Wikipedia:Reference desk/Archives/Mathematics/2023 August 4

= August 4 =

Infinite debt
This might be a really obvious question(s) but I might as well ask.

I was thinking of scenarios of infinite debt and was overall curious how it would work. For example, if you a had mortgage on your home for aleph null ¤ and you had to pay 1/30 of it every year for 30 years. I had one other question that might be in the wrong place but I may as well also ask it here.
 * 1) Wouldn't you have to pay it all in one go because a thitieth of aleph null would be aleph null in the same way all even numbers and natural numbers are equal to aleph null.
 * 2) Couldn't I give the bank an infinitely small fraction of infinity, pay off my debt, and still have my money.

How would the debt be passed on, wouldn't the entire world including the creditor be in debt? ✶Mitch 199811  ✶  02:53, 4 August 2023 (UTC)


 * Algebraic operations are generally only defined on restricted classes of numbers. Some can be generically defined as the solution of an equation. Subtraction can be defined by: $$a{-}b$$ is the unique solution for $$x$$ of the equation $$a=b+x.$$ Division can be defined by: $$a/b$$ is the unique solution for $$x$$ of the equation $$a=b\times x.$$ In the domain of the natural numbers, the equations $$1=2+x$$ and $$1=3\times x$$ have no solutions, so there the expressions $$1{-}2$$ and $$1/3$$ are undefined. For the first, one has to move to the integers in order to obtain a solution, and for the second to the rationals. In the latter domain we still have a problem with the equation $$1=0\times x,$$ so $$1/0$$ is also undefined. We see a different problem with assigning a meaning to $0/0$; here the problem is that while $$0=0\times x$$ is solvable, it does not have a unique solution.
 * Working in the domain of cardinals, the algebraic operations that are commonly defined are addition, multiplication and exponentiation. Since natural numbers are also cardinal numbers, $$30$$ is a cardinal number, and we can consider the meaning of $$\aleph_0/30$$ by considering the equation $$\aleph_0=30\times x.$$ As it is, we are lucky: it has the unique solution $$x=\aleph_0.$$ But now, what about the equation $$\aleph_0=\aleph_0+x$$? Any number $$x$$ such that $$x\le\aleph_0$$ solves the equation, so we cannot define $$\aleph_0{-}\aleph_0$$ any more than we can define $$0/0.$$
 * The conumdrum whether (1) or (2) applies stems from an unwarranted extension of familiar algebraic laws, such as $$p{\times}a-q{\times}a=(p{-}q)\times a,$$ to a domain where they fail to apply, making the meanings of the questions undefined. --Lambiam 08:55, 4 August 2023 (UTC)
 * "Owe your banker £1,000 and you are at his mercy; owe him £1 million and the position is reversed."John Maynard Keynes. Owe the bank aleph null, you probably own the Universe. -- Verbarson talkedits 14:36, 4 August 2023 (UTC)
 * Listen I was doing some crypto/stock market/[insert risky market] junk last night, I needed to make sure I can get out of this situation⸮ ✶Mitch  199811  ✶  15:29, 4 August 2023 (UTC)


 * So, maybe I'm mistaken here, but if Aleph-null is the infinite set of all natural numbers, all subsets non-empty infinite subsets of the natural numbers have a one-to-one correspondence with the natural numbers, so functionally all subsets of the natural numbers are also the same size as Aleph-null. Meaning that if you pay off 1/30 of your Aleph-null sized mortgage, you have functionally paid it all off; if I owe the bank Aleph-null dollar bills, I can just take a second Aleph-null set of dollar bills, pull out every thirtieth bill, and pay those to the bank.  My mortgage is now functionally paid off, right?  And I still have an infinite number of dollar bills.  -- Jayron 32 14:54, 4 August 2023 (UTC)
 * Ahem, the empty set is also a subset of the natural numbers. --Lambiam 15:12, 4 August 2023 (UTC)
 * So corrected. -- Jayron 32 16:26, 4 August 2023 (UTC)
 * There's a version of your problem a Ross–Littlewood paradox where at each step ten balls are added to a vase and one taken out. One possible answer is that after an infinite number of steps the vase is empty ;-) NadVolum (talk) 15:39, 4 August 2023 (UTC)