Re: The set of necessary FISONs

WM submitted this idea :

On 05.02.2025 00:43, Richard Damon wrote:

On 2/4/25 12:04 PM, WM wrote:

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Note also, Peano doesn't "create" the Naturals with induction,

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He does. 1 or 0 ∈ ℕ, and if n is there, then n' is there.

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Which doesn't CREATE the set N, it shows that some other set is N.

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If there is another set, then we could use it without need of Peano.

 There are many ways to create the set of Natual Numbers.

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Peano, Dedekind, Cantor, Zermelo, Schmidt, v. Neumann, Lorenzen did it. By induction. In all cases "if n ∈ ℕ then n+1 ∈ ℕ" is the fundamental property.

 It isn't the induction axiom that does it though.

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A proof by induction consists of two cases. The first, the base case, proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k then it must also hold for the next case
n = k+1. [Wikipedia]

 The set of Natural numbers (in Peano) are created by the OTHER axioms,

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They are necessary only because Peano uses the clumsy notion of successor. Nevertheless he fails, because he describes only sequences like 1, π, π^π, π^π^π, ... Lorenzen for instance does not need any other axiom than the induction described above.

Induction is what gets you from each and every to all. The successor function is not induction.