MHB Easy question regarding symbols in discrete mathematics

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The set of symbols that constitute strings in discrete mathematics is denoted by the symbol Σ, representing the alphabet. In contrast, Σ* denotes the set of all finite strings that can be formed from the symbols in Σ, including the empty string. The Kleene Star (*) indicates that the strings can consist of zero or more symbols. Therefore, Σ refers to the basic set of symbols, while Σ* encompasses all possible combinations of those symbols. Understanding this distinction is fundamental in the study of formal languages and automata theory.
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is the set of symbols that make up strings denoted by the symbol Σ or Σ* , also what is this difference?
 
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Usually the set of symbols, or the alphabet, is denoted by $\Sigma$. Then $\Sigma^*$ denotes the set of all finite strings in the alphabet $\Sigma$.
 
The star (*) is the so called Kleene Star.
It means zero-or-more.

So indeed, Σ is the alphabet, while Σ* is the set of strings consisting of zero or more symbols from the alphabet.
 
I'm taking a look at intuitionistic propositional logic (IPL). Basically it exclude Double Negation Elimination (DNE) from the set of axiom schemas replacing it with Ex falso quodlibet: ⊥ → p for any proposition p (including both atomic and composite propositions). In IPL, for instance, the Law of Excluded Middle (LEM) p ∨ ¬p is no longer a theorem. My question: aside from the logic formal perspective, is IPL supposed to model/address some specific "kind of world" ? Thanks.
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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