From d68e3e12edfcbc4e8cd528b498b0102ed10342de Mon Sep 17 00:00:00 2001 From: Youwen Wu Date: Sat, 19 Oct 2024 12:43:32 -0700 Subject: [PATCH] update footnote --- work/2024/phil-1/paper-1/main.typ | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/work/2024/phil-1/paper-1/main.typ b/work/2024/phil-1/paper-1/main.typ index 00fae1b..a5dc1cd 100644 --- a/work/2024/phil-1/paper-1/main.typ +++ b/work/2024/phil-1/paper-1/main.typ @@ -214,14 +214,14 @@ usual method. We have $0.333 times infinity + 0.333 times 1 + 0.333 times infinity is not a number in the traditional sense. It makes no sense to add or subtract infinite values. For instance, try and subtract the total amount of integers ($infinity$) from the total amount of real numbers (also $infinity$) -#footnote[Famously, the infinity of $RR$ is "larger" than the infinity of $ZZ$ -in the sense of cardinality, where $frak(c) > aleph_0$ (G. Cantor). However, -our familiar algebraic operations of $+$ and $-$ are still not defined on them. -Perhaps we could pursue a line of reasoning to rigorously define algebra with -infinity using the hyperreals $attach(RR, tl: *)$, but that is out of the scope -of this paper.]. Clearly, this notion is meaningless and we cannot obtain a -solution. So, we consider $infinity - infinity$ an _indeterminate form_. So, -the expected utility is now _undefined_. +#footnote[Minor digression: famously, the infinity of $RR$ is "larger" than +the infinity of $ZZ$ in the sense of cardinality, where $frak(c) > aleph_0$ (G. +Cantor). However, our familiar algebraic operations of $+$ and $-$ are still +not defined on them. Perhaps we could pursue a line of reasoning to rigorously +define algebra with infinity using the hyperreals $attach(RR, tl: *)$, but that +is out of the scope of this paper.]. Clearly, this notion is meaningless and we +cannot obtain a solution. So, we consider $infinity - infinity$ an +_indeterminate form_. So, the expected utility is now _undefined_. Consider the following Indeterminate Utilities argument: