about where or how the last is instantiated, let

it float above reality in a platonic mathematical realm.

I am still measure zero in the integers. I can fit pure occasionalism in that mesh. I can

fit any degree of apparent indeterminism you can imagine. I can fit all possible physical

theories, true approximate or completely false. But it is all discrete and computable and

moreover, finite. All the realized computations of all the physically realized intelligences

in the history of the slice of the universe observable by us and all of our descendents or

successors for hundred of thousands of billions of years, along with all their aides or

computational devices, cannot begin to span that possibility space – but it is strictly

finite. So, what is it I am supposed to detect operationally, that I can’t fit into a theory

within that mesh, or above it? Notice, I didn’t even posit determinism let alone locality or

the truth of any given theory. It is enough if I can characterize a state by millions of values

at each of an astronomical number of locations. If supposedly I can’t, then no operational

theory is possible period. If any operational theory is possible, it will be strictly less

fine or exhaustive than the thought-experiment mesh given, and strictly more determined or

restrictive, as to transitions that actually occur according to that theory. I further note

that the mesh given is already completely intractable computationally, not because it is

formally noncomputable or has halting problems, let alone because of higher order infinities.

No, it is intractable already, for all finite intelligences and anything they will ever know,

without a single countable infinity. Naturally this does not preclude the possibility of

tractable, even more finite models. But it does show that intractability arises for reasons

of pure scale, within finitude.

Finite simply doesn’t mean small, nor simple.