MHB Transfinite Induction: Theorem or Axiom?

Thread starter facenian1
Start date Dec 11, 2018
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Induction Transfinite

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Transfinite induction can be classified as either a theorem or an axiom, depending on the logical system in use. In any system, a statement can be treated as an axiom, though this may require dropping another axiom to maintain minimality. Conversely, it can also be considered a theorem, which might necessitate the addition of another axiom for proof. The classification ultimately hinges on the foundational framework being applied. Thus, the status of transfinite induction is context-dependent.

Dec 11, 2018

facenian1

Is transfinite induction a theorem o an axiom?

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Dec 11, 2018

HOI

That depends. In any logical system you can always take any given statement as an "axiom" (you might need to drop at least one other axiom to keep a "minimal" set of axioms) or as a theorem (you might need to add at least one other axiom to be able to prove the theorem).

Thread 'Formal derivation of statement from Peano Arithmetic system'

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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