HA cannot prove ∃yG(y) because that would assert HA proves (0=1), contradicting the consistency assumption together with the fact that HA proves all true quantifier-free sentences.
A placeholder for some property or formula that applies to a thing — think of it like a test where you plug in different values to see if they pass.
HA(Constructive mathematics)
Heyting Arithmetic — the standard formal system for intuitionistic/constructive arithmetic
Quantifier-free sentences(the type of sentences HA is known to prove correctly)
Mathematical statements that don't use 'there exists' or 'for all' — basically, statements you can check by plugging in specific numbers rather than reasoning about infinite collections.
∃y (existential quantifier)(as used in formal logic notation)
A symbol meaning 'there exists at least one'—it says that at least one thing satisfying the condition is out there.
Here is a proof that the rule “If \(\forall x (A \vee B(x))\) is a theorem, so is \(A \vee \forall x B(x)\)” (where \(x\) is not free in \(A)\) is not admissible for \(\mathbf{HA},\) if \(\mathbf{HA}\) is consistent. Gödel numbering provides a quantifier-free formula \(G(x)\) which (numeralwise) expresses the predicate “\(x\) is the code of a proof in \(\mathbf{HA}\) of \((0 = 1).\)” By intuitionistic logic with the decidability of quantifier-free arithmetical formulas, \(\mathbf{HA}\) proves \(