From e596fe72d807a3db1014de2a924aa9ff89d04faf 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.pdf | Bin 454582 -> 454603 bytes work/2024/phil-1/paper-1/main.typ | 16 ++++++++-------- 2 files changed, 8 insertions(+), 8 deletions(-) diff --git a/work/2024/phil-1/paper-1/main.pdf b/work/2024/phil-1/paper-1/main.pdf index fa245526251b7e4776b56b2b1ce0fcbf26d6b452..7198f4eab8f8af39e687b8be08fa20ff438b7663 100644 GIT binary patch delta 4787 zcmZ{gbyU>dy2qu4E*V5}2x$~%7@QeGN_rTO5Cl+FzTx23D{-O69$!q9gdCbL7Una^$E1dWVLx8I70I0! ztZF>R^z1;cPD@v&Jjn^(C)-olB(2G{n$gHG+lv>#?*101eKX)$X6#~r%{9xkqn_d2 zcJ}CF#3qEU`O;CDBKEifRrOtG<~GejnKzG3pU`r)s{-T)wY+2UI^Z&G_(N+G`V{1L z^W8I|G^CqK_ndd!$kX&QuG8iR_pb+(t!Q66CEOcNdYP;to6{`0AU=3;I6XrRtP1)2 z|JJSqnvqDe92lX$ssh^*Se;4t{sZ^@?*oc;@ivZPrj8nc&?t&YpF~- ze5YD7tY;=+&9r|rC$XSPw3+qn4xi`+W0tnJmN35r{k(APV}7|kes&wl@}?J45yHQT zX|$xXJp>=i^MdgkjA~QQNxJoZeXD&1tThH`8*C9lSthG9NjudI?z-@8sqeQLdUxE` z;EK=5XD|4vRk~>^MJkySsBnKIvd9rMm5NSKF>h)}ph+GJf=>I2x`RvnTKo|6CLYgP zdE!+Ag^KNwE~rC3`H8lxGS1FKhSC-p{99_D=3TB6*h3HF53PtIWJ;$YgBcT`u7&5l z7AbTkCV!<^e5EtSHq)y}AZAQ^2M&Su@XQlI6CJBChii2g}4n$#+-`nAdu zGJr#gEtPe}bbE08e(4u>O*Ai)E429I1)FPvhoo)1#c^-dd&vp=6PXQuNHkbxQ>%4e zO#J=Wg%>iKznPFz^Rxe?PAhvb~vj9Ys_m&0d|AbkpX_x0)x$ClJ%YkXA4 z&xdSN-s#$?fXc6|`G|}_(0j&SeCDo1r3Bum6>EEbMo+b?al+Cx{X5ctvbJK7x8DRG zNY}G9QgmV+rbs;JP8M z1%xQcT9>d5sbXf0>ECsnb%TtKQe|7 zR2xU_LnFCXnYtMU+#r8j-o8S5Z=b4s7hCt0nOjXMfn~WXNwH3`T6Z_WNC$Cg6&>pH z6sQ&J+oAv)z6IVYr5ZaY7#{bgc>1S`Sl+5C@aYb74AMN5INQQpt zsag1Q*t{2IbyddvzIB|xKXFuJo9sUQD|5=(#QvXHLs zgU$ky>Ly35r2MRxWtgD4@L`B2LNWLRFM7&*Hsi3j^kj6U# z)-ffb4A7D%WIp>Y%m^MtH6pmAg+@2Wnn77~s#}(TztMi4F!%L??|5vU3PPD$=fOhg6V5n_Tn^%t$@h1lvoLisyY8GrXVL1wBLT@_8rg8n&@p$6(0>2Lx0{!i z)(jDD`rg9r+Oo=njzZRu-vk;q>`l^GOniC@P~|r!#~8rZ?k|RSP)7U`xnP~P+X`mLx#f4KFK-#3Gd)zE8n&Pt zfHGe#XV#t>9F{KCsGM6RGH*K|INyxnwY7PJ^>rvmzb0BSl(+OiNA$CE@)5awF$~H8 z7NQ#y9O33Au2(#c)i!I?}?z@_)OMPo+_34oJLr=l`wDEiI*|O8zCD`4`o@#bVTY6sBphf5dwQD< zE-vMSu|x=q@os={aK+6<4e@(xZQ=y-we}JPFsHajzv^?7K$&tdj(_mO=lN3ry9W3T zOImE=X)++RFiK6U7ybC*%njkNxcOdcBje=r(-7idbwCnfzH-gH@7?Oz{jVP%N*EO8 zjI-P@RB-ci`Q^5to|QkK9cux37ccP{iuq!i!&7>5EqMF83e+Hg)0(`8O}F{{V^7aI zmv*yds8MriH^hj@-Hd5}xQc1*spL5;pMRufLa(e#b zpg9uIOq6!%Smcf|L$ay-#AquY;ECH9w~2ii@1hRa7PNymy~Sh0yk97C+pB|p4ig}D zUG2E(W}ITfV=p^*8(xY-vozy}Ujy2wf4Zr59dwEM2~@V(KMQ|c@#YjDw7RM%td7q8 z{HWcTa{WWR;5^AK*I-l~;YE7fZ{qu;B_X97ZgnMn7F0Q#Qy8K>irQ0@V51Uk!d_xU z2|EGt>uxB2Uh6`PxMR{DE)*=T!$l}&xyg!{cLzK_X>8xgWnh-qnFK>kH3q8v>G8I+ zIGA1LK@EIfQa+!3a0JNqxZETQHRLjXQOS}Y-a0y~w6|;fNgQ(Jl&u+l$aI6^@ z;v+~VbH!zC3bi*e%FWhl+~2N{deNv>@Lt;NxjT&}>`hipTL$1*Mn1%hzr$gZ($u9R~tF64qkzLAdu*V+d>0c#||^>*0?M?<|gg zj!N0^yFJSN#sQtucPm&Qy1J-CeO&D@53^8WLBF#S2eX2lXOc1B_I*fRZd$%v8k%j3 zwDVT2$mL7|V=fm+W7N--hF_T*_$3)eMZ4x>mlgLhVL-*cZQQz57;R#TCXqZH7cVtF z`{l~E5dI=!4EA<-*pN}c6koyAs=|p{yR3(dm8(%|WZ_O3jO-#=MqdBW`&6PoHK67; 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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: