Re: Simple enough for every reader?

On 29.05.2025 12:07, Mikko wrote:

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

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.

Anyhow a reader in sci.logic should understand it.

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

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

I am assuming Cantor's infinity. This is expressed by ℵo.

 

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].

P[n]: {1, 2, 3, ..., n} has infinitely many (ℵo) successors.
P[n+1]: {1, 2, 3, ..., n, n+1} has infinitely many (ℵo) successors.
Do you doubt ℵo - 1 = ℵo?

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.

It is basic mathematics as you learn it in the first semester.

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.

Induction is applied to every natural number of the Peano set. The proof shows that it cannot be applied to every natural number of the Cantor set.

 

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.

Here nothing gets complicated, but all remains very simple.
All Cantor's natural numbers can be manipulated collectively, for instance subtracted: ℕ \ {1, 2, 3, ...} = { }. Here all have disappeared.
Could all Cantor's natural numbers be distinguished, then this subtraction could also happen but, caused by the well-order, a last one would disappear. Contradiction.
Regards, WM
  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.