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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 $0.333 times infinity + 0.333 times 1 + 0.333 times
infinity is not a number in the traditional sense. It makes no sense to add or
subtract infinite values. For instance, try and subtract the total amount of
integers ($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: