Peano Axioms

• PA1: 1 is a natural number.
• PA2: For any natural number n, succ(n) is also a natural number.
• PA3: For any natural number n, succ(n) is not equal to 1. It means that 1 is not the successor of any number.
• PA4: Different natural numbers have different successors: if a ≠ b, then succ(a) ≠ succ(b). Hmm, this is different from the description in “Mathematical Girl”. In “Mathematical Girl”, it says that if succ(a) ≠ succ(b), then a ≠ b. Both versions prevent merging (blu3mo). It seems that if succ(a) ≠ succ(b), then a ≠ b and if a ≠ b, then succ(a) ≠ succ(b) are equivalent.
• No, “Mathematical Girl” says that if succ(a) = succ(b), then a = b. (対偶) I understand.
• PA5: If 0 has a certain property and a has that property, then its successor suc(a) also has that property. Therefore, all natural numbers have that property. I don’t really understand the necessity of PA5 even after reading “Mathematical Girl”. Why can’t we use mathematical induction if it’s not defined in PA5? This is the opposite. We can use mathematical induction because PA5 is defined (takker). PA5 implies mathematical induction. Conversely, in a world where PA5 does not exist, it is uncertain whether we can use mathematical induction. I don’t understand the part that says “it is uncertain whether we can use mathematical induction” (blu3mo). Mathematical induction is just connecting deductive reasoning based on predicate logic, so we should be able to use it without it being defined (blu3mo). Conversely, if we can’t do this, then we can’t say that “1 is a natural number” (PA1) and “succ(n) is a natural number” (PA2) imply “suc(1) and suc(suc(1)) are also natural numbers” (blu3mo) (blu3mo) (blu3mo) (blu3mo) (blu3mo) (blu3mo). But we can say that (takker). The act of combining PA1 and PA2 to derive $\mathrm{succ}(1)\in \mathbb{N}$ is logically correct. Ah, I may have explained it poorly. Mathematical induction and PA5 are equivalent (takker). PA5: $\forall B((1\in B\wedge \forall n\in B;\mathrm{succ}(n)\in B)⟹B=A)$. Mathematical induction: For any logical formula P, $P(1)\wedge \forall n\in A;(P(n)⟹P(\mathrm{succ}(n)))⟹\forall n\in A;P(n)$. PA5 $⟺$ mathematical induction. In the quote, PA5 is expressed in the same way as mathematical induction.
• It seems that (blu3mo) doesn’t understand the part that (blu3mo) had doubts about, or maybe the question and answer don’t match (takker). It seems that (blu3mo) doesn’t understand the part that (blu3mo) had doubts about.
• One of the Peano axioms includes the principle of mathematical induction, so what theorems can’t be proven without using mathematical induction? - Quora I see, so it’s also saying that there are no natural numbers other than the successor of 1. In that case, by incorporating the logic that only applies to the successor of 1 into the definition, it is excluding everything else in a paradoxical way. Can’t we just write it more simply as “Only the successor of 1 and its successors are natural numbers”? Ah, but if we do that, we can’t avoid recursive expressions. In that case, we need PA5 to define the inference.
• By the way, if we introduce the negation of PA5 instead of PA5, it becomes interesting (takker).
• Interesting (blu3mo).