vidi-plates
 


kruel

"officially the largest number in the universe."

This one all began during my final year of OU studies. There was a comment in the course-work along the lines ~ the set of all numbers is infinite. This line was literally in the first page of material for the year. It probably sounds reasonable and on another day I might have just read it and accepted it, but for some reason it just struck me as an odd statement of fact. Is it really an objective fact?

I eventually decided that if there were an infinite number of numbers, because we can always add one, then there must also be an infinite number of hats, because we can always create one more hat. Physical hats seem a lot different to numbers but I'm also including any number of hats we can imagine, which opens the door to a lot of hats. Just imagine a bowler then keep adding an inch to its height. Even with all the hats we can make, with all the carbon in the universe, combined with all the hats we can imagine with our synapses, it's finite. It has to be. When the universe is over there will be a value for the total number of hats that existed in it. Numbers are no different.

This may seem spurious, and I'll push the view a little bit further in the below ramblings, but it makes sense to me. Incrementing something indefinitely does not make it infinite. So because I'm certain there aren't an infinite set of numbers it means there must be a cogent concept of the current largest number. If I could create that then I could maybe demonstrate an objection to infinite. That's kruel. The largest number in the universe - and it's getting bigger every second.

- created in 2014



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 > what is kruel? < 

  • kruel consists of three auto-incrementing digits which express a numeric value.
  • the first kruel number expressed the numeric value of 'Graham's number', which is a number so large it cannot be represented, using standard number systems, in the observable universe.
  • kruel numbers are incremented using the kruel operator.
    • the kruel operator multiplies a kruel number with a kruel number any number of times between 2 and the present largest numeric value of kruel.
    • the kruel operator has been operating on kruel since the first kruel number sometime between 2011 and 2015.
  • isn't kruel + 5 larger than the present kruel number? kruel is an explicit-number-system which means standard operations are not permitted on kruel when the resulting value hasn't already been expressed by a previous kruel - and as the kruel operator increments kruel by random intervals it's not actually possible to apply any operator to kruel other than the kruel operator.
  • what if someone created a new number system which is like kruel but adds an extra 2 every millisecond? kruel is an omni-aware-number-system which means kruel becomes automatically as large as any other value as expressed in any other number system, if larger - then instantly applies the kruel operator.
  • surely these kruel digits are not unique and kruel will, at some point, display three digits it has already displayed? kruel is a context-dependant-number-system which means the numeric value of each a kruel number is bound to any amount of preceding kruel numbers i.e. if the denary-number-system was a context-dependant-number-system then the number 3 in the following sequences [1, 2, 3] and [1, 5, 3] would express different values.

 > more infinite problems < 

Aside from the similarities of hats and numbers with the former most certainly not being infinite I'd like to introduce two more reasons to be skeptical of infinite numbers.

Zeno's paradox is interesting and can, I think, be re-phrased as 3 questions:

[Q1] How long would it take to travel through an infinite amount of points?
[Q2] How many times can you divide a distance A to B by two to create a new half-way point?
[Q3] How long would it take to travel through each half-way point between A and B?

Obviously we know the third answer is not an infinite amount so the solution is to say...

[A2] an infinite amount of times.

...is wrong. Space cannot be divided indefinitely. This solution is well known and naturally makes sense but I think it beautifully illustrates a damaging disconect between numbers, the real world and the usefulness of infinite. Dividing a distance by two is not the same as dividing space by two.

In my phrasing of [Q2] I think the answer is actually correct, you can divide a distance by two for as long as the universe is around, because distance is just a number we have decided expresses a measurement in space, and numbers can be divided by 2 forever. Seems sound but the number is an abstraction. It isn't real. If I have a two foot piece of wood and cut it exactly in the middle, I haven't got two pieces of wood 1 foot in length each, because I never had a two foot piece of wood in the first place. I had a piece of material comprising of x number of atoms which I gave the labels 'wood' and '2 foot', and I now have 2 pieces of material comprising of x number of atoms.

It is because our model of the world expects numbers, and their behaviour, to bind exactly to properties in the real world that Zeno's paradox can trip us up. I think this should make us be mindful of the differences between manipulating numbers and interactions in the real world.

The other reason:

A zero mass universe is what many consider to be the state of the universe moments prior to the big bang. It is obviously well known that as the volume of an object decreases its density increases. This can be visualised by imagining people on islands. With a mass of one hundred people on an island the size of Australia the population isn't very dense, but the same mass of people on an island the size of a stamp would have an incredibly dense population.

The equation for density is mass divided by volume. So in the above example we could say:

one hundred people / size of Australia (km2) = 100 / 8,000 = 0.0125
one hundred people / size of stamp (km2) = 100 / 0.00002 = 5,000,000

Which is super straight forward, but what happens when the size of the island is zero with one hundred people:

one hundred people / no island = 100 / 0 =

So the maths tells us the zero sized island must have infinite density, and obviously this is the mathematical outcome no matter how many people there are. How can that be? How can one hundred people have the same density of one million people in an area of zero volume? When the island has zero volume the reality is people get wet. Again I think this shows a disconnect between numbers and the real world. So when maths describes an infinitely dense universe what is really happening here is maths is breaking down and infinite becomes the escape.

It's weird and I think we should be skeptical and question all statements which rely on infinite.

 > conclusion < 

So having said all that, I do accept I know nothing about maths and this will make those that do sick, but for those who know as little as I do I hope this sows some seeds of skeptisism towards infinite.

Finally I'd like to add I don't think nothing could ever be inifinite, I just think nothing can be infinite that didn't already start off infinite.



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