Re: The set of necessary FISONs

On 20.02.2025 12:18, Alan Mackenzie wrote:

WM <invalid@no.org> wrote:

The set of all natural numbers which can be defined is (potentially in-)
finite. infinitesimally smaller than ℕ.

 The set of all natural numbers is N, by definition.  N is infinite.
Lose the useless qulifications.  They're not helping anybody.

They are helping those who wish to understand.

 

Proof: If UF = ℕ is assumed, then F(1) can be omitted without changing
the union of the remainder.

 I can accept this, though it needs a bit more rigour.  Under what
conditions can a FISON be omitted from a union of them without changing
that union?  Only when there is a subsequent FISON in the union.

No, that is not true for the union ℕ. Every FISON that is smaller than ℕ can be omitted because it is useless. Note that we are not looking for the real union of FISONs but for those FISONs which are clearly useless to accomplish the union ℕ.

 

And if F(n) can be omitted without changing this union, then also
F(n+1) can be omitted without changing this union.

 This is questionable,indeed.  What precisely is the nature of the
relationship between F(n) and F(n+1) which allows this?

The FISONs are v. Neumann's natural numbers except that he started with 0. Therefore they have the relationship of the natural numbers. They are defined by induction.

You have failed to prove this inductive step.

|ℕ \ {1, 2, 3, ..., n}| = ℵo
==>
|ℕ \ {1, 2, 3, ..., n+1}| = ℵo - 1 = ℵo.

Again, you can omit F(n+1) when and only when there is a bigger FISON in
the union.

Again, that is only true if the real union is asked for.

 

That makes the omitted FISONs the inductive collection of all FISONs
and proves the implication: If UF = ℕ, then { } = ℕ.

 You have entirely failed to prove that.  A problem is that you are
defining members of a set only by their relationship to other members.

That is induction.

On finishing your alleged induction, that relationship no longer holds -
You end up with a set of FISONs none of which can be omitted, since
there isn't a bigger FISON in the complementary set of remainders.  So
the whole mechanism collapses like a bubble bursting.

There is no finishing when the natural numbers are defined by induction. Why should there be a finishing for FISONs which in fact are the same natural numbers, only maintaining all their predecessors.

A large part of the problem is that you are using FISONs as though they
were some sort of primitive.

They represent natural numbers according to v. Neumann.

See how long ago it is since you actually
defined what you mean by FISON on this group, if you ever have.

IIRC it was Virgil who proposed this abbreviation. I see that I have used it in 2011 already.
Here they are:
{1}
{1, 2}
{1, 2, 3}
...
The usual property is that addition of one is always possible. My invention however is stronger: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
Regards, WM