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The induction principle is used to prove a sequence of propositions $P_0, P_1, P_2, P_3, ...$.

And it proceeds as follows

(i) Verify $P_0$. (ii) Assuming truth of $P_k$ for some $k$, verify $P_{k+1}$.

In this way, denoting by $S$ the set of those $k$ for which $P_k$ is true, then $S$ verifies both (i) and (ii) in the induction principle. Hence S coincides with N, so all propositions $P_k$ are true.

Can one describe this using mathematical symbolisms rather than prose ? What does one do exactly ?

Veak
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  • You can replace everything with notation but it often makes it harder to read and solve. Introducing notation requires justification. – CyclotomicField Aug 08 '23 at 21:48
  • In second order logic, it can be written as an axiom about sets of natural numbers. In first order logic, you are stuck with an "axiom schema." That is, an infinite collections of axioms that follow a pattern. – Thomas Andrews Aug 08 '23 at 21:52
  • There is a reason that we use function-like notation rather than sequence notation, $P(n)$ rather than $P_n.$ $P(n)$ has a particular meta-meaning in first order logic. – Thomas Andrews Aug 08 '23 at 21:56
  • First order logic, second order logic, Not heard about that. Is there anything good to read of it? – Veak Aug 08 '23 at 22:05
  • Is there a symbol in mathematical logic that specifies truth of a proposition P ? – Veak Aug 08 '23 at 22:06
  • Typically the truth of proposition $P$ is represented by $P$ itself. So induction can be written as something like $((P_0 \land \forall k (P_k \implies P_{k+1})) \implies \forall n P_n$. – ConMan Aug 09 '23 at 00:03
  • I like the expression. As something that I had in mind would look like. – Veak Aug 09 '23 at 00:16

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