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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 5 Feb 2025 10:26:41 +0100
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On 05.02.2025 00:43, Richard Damon wrote:
> On 2/4/25 12:04 PM, WM wrote:

>>>>> Note also, Peano doesn't "create" the Naturals with induction,
>>>>
>>>> He does. 1 or 0 ∈ ℕ, and if n is there, then n' is there.
>>>
>>> Which doesn't CREATE the set N, it shows that some other set is N.
>>
>> 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.

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.

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,

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.

Regards, WM