Comment by dredmorbius | Hacker News Reader

Comment by dredmorbius | Hacker News Reader

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The number of potentially awardable points is infinite. They literally come from the set of positive integers.

The potential awardable events --- baskets sunk in a game, multiplied by their value (1, 2, or 3 points per basket) --- is finite and bounded. But should there ever be a means to score more baskets, the points will be there.

You seem to be confusing awardable events (baskets) with points (numbers from the set of all positive integers). It's the latter which are infinite.

Put another way, even if the total number of baskets scored in all games for all time has some finite maximum, by changing the awarded points per basket the total points in a game could increase.

Rather than 1, 2, and 3 points we might award 2, 4, and 6.

Or 10, 20, and 30.

Or 1*10^100, 2*10^100, and 3*10^100.

There are always more points available to match to baskets and their value. There is no "point store" that can be depleted.

(I'm not arguing that awarding a googol points per free-throw is desirable or would improve the game. I'm saying it's possible to set the scoring to do this. Pinball games seem to follow a similar scoring logic, for comparison.)

There are a finite number of particles in the observable universe and each particle can store a finite amount of information. Therefore anything that requires information or anything that requires matter or energy will have a finite amount of availability. The point store as you put it is the observable universe.

Estimates are that there are 10^78 -- 10^82 atoms in the observable universe.

10^100 would be a number greater than either of these values. That's the named large number googol.

A googolplex is 10^10^100 ... that is, ten raised to the power of ten raised to the power of a googol.

There are notations for describing yet larger numbers.

One of these is Knuth's up-arrow notation, in which an upward-pointing arrow (↑) indicates iterated exponentiation.[1]

  2↑4 = 2 * (2 * (2 * 2)) = 2^4 = 2^16
  2↑↑4 = 2↑(2↑(2↑2) = 2^16 = 65,536

2↑↑↑4 is harder to describe, especially with the typographic limitations of HN.

It is 2↑↑(2↑↑(2↑↑2)), which is 2^2, itself raised to the power of 2, and so forth, 2^65,536 times. (The Wikipedia article represents this more clearly, though omitting ... some detail...)

The number is immensely large. It is immensely larger than the number of atoms, or subatomic particles (roughly 1 electron and 3 quarks per atom), or Planck lengths (~10^53), or Planck volumes (~10^53^3), in the observable universe. And yet it can be represented.

And it's hardly the largest representable number.[2]

The up-arrow notation itself can be more concisely represented as ↑^n, where n is the number of arrows: ↑^2 = ↑↑, ↑^3 = ↑↑↑, etc.

And all of these are simply representations of integer numbers. There are infinitely many integers, and given a need, a sufficiently concise set of expressions can be constructed to describe these.

Infinity is like, really big, man.[3]

There is no bound on points. There is no bound on monetary units which can be created. There is a bound on what those can represent, but that is a separate argument. Money != value. Money is a unit of measurement of value.

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Notes:

1. See generally: https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

2. See generally: https://en.wikipedia.org/wiki/Large_numbers

3. I won't even get to different sizes of infinities, though that's a fun exercise. See for example Cantor's Diagonal Argument: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

You can’t store more information than a fixed multiple of the number of atoms in the observable universe. The set of integers is infinite but there is a bound on the size of the set of integers we will ever be able to store.

That is a question of precision rather than magnitude, which is to say, a separate matter.

One could say each basket is worth aleph null points and therefore there are infinitely many points. But it wouldn’t be meaningful without precision. The Fed tomorrow could revalue the currency and make everyone millionaires but it wouldn’t be meaningful because the amount of money a person has is just a number. It’s the relative difference in amounts that’s meaningful. There is an upper bound to the amount of wealth a person can have. One can’t have more wealth than a fixed multiple of the energy of the observable universe.

In practical terms if too much money is printed then the currency quickly loses value. We can’t just print any number of dollars. At a given time there is a number beyond which the currency has no meaning, Your initial assumption about the economy is just plain wrong.

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