Re: The set of necessary FISONs

On 2/3/2025 1:36 PM, WM wrote:

On 03.02.2025 19:06, Jim Burns wrote:

On 2/3/2025 7:41 AM, WM wrote:

How can Peano create the complete set by induction?

>
Peano describes a set with induction.

>
Without axioms
nothing must be used oin formal mathematics.

Axioms describe the domain.
Describe a finite ordinal:
⎛ Sets of them are minimummed or empty.
⎜ Each has an immediate predecessor or is zero,
⎜ and each of the priors of each
⎝ has an immediate predecessor of is zero.
That description is the axioms of the finite ordinals.
(There are other ways to describe them.)

Therefore Peano, Zermelo, or v. Neumann
create ℕ as well as the set of all FISONs
for use in set theory.

Axioms describe.
Magic spells create.

It is a complete set which is described.
(We don't use any other, "incomplete" sets.)

>
Therefore all FISONs can be removed from
the set of all FISONs.

We can describe the removal of all of them, sic: {}

All natural numbers can be added by induction to a set A.

Either all natural numbers are in A,
or they aren't all in A.
Those are all the choices.
A set with all natural numbers in it
has certain properties.
⎛ For example, enough in.set swaps
⎜ can leave Bob not.in the set,
⎝ even though no swap is out.of.set.

1 is added to A, and
if n is added to A, then n+1 is added to A.

Describing A:
⎛ 1 is in A.
⎝ If n is in A, then n+1 is in A
ℕ is the unique set subset to each such set A
ℕ is the minimal inductive set.
⎛ If you deny that,
⎜ you're only absconding with the name 'ℕ'
⎜ Everything I claim about ℕ remains true of
⎝ the minimal inductive set, whatever it's called.

All FISONs can be subtracted from the set of all FISONs
by the same procedure.
F(1) is subtracted.
If F(n) is subtracted, then F(n+1) is subtracted.

For each FISON,
there is a larger FISON not larger than U{FISON}
U{FISON} is larger than any FISON.
The sum of any two FISONs is a FISON.
U{FISON} is larger than the sum of any two FISONs.
Each end.segment U{FISON}\{1,...,j}
is larger than any FISON.
Such behavior is unlike that of finite sets.
We could decide that  that isn't their behavior,
but, if we decide that, everything turns to gibberish.
The reasoning is implacable, and does not disappear
because we have decided against it.
We could decide that these sets aren't finite.
What I mean here by 'finite' is what we mean.
You (WM) mean something else.
There exists a general preference to avoid gibberish.
This preference is what you (WM) call "matheology".

But the claims are silent about what wasn't described.
Peano describes _the elements_ of ⋃{FISON}
⋃{FISON} isn't an element of ⋃{FISON}

>
Peano creates

...describes...

the set of all natural numbers as well as
the set of all FISONs.