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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 5 Feb 2025 07:07:58 -0500
Organization: i2pn2 (i2pn.org)
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On 2/5/25 4:26 AM, WM wrote:
> 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.

No. Induciton is part of the Axiom system, it isn't the part that 
"creates" the Set of Natural Numbers.

Induction is the way to test if another Set, the set that satisfies some 
relationship, is equal to the set of the Natual Numbers.

Peano et all didn't "invent" the Natural Numbers, as they existed long 
before them. What they did was FORMALIZE the definition.

I suppose to you, the made it a mathology.


>>
>> 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]

Right, and the RESULT of a proof by induction, is that the relationship 
holds for all Natural Numbers. It doesn't "create" the set of Natural 
Numbers.

The Successor Axiom is more important for that, the others mostly steer 
the set to be what we want from the set of Natural Numbers.

One thing to note, Induction is a second order logic form, and there are 
variations of Peano that drop it, and replace it with a first order rule 
to do something related, and those all have the Natural Numbers too, so 
it can't be the axiom of Induction that "creates" the Natural Numbers.

>>
>> 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.

Since you don't understand what they did, you aren't a good person to 
ask to judge the result.

> 
> Regards, WM
> 
> 
> 

You are just proving your stupidity, and that you are too dumb to see 
your stupidity becuase you start with the lie that you think you know 
what you are doing.