Let τbe a tautology, i, j, m ∈N a t, and ϕ, ψ, χbe S C sentences. Define a sequence of sets of s

step 1 Hello, so since nothing is a model of both Sigma 1 and Sigma 2, we have that Sigma 1, Union, Sig

Let τbe a tautology, i, j, m ∈N a t, and ϕ, ψ, χbe S C...

Let $\tau$ be a tautology, $i, j, m \in N a t$, and $\phi, \psi, \chi$ be $S C$ sentences. Define a sequence of sets of sentences as follows: $$ \Gamma_0=|\tau|, $$ and, for any $m>0$, $\Gamma_m=\left\{(\phi \wedge \psi) \mid\right.$ there exist $i, j<m$ such that $\left.\phi \in \Gamma_i \& \psi \in \Gamma_j\right\}$. Now let $\Delta=\cup\left\{\Gamma_k \mid k \in N a t\right\}$. Then for any $\chi \in \Delta, \chi$ is tautologous. [Hint. $\chi$ must be in $\Gamma_k$ for some $k$. Why? Now use Course of Values Induction to prove that $\chi$ in $\Gamma_k$ is tautologous.]

Let $\tau$ be a tautology, $i, j, m \in N a t$, and $\phi, \psi, \chi$ be $S C$ sentences. Define a sequence of sets of sentences as follows:
$$
\Gamma_0=|\tau|,
$$
and, for any $m>0$,
$\Gamma_m=\left\{(\phi \wedge \psi) \mid\right.$ there exist $i, j<m$ such that $\left.\phi \in \Gamma_i \& \psi \in \Gamma_j\right\}$.
Now let $\Delta=\cup\left\{\Gamma_k \mid k \in N a t\right\}$. Then for any $\chi \in \Delta, \chi$ is tautologous. [Hint. $\chi$ must be in $\Gamma_k$ for some $k$. Why? Now use Course of Values Induction to prove that $\chi$ in $\Gamma_k$ is tautologous.]

Key Concepts

Recursive Definitions

A recursive definition specifies an object in terms of simpler instances of itself. In the context of the given problem, the sequence of sets is defined recursively, starting with a base set and then constructing new sets by applying logical operations to elements of previously defined sets. This methodical construction is fundamental in both set theory and logic, as it helps build complex structures from simple, well-understood components.

Logical Connectives

Logical connectives are symbols or words used to connect propositions to form more complex logical expressions. In the context of the problem, the conjunction (represented by ‘?’) is used to combine sentences from earlier stages in the recursive process. Understanding how logical connectives operate is crucial to analyzing and constructing compound logical statements, and plays an important role in determining the overall truth value of those statements.

Tautology

A tautology is a logical sentence that is true under every possible interpretation regardless of the truth values assigned to its components. In mathematical logic, tautologies are important because they represent universal logical truths, and proving that a sentence is tautologous ensures its validity in all contexts within the logical system.

Course of Values Induction

Course of Values Induction is a form of mathematical induction where the induction step assumes a property holds for all values less than a given value to prove it holds for that value. This technique is particularly useful when the definition of an object or set at one stage depends on all previous stages, as is the case in recursively defined sequences of sets.

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