Re: Simple enough for every reader?

On 2025-05-28 15:13:54 +0000, WM said:

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.

It is in certain mathematical structures but not in all. For example it
is an axiom of Peano arithmetic but not RR arithmetic. Therefor it is
important to specify which theory is used.

An induction proof must prove P[0]

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

But you navn't proven that this infinity is not begger than some other
infinity.

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.

To me this does not look like P[n] -> P[n+1].

that for all x P[x].

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

It is not wrong because you failed specify the theory you are using.
As I said the theory must be specified.
In Peano arithmetic the induction axiom is applicable to everything.
If you want something else you must specify some other theory, perhaps
some 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.

 Then call it a collection.

Things get soon complicated if we allow other than objects, first order
functions and first order predicates. A set of objects is equivalent to
a first order predicate so it does not complicate too much. But a
collection that is not a set would require another book and I don't
think I would read it.
--
Mikko