Re: Simple enough for every reader?

On 28.05.2025 10:25, Mikko wrote:

On 2025-05-27 15:09:30 +0000, WM said:

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.

Why do you think has the induction axiom been devised at all? Right, because the sequence of natural numbers has this property. When Pascal and and Fermat first used induction, there was no axiom but the property of natural numbers had been recognized.

There is no induction in plain logic.

But it is in the mathematics we apply.

 An induction proof must prove P[0]

I have said: {1} has infinitely many (ℵo) successors.

and P[n] -> P[n+1] before it can infer

I did not expect that you need this explanation:
If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2, 3, ..., n, n+1} has infinitely many (ℵo) successors because here the number of successors has been reduced by 1, and ℵo - 1 = ℵo. There is no way to avoid this conclusion if ℵo natural numbers are assumed to exist. And that is the theory that I use.

that for all x P[x].

Just that is wrong because it is not true for all natural numbers but only for definable ones.

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

 There is nothing potential in a set.

Then call it a collection.

If there are infinitely many members
in a set then the set is infinite, otherwise it is finite.

Wrong. The set of known prime numbers is finite without a fixed last number. It exists in mathematics and is a potentially infinite collection.

 

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

 Infinity is not a number but a feature some sets have

Wrong again. ω is an infinite ordinal number. Cantor has devised it and has called it an infinite whole number. In fact it is a whole number because ω + 1 is also a whole number, but fractions could be added, according to Cantor.
Regards, WM