From 575c0942f7236385a853c844b9bcf9673be5e30f Mon Sep 17 00:00:00 2001 From: Zentrik Date: Mon, 20 May 2024 10:31:20 +0100 Subject: [PATCH] LST: Fix typos --- LogicAndSetTheory/05_set_theory.tex | 10 +++++++--- LogicAndSetTheory/logicandsettheory.pdf | Bin 324938 -> 326122 bytes 2 files changed, 7 insertions(+), 3 deletions(-) diff --git a/LogicAndSetTheory/05_set_theory.tex b/LogicAndSetTheory/05_set_theory.tex index 5e014d8..3069c10 100644 --- a/LogicAndSetTheory/05_set_theory.tex +++ b/LogicAndSetTheory/05_set_theory.tex @@ -14,7 +14,7 @@ \subsection{Axioms of \texorpdfstring{$\mathsf{ZF}$}{ZF}} We now define the axioms (there are 2 + 4 + 3 axioms) of $\mathsf{ZF}$ set theory. \begin{enumerate} - \item \vocab{Axiom of Existensionality (Ext)} \\ + \item \vocab{Axiom of Extensionality (Ext)} \\ `If two sets have the same members, then they are equal' \begin{align*} (\forall x)(\forall y)((\forall z)(z \in x \Leftrightarrow z \in y) \implies x = y) @@ -46,6 +46,10 @@ \subsection{Axioms of \texorpdfstring{$\mathsf{ZF}$}{ZF}} \begin{example} For instance, $p(\varnothing)$ is the sentence $(\exists x)((\forall y)(\neg y \in x) \wedge p(x))$. \end{example} + + Strictly speaking, this axiom is not needed as it follows from (Sep). + Indeed, in a structure $V$, we can pick any set $x$ and form the set $\{y \in x : \neg (y = y)\}$ by (Sep). + However, if in first-order logic we allow the empty set as a structure, then (Emp) is needed (or some axiom asserting the existence of some set). \item \vocab{Pair-set Axiom (Pair)} \\ `We can form unordered pairs' \begin{align*} @@ -148,7 +152,7 @@ \subsection{Axioms of \texorpdfstring{$\mathsf{ZF}$}{ZF}} We can define abbreviations: \\ ``$x$ is finite'' for $(\exists y)(y \in \omega \wedge (\exists f)(f : x \to y \wedge \text{`$f$ is bijective'}))$; \\ - ``$x$ is countable'' for $(\exists f)(f : x \to \omega \wedge \text{`$f$ in injective'})$. + ``$x$ is countable'' for $(\exists f)(f : x \to \omega \wedge \text{`$f$ is injective'})$. \item \vocab{Axiom of Replacement (Rep)} \\ (Inf) says that there exist sets containing $0, 1, 2, 3, \dots$ @@ -171,7 +175,7 @@ \subsection{Axioms of \texorpdfstring{$\mathsf{ZF}$}{ZF}} \begin{example} ~\vspace*{-1.5\baselineskip} \begin{itemize} \item For instance, $V$ is a class, taking $p$ to be $x = x$. - \item The set of sets of size $1$ in a