VerifiableStatements: notes and typo corrections from reading group

Author: gcarcassi

pm_VerifiableStatements.tex

@@ -895,7 +895,10 @@ \section{Statements}
 
 		\begin{proof}
 			Let $x \in \logCtx$ be a statement that determines and only determines all truth values of all statements in a theoretical domain $\tdomain$. This is equivalent to determining the truth values and only the truth values of all elements of a basis $\basis \subseteq \edomain$. As we can find a countable basis, the statement $x$ is equivalent to the countable conjunction of statements of $\basis$ or their negation. Therefore $x \in \tdomain$ as it is generated by the statements of the basis by negation and countable conjunction. But a statement in $\tdomain$ that determines all truth values of the statements in $\tdomain$ is a possibility by definition. Therefore $x$ is a possibility.
-		\end{proof}
+		\end{proof}	
+		
+		\todo{ngldskgfjb}{Add a justification that shows how $S$ determines the truth given the formal definition.}
+		
 	\end{mathSection}
 
 	There is one possibility that is often forgotten and sometimes needs special handling. Suppose one is trying to identify an illness by going through a series of known markers. It may happen that no match for the disease is found because we are dealing with a new kind of illness. In the same way, we may fail to measure the value of a quantity because it lies outside the sensitive range of our equipment. In other words, it may be possible that none of our tests succeed and none of the verifiable statements is verified. We call this possibility the residual because it's what remains after we went through all the cases we already know.
@@ -936,15 +939,15 @@ \section{Statements}
 		\end{thrm}
 
 		\begin{proof}
-			Let $\basis = \{\stmt[e]_i\}_{i=1}^{\infty} \subseteq \edomain$ be a countable basis. Let $2^{\mathbb{N}}$ denote the set of infinite binary sequences. We define the function $F:X\to\Bool^{\mathbb{N}}$ such that $F(x) = \{F(x)_i\}_{i=1}^{\infty}$ is given by: 
+			Let $\basis = \{\stmt[e]_i\}_{i=1}^{\infty} \subseteq \edomain$ be a countable basis. Let $\Bool^{\mathbb{N}}$ denote the set of infinite binary sequences. We define the function $F:X\to\Bool^{\mathbb{N}}$ such that $F(x) = \{F(x)_i\}_{i=1}^{\infty}$ is given by: 
 			$$
 			F(x)_i = 
 			\begin{cases}
 				\TRUE & x \comp \stmt[e]_i \\
 				\FALSE & x \ncomp \stmt[e]_i
 			\end{cases}
 			$$
-			For each $x \in X$ we have $x = \bigAND\limits_{i=1}^{\infty} \NOT^{F(x)_i} \stmt[e]_i$. Suppose $x_1 \neq x_2$, then $F(x_1)_i \neq F(x_2)_i$ for some $i$, therefore $F$ is injective. We then have $|X| \leq |2^{\mathbb{N}}|=|\mathbb{R}|$. $X$ has at most the cardinality of the continuum.
+			For each $x \in X$ we have $x = \bigAND\limits_{i=1}^{\infty} \NOT^{F(x)_i} \stmt[e]_i$. Suppose $x_1 \neq x_2$, then $F(x_1)_i \neq F(x_2)_i$ for some $i$, therefore $F$ is injective. We then have $|X| \leq |\Bool^{\mathbb{N}}|=|\mathbb{R}|$. $X$ has at most the cardinality of the continuum.
 		\end{proof}
 	\end{mathSection}
 
@@ -996,7 +999,7 @@ \section{Statements}
 
 	Let's go back to our verifiable statements and possibilities. For example, consider $\stmt_1=$\statement{the mass of the photon is less than $10^{-10}$ eV}. This can be expressed as $\stmt_1=\bigOR\limits_{0\leq x<10^{-10}}$\statement{the mass of the photon is precisely x eV}: the precise value must be in the given range of possibilities. Consider $\stmt_2=$\statement{the mass of the photon is greater than $10^{-20}$ eV}$=\bigOR\limits_{x>10^{-20}}$\statement{the mass of the photon is precisely x eV}. The conjunction is the intersection of the possible values: $\stmt_1\AND\stmt_2=$\statement{the mass of the photon is between $10^{-20}$ and $10^{-10}$ eV}$=\bigOR\limits_{10^{-20}< x<10^{-10}}$\statement{the mass of the photon is precisely x eV}. The disjunction is the union of the possible values: $\stmt_1\OR\stmt_2=$\statement{the mass of the photon can be anything}$=\bigOR\limits_{x\geq0}$\statement{the mass of the photon is precisely x eV}.
 
-	This is something that works in general. In Proposition \ref{pm-vs-disjunctiveNormalForm} we saw that, if a statement is a function of other statements, it can be expressed as the disjunction of minterms of the arguments. A verifiable statement is a function of basis, so it can be expressed as the disjunction of minterms of a basis. But we have also seen that the minterms of a basis are the possibilities, so each verifiable statement can be expressed as the disjunction of possibilities.
+	This is something that works in general. In Proposition \ref{pm-vs-disjunctiveNormalForm} we saw that, if a statement is a function of other statements, it can be expressed as the disjunction of minterms of the arguments. A verifiable statement is a function of a basis, so it can be expressed as the disjunction of minterms of a basis. But we have also seen that the minterms of a basis are the possibilities, so each verifiable statement can be expressed as the disjunction of possibilities.
 
 
 	Therefore each statement in the experimental domain defines a set of possibilities, which we call a verifiable set. Since certainty and impossibility are in the domain, the empty set and the full set of possibilities are verifiable sets. Since we can take finite conjunction and countable disjunction of verifiable statements, we can take finite intersection and countable union of verifiable sets. The collection of all verifiable sets forms a topology on the set of possibilities.
@@ -1139,6 +1142,7 @@ \section{Statements}
 	Now suppose two possibilities are experimentally distinguishable as we defined in Definition \ref{pm-vs-defExperimentallyDistinguishable}. Then, by \ref{pm-vs-experimentallyDistinguishableIsDisjointApproximation}, we can find two disjoint approximations. In the example before, the two verifiable statements were in fact incompatible. This means that, given two approximately verifiable possibilities, we can find two disjoint verifiable sets each containing only one possibility: if all possibilities are experimentally distinguishable then the natural topology is $\mathsf{T}_2$.
 
 	\begin{mathSection}
+		\todo{gjoihtoeir}{It should be provable that the natural topology is sober}
 		\begin{prop}
 			The natural topology of a set of possibilities is Kolmogorov (or $\mathsf{T}_0$).
 		\end{prop}