Re: The set of necessary FISONs

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

On 02.02.2025 19:23, Jim Burns wrote:

On 2/2/2025 6:25 AM, WM wrote:

F(1) can be discarded.
If F(n) can be discarded, then F(n+1) can be discarded.
>
Note:
Mathematical induction is a method for proving that
a statement P(n) is true for every natural number n
that is, that
the infinitely many cases P(0),P(1),P(2),P(3),...
all hold. [Wikipedia]

>
For each k ∈ ⋃{FISON}:

>
No, for *all* k ∈ ⋃{FISON}.

For EACH k ∈ ⋃{FISON}
certain claims are known.without.exception of k.
By 'known.without.exception', I mean that
it's not necessary to know which of ⋃{FISON} k is.
It's sufficient to know that k is of ⋃{FISON}
in order to know that claim of k.
We say: that is a valid claim.
Explicitly or implicitly, its validity is
_with respect to_ a domain. Here, it's ⋃{FISON}
We say: k is not between 2 and 3.
That is a valid claim with respect to ⋃{FISON}
⋃{FISON} isn't the only domain. In some others,
it ISN'T valid to say k is not between 2 and 3.
But, then, it must be that the "same" claim
(the same words) isn't the same claim after all
(different meanings) with respect to
different domains.
Induction says: certain claims are
valid (known.without.exception) with respect to
the elements of ⋃{FISON}
What domain it is with respect to
is an essential aspect of a claim's meaning.
Induction says:
because  0≠0+1  is true
and  k≠k+1 ⇒ k+1≠k+2  is valid
k≠k+1 is valid.
Valid (known.without.exception) with respect to
the elements of ⋃{FISON}
EACH element of ⋃{FISON} without exception,
but, for non.elements, this claim is silent.

Peano creates the set ℕ by induction.

No.
Peano describes the set ℕ as such that
induction is valid with respect to ℕ

I remove the set of FISONs by the same induction:
n ==> n+1.
>
What us the difference?

What.you.think.Peano.does  Peano doesn't do.
What.you.think.our.claims.are.with.respect.to
our claims are not with respect to.

⋃{FISON} = ⋃({FISON}\{F(k)})
>
P(k)  :⇔  ⋃{FISON}=⋃({FISON}\{F(k)})
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For each k ∈ ⋃{FISON}:  P(k)
>
Because
none in linearly.ordered {FISON} is last.

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None of the natural numbers is last.

Yes.

How can Peano create the complete set by induction?

Peano describes a set with induction.
It is a complete set which is described.
(We don't use any other, "incomplete" sets.)
----
We don't handle k or ⋃{FISON}
We handle descriptions of k or ⋃{FISON}
We know that a description is
a valid claim with respect to those.it.describes.
Even if those.it.describes are infinitely.many.
We know because that's what a description is.
Consider the three claims, in order: P(k) P(k)⇒Q(k) Q(k)
Q(k) is not.first.false.
⎛ If one of P(k) P(k)⇒Q(k) is false,
⎜ Q(k) is not first.false.
⎜ If neither of P(k) P(k)⇒Q(k) is false,
⎝ Q(k) is not false, and not.first.false.
We say:
Q(k) is a valid inference from P(k) P(k)⇒Q(k)
Valid (not.first.false) inferences and
valid (true.without.exception IN THE DOMAIN) claims
are the electricity which powers mathematics.
Something is described.
Being a description, it is valid for the described.
Claims are assembled, claims such that each is
true.without.exception.or.not.first.false.
Each of those claims must be true.without.exception.
But the claims are silent about what wasn't described.
Peano describes _the elements_ of ⋃{FISON}
⋃{FISON} isn't an element of ⋃{FISON}