class, e.g. take $p$ to be $(\exists y)(x = \{y\})$ + \item The set of sets of size $1$ is a class, e.g. take $p$ to be $(\exists y)(x = \{y\})$ \item There is a class of infinite sets, taking $p$ to be `$x$ is not finite'. \item For any $t \in V$, the collection of $x$ with $t \in x$ is a class; here, $t$ is a parameter to the class. \item Every set $y \in V$ is a class by setting $p$ to be $x \in y$. diff --git a/LogicAndSetTheory/logicandsettheory.pdf b/LogicAndSetTheory/logicandsettheory.pdf index 06fb6f9c6527a453c83ff81bbf78177c7163e6ac..0cee45b602aef9d0c63d0564bff0fd526237630d 100644 GIT binary patch delta 107250 zcmV)0K+eC)>J#eo6R^Y+e>grq3UhRFWnpa!c-rlmJ8#@L5Xbv`3YAG=hBKTYMIcZS zAP$hL4%jNe6^qz=MUW!jzsc|=iPYNJhxTmX1Va!_@iAKd_@5c7`_)0G-+sX0(?}nqh2(cJ>7G(}&z^d^Y^E`_g^djn*!$0_zmY zE7oS|!*C}9FG}S7Gf)3#r9b|*B)_S4$Ceyi+xvWuj8)dVmLsk>;(bF2{U( z%F<%l!C%i3YlUQ6f0pdX5^^R%?rdw0LTfq{mN)>&QXrqYgvV@(q@4f}dj-00Y}$ai z5l;2W%NhY(kCQdZd2VrXwKVCB(jj3L?}3-_j85PLdIjD$lWm;58nS8#0$F%HSXXe!ns=HO}H zMplnA@=`E=LS!Xnw5?GaMmI;Rp}@x2y6JurV^FQ9WLr|Y1MH;^^5{mKp^s=;Q79t- zSrIW~U8oUqf6O7(G7KiRl6P@WTC<#$5R~s}IjvmgqL(_paR?PL%zm=Cad778}4>Ue}Hf+MFQ3OjmX!-W=)Y>=i>%=t1P*e&FMs1!rbVTBVZ5diZbYulU^Jz z#?$N3e=(a`Pi%$qL}%?-k%QLO4b3&6s+ZrzGrp^#La774fNFLw&E>nrm`4T5fUZJMhZftY$e=M-!#9`bmmq=>y_VtY0E_glrrbeM( zCpSV1%CTtyGtUS!<)fb@#sg4J%%f)y;_C+!dQ_-Cu793Y&v3cqvQDE@+$*3l5Ry;%|Sn=x<`EE)59U6@*w-)vQf~F z4W~p;9)17)<~Jb3v$T`J1QUbW7l+yx0k_&00tJKuG&i^5n*yo^f0Ts5&=Qay4;NZ^ zx)k_yq1Hkvh#l1^jc!jz`3-)FjxcWu*#cPeE;R9JSI4SAHK&mMpdQi#;-{trhd9r= z-v{(uy?`1^6~yt5dyBf08>o4`d(sR*OSs8Lut5nHq>pQd`7B1KedQm|1ycmjHQN5d z1uwgRGA)Gilh6{`f951LG5?jD#Vk&fN%(1Gte*Lu!u-ZhV~ko0vsv%^jhEJVy&Dnz z!jT9I*gD6bSQ2UpD%|2i?dIb4qTd2XZN!RCea(ERq+f2}b|cNbYz~g(2p$6~E*IGj z(W5=*let*PHm6>5P;!07Rx^fc} z_e7(3=uJip7wIqSI+E&>yT%gZUx+KOf$nJ1QD-a9Un-;e#1&U-O^k}eEpAktn2The z;KU31sbgyvzgaM#-jzpovriw6t?%Zjo_Olhcrc!;KN&i&1N=5Th$&lZxRC0l=wKkX z7aXWo_9Jl%f0*iJ0|WGlJzzk3SWHZU2L8%g-g7<|(8t3{&R$;1sF{`~!xrG#4`GzX z$OC~u7zVZLRlac@tkly^wY5)19c>(wL>Y4aJr&7ZS&Euu6;QG!$E7+bIAV~=@lR=2 ztcf!@|Em01m61!J#GjgWsyQcG=gyVyMBHd%2m8^7YkTv! z_kQuze>L$w@T#=R4dT5ZHTazQNBA81$BUDk-#7cXANNlroV%5L=+Ez;CKA?EZ}KmW zL|AMq-QM;`8+kzpi?gKC9jfBdP8dOdj3wyh?{zVj?^2+(-_?W?2P z2tE=GM%`gA+f6!>oD@Agb8{gn(>Li+tK6sK_;`_;sHsIrK6uu6;b`Yi3kUSqJR3DX zi0wm?PWOQmvit8m==}zaGJ4hNqa03wVQA=h1XYjTfiDM^J4}xO+(er=p>1tWzw*{; 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