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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Wed, 28 May 2025 11:25:29 +0300
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On 2025-05-27 15:09:30 +0000, WM said:

> On 27.05.2025 14:01, Mikko wrote:
>> On 2025-05-26 13:38:00 +0000, WM said:
>> 
>> No alternative view is known to be better.
> 
> That does not make this view good.

It does. Perhaps not good enough for all purposes but certainly good
enough to be called good.

>>> But even pure mathematics proves that most natural numbers will never 
>>> be definable:
>>> 
>>> {1} has infinitely many (ℵo) successors.
>> 
>> (2) there are infinitely many (ℵo) possible definitions.
> 
> No. All natural numbers can be manipulated collectively, for instance 
> subtracted: ℕ \ {1, 2, 3, ...} = { }. Here all have disappeared.
> 
> Could all natural numbers be distinguished by individually defining 
> each one, then this subtraction could also happen but, caused by the 
> well-order, a last number would disappear.
> 
>>> If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2, 
>>> 3, ..., n, n+1} has infinitely many (ℵo) successors, for every n that 
>>> can be defined.
>> 
>> You can't formulate that as a logically or mathematically valid proof.
>> 
> It is a valid proof by induction. Claim it for all natural numbers. Get 
> a contradiction. But perhaps you prefer geometry?

No, it is not. In order to use an inductive proof you must first specify
the theory you are using, and that theory must have an induction axiom.
There is no induction in plain logic.

An induction proof must prove P[0] and P[n] -> P[n+1] before it can infer
that for all x P[x]. You have not even identified what P you are talking
about.

For discussion of natural numbers a good basis is Peano axioms. Another
good basis is Cantor's construction, perhaps in ZF set theory.

> The set of finite initial segments of natural numbers is potentially 
> infinite but not actually infinite.

There is nothing potential in a set. If there are infinitely many members
in a set then the set is infinite, otherwise it is finite.

>  (Actual infinity is a fixed number greater than all natural numbers.)

Infinity is not a number but a feature some sets have and some don't.
If a set has that feature then all bigger sets have that feature,
too. Typical set theories have infinite sets of different sizes.

-- 
Mikko