Sujet : Re: The set of necessary FISONs
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.mathDate : 16. Feb 2025, 00:52:53
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <89d12ce1-5183-40f6-aa37-e4c1522fbc3e@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
User-Agent : Mozilla Thunderbird
On 2/15/2025 7:49 AM, WM wrote:
On 14.02.2025 20:22, Jim Burns wrote:
On 2/14/2025 11:40 AM, WM wrote:
On 14.02.2025 16:42, Jim Burns wrote:
On 2/14/2025 7:52 AM, WM wrote:
Induction creates infinite sets.
>
Zermelo set theory [I...VII] describes
a domain of discourse with
Z an inductiveᶻ set
>
Where all elements are covered by induction.
>
No,
we don't know that, in inductiveᶻ Z,
each element is covered by inductionᶻ.
>
Induction proves the existence of
the inductive set of all its members.
Induction proves,
from the subset {k:A(k)} of all j such that A(j)
being any inductive subset,
that the subset {k:A(k)} of all j such that A(j)
is the only.inductive subset.
That is something which induction can only prove
about a set which HAS an only.inductive subset.
ℕ has an only.inductive subset ℕ
If Z is inductive, then
⋂𝒫ⁱⁿᵈ(Z) has an only.inductive subset ⋂𝒫ⁱⁿᵈ(Z)
Being covered by induction is
being in the only.inductive.subset.
>
That is here the set of all FISONs.
Consider the set {F} of all FISONs
and the set {F:A(F)} of all FISONs F′ such that A(F′)
Suppose we prove
/ first.{F} ∈ {F:A(F)}
\ if F′ ∈ {F:A(F)} then F′+1 ∈ {F:A(F)}
{F:A(F)} is one of the inductive subsets of {F}
{F} has only one inductive subset: {F}
We have proved {F:A(F)} = {F}
As a consequence of {F:A(F)} = {F}
∀F′ ∈ {F}: A(F′)
That is here the set of all FISONs.
{F} is not.in the only.inductive subset of {F}
Induction proves ∀F′ ∈ {F}: A(F′)
Induction doesn't prove A({F})
Each FISON is.
Some sets are and some sets aren't.
>
Which set do you have in mind?
{F} is a set of omissible FISONs.
{F} isn't an omissible set of FISONs.
Without the leap, there is no conflict.
{F} is a set of omissible FISONs.
{F} isn't an omissible set of FISONs.
Without the leap, there is no conflict.
>
Without this leap there are no infinite sets.
No.
finiteᵂᴹ infiniteᵂᴹ matheologicalᵂᴹ
↕ ↕ ↕
finite finite infinite
You (WM) imagine a last finite step,
into the infinite.
However,
stepping.from the finite
makes that which is stepped.to finite.
The set of necessary FISONs
Re: The set of necessary FISONs