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Youwen Wu 2024-10-19 12:43:32 -07:00
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@ -214,14 +214,14 @@ usual method. We have $0.333 times infinity + 0.333 times 1 + 0.333 times
infinity is not a number in the traditional sense. It makes no sense to add or infinity is not a number in the traditional sense. It makes no sense to add or
subtract infinite values. For instance, try and subtract the total amount of subtract infinite values. For instance, try and subtract the total amount of
integers ($infinity$) from the total amount of real numbers (also $infinity$) integers ($infinity$) from the total amount of real numbers (also $infinity$)
#footnote[Famously, the infinity of $RR$ is "larger" than the infinity of $ZZ$ #footnote[Minor digression: famously, the infinity of $RR$ is "larger" than
in the sense of cardinality, where $frak(c) > aleph_0$ (G. Cantor). However, the infinity of $ZZ$ in the sense of cardinality, where $frak(c) > aleph_0$ (G.
our familiar algebraic operations of $+$ and $-$ are still not defined on them. Cantor). However, our familiar algebraic operations of $+$ and $-$ are still
Perhaps we could pursue a line of reasoning to rigorously define algebra with not defined on them. Perhaps we could pursue a line of reasoning to rigorously
infinity using the hyperreals $attach(RR, tl: *)$, but that is out of the scope define algebra with infinity using the hyperreals $attach(RR, tl: *)$, but that
of this paper.]. Clearly, this notion is meaningless and we cannot obtain a is out of the scope of this paper.]. Clearly, this notion is meaningless and we
solution. So, we consider $infinity - infinity$ an _indeterminate form_. So, cannot obtain a solution. So, we consider $infinity - infinity$ an
the expected utility is now _undefined_. _indeterminate form_. So, the expected utility is now _undefined_.
Consider the following Indeterminate Utilities argument: Consider the following Indeterminate Utilities argument: