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

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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.

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.

I had one other question that might be in the wrong place but I may as well also ask it here.

How would the debt be passed on, wouldn't the entire world including the creditor be in debt? ✶Mitch199811✶ 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 $x\leq \aleph _{0}$ 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  ^talk_edits 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⸮ ✶Mitch199811✶ 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. --Jayron32 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. --Jayron32 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)