diff --git a/content/incompleteness/arithmetization-syntax/coding-symbols.tex b/content/incompleteness/arithmetization-syntax/coding-symbols.tex index ed780c4e..a94e5ccd 100644 --- a/content/incompleteness/arithmetization-syntax/coding-symbols.tex +++ b/content/incompleteness/arithmetization-syntax/coding-symbols.tex @@ -95,7 +95,7 @@ c_0},\scode{)}}. \] Here, $\scode{\eq}$ is $\tuple{0,7} = 2^{0+1}\cdot -3^{7=1}$, $\scode{\Obj v_0}$ is $\tuple{1,0} = 2^{1+1}\cdot3^{0+1}$, +3^{7+1}$, $\scode{\Obj v_0}$ is $\tuple{1,0} = 2^{1+1}\cdot3^{0+1}$, etc. So $\Gn{=(\Obj v_0,\Obj c_0)}$ is \begin{multline*} 2^{\scode{=} + 1}\cdot 3^{\scode{(}+1}\cdot 5^{\scode{\Obj v_0}+1} diff --git a/content/incompleteness/introduction/definitions.tex b/content/incompleteness/introduction/definitions.tex index a5683c6c..82ac87f3 100644 --- a/content/incompleteness/introduction/definitions.tex +++ b/content/incompleteness/introduction/definitions.tex @@ -223,7 +223,7 @@ $y_1$, \dots, $y_n$ are all the free variables of~$!A$ and the initial quantifiers of~$!B$ bind the variables~$y_1$, \dots,~$y_n$. Once we have extracted this~$!A$ and checked that its free variables match the -variables bound by the universal qauntifiers at the front +variables bound by the universal quantifiers at the front and~$\lforall[x]$, we go on to check that the antecedent of the conditional matches \[ diff --git a/content/incompleteness/introduction/historical-background.tex b/content/incompleteness/introduction/historical-background.tex index 2cd7831b..907b5c8f 100644 --- a/content/incompleteness/introduction/historical-background.tex +++ b/content/incompleteness/introduction/historical-background.tex @@ -33,7 +33,7 @@ thorough and systematic study of the syllogism. Aristotle's logic dominated scholastic philosophy through the middle -ages; indeed, as late as eighteenth century Kant maintained that +ages; indeed, as late as the eighteenth century, Kant maintained that Aristotle's logic was perfect and in no need of revision. But the theory of the syllogism is far too limited to model anything but the most superficial aspects of mathematical reasoning. A century @@ -71,7 +71,7 @@ Greeks. Euclid's \emph{Elements}, written around 300 B.C., is already a mature representative of Greek mathematics, with its emphasis on rigor and precision. The definitions and proofs in Euclid's -\emph{Elements} survive more or less in tact in high school geometry +\emph{Elements} survive more or less intact in high school geometry textbooks today (to the extent that geometry is still taught in high schools). This model of mathematical reasoning has been held to be a paradigm for rigorous argumentation not only in mathematics but in diff --git a/content/incompleteness/introduction/undecidability.tex b/content/incompleteness/introduction/undecidability.tex index 8ce7f4ae..a6589b48 100644 --- a/content/incompleteness/introduction/undecidability.tex +++ b/content/incompleteness/introduction/undecidability.tex @@ -62,7 +62,7 @@ relations; this means that all theories that include $\Th{Q}$, such as $\Th{PA}$ and $\Th{TA}$, also do, and hence also are not !!{decidable}. (Since all these theories are true in the standard -model, they are all consistent.)) +model, they are all consistent.) We can also use this result to obtain a weak version of the first incompleteness theorem. Any theory that is !!{axiomatizable} and