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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Thu, 27 Feb 2025 23:01:24 +0100
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On 27.02.2025 21:41, Jim Burns wrote:
> On 2/27/2025 2:50 PM, WM wrote:
>> On 27.02.2025 19:19, Jim Burns wrote:
>>> On 2/27/2025 5:45 AM, WM wrote:
>>>> On 26.02.2025 23:17, Jim Burns wrote:
> 
>>>>> This next bit you (WM) might like, for a change.
>>>>> It looks like the pseudo.induction.rule which
>>>>> you have been trying to use.
>>>>
>>>> It is induction.
>>>
>>> This is what you (WM) have called induction:
>>> ⎛ Each inductive predicate A
>>
>> No, I call induction
>> a very restricted number of predicates.

>> I prefer Wikipedia:
>> ∀P (P(1) /\ ∀k(P(k) ==> P(k+1)) ==> ∀n (P(n)).

Correct. But not necessary in its generality for my purpose.
>>
> </WM>
> Date: Thu, 6 Feb 2025 17:55:57 +0100
> Message-ID: <vo2pit$31hlr$1@dont-email.me>
> 
>> If A(n) is useless for UA = ℕ,
>> then A(n+1) us useless too.
>> No reason to extend this simple concept.
> 
> You extend ∀n:Aᴺ(n) to Aᴺ(ℕ)
> but you only claim it, you don't justify it.

I use Zermelo's approach without wich there is no set theory.
> 
>> I do it order to avoid the following waffle:
> 
> How very Orwellian of you.
> I justify my claims. Doing so is 'waffle'.
> You don't. Abstaining from doing so is
> 'mathematics' and 'logic' and 'geometry'.
> 
>>> What that version of 'induction' seems to say
>>> is false if it's read literally.
>>> It's false that
>>> each inductive predicate is true.without.exception
>>> _in each domain without exception_
>>> Z₀ is a subset of a set Z holding 0 and all the {a}
>>
>>>> By the same induction
>>>> I prove UF = ℕ  ==> Ø = ℕ.
>>>
>>> What you use to prove that is
>>> ∀n:Aᴺ(n) ⇒ A(ℕ)
>>
>> That is how Zermelo guarantees Z₀.
> 
> Zermello's Infinity guarantees a superset Z of Z₀

How is that accomplished?

>  From Z, it follows,
> by Powerset and by Separation,
> that Z₀ exists.
> 
> ∀n:Aᴺ(n) ⇒ Aᴺ(ℕ)
> is your fantasy.

It is Zermelo's approach.
> 
> You would find your posts greatly improved
> by criticizing (if you can) _our_ reasoning,

You deny Zermelo's approach. His Z is ensured by induction.

>> There is no difference in some cases like these:
>> When all n are added by induction to the empty set,
>> then we have constructed ℕ.
> 
> When we have shown that there is

Gibberish! Simply agree that Z is ensured by induction.

> the intersection of all inductive subsets of
> an inductive set,
> then we have constructed ℕ.

We don't even need the intersection if we reduce Zermelo's approach to 
Lorenzen's approach: I is a natural number, and if x is a natural 
numbers then x+1 is a natural number.
> 
> In this context,
> a 'construction' is a proof of existence.

Induction is a proof of Z.
> 
>> When all n are subtracted by induction from ℕ
>> then we have created the empty set.
>> Do you agree?
> 
> I am trying to reach some expressions
> with which I can answer you and be understood.
> I'm not there yet.

Simply say yes.

Regards, §WM