Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 10/17/2024 2:22 PM, WM wrote:

On 17.10.2024 00:39, Jim Burns wrote:

No natural number is
the first to not.have a natural.number.double.

>
True.

What we mean by 'natural number' is
'finite ordinal'.
⎛ Each ordinal ξ, finite or not,
⎜ has successor ξ+1 = ξ∪{ξ}
⎜
⎜ Each set S of ordinals, finite or not,
⎜ holds first.S or is {}
⎜
⎜ Each _finite_ ordinal k
⎜ has predecessor k-1: (k-1)+1 = k or is 0
⎜ and each prior ordinal j < k
⎝ has predecessor j-1: (j-1)+1 = j or is 0

When doubling all natural numbers
we obtain only natural numbers.

>
That is impossible.

>
There is no first natural number from which we obtain
(by doubling) anything not.a.natural.number.

>
True.
>

The only set of natural numbers with no first
is the empty set..

>
No, the set of dark numbers is
another set without smallest element.

A nonempty set without a first element
is not a set of only finite ordinals.
What we mean by 'natural number' is
'finite ordinal'.

There is no ▒▒▒▒▒ natural number from which we obtain
(by doubling) anything not.a.natural.number.

>
Correct is:
There is no such _definable_ natnumber.

There are sets of natural numbers with first elements
and there is the empty set.
The first.element.free set of
natural numbers from which we obtain
(by doubling) anything not.a.natural.number
is not the first option.
That set is the second option: the empty set.
It is empty.
There is no ▒▒▒▒▒ natural number from which we obtain
(by doubling) anything not.a.natural.number.

There is a general rule not open to further discussion:

⎛ Things which aren't natural numbers
⎝ aren't called natural numbers.
What you (WM) are doing is like
'proving' that transcendental integers exist
by declaring a rule "not open to further discussion"
that π is an integer.

When doubling natural numbers we obtain
even numbers which have not been doubled.
In potential infinity we obtain
more even natural numbers than have been doubled.
In actual infinity we double ℕ and obtain
neither ℕ or a subset of ℕ.