Re: The set of necessary FISONs

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Sujet : Re: The set of necessary FISONs
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.math
Date : 12. Mar 2025, 11:54:23
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <1a0b4566-e625-4b18-8139-a34c8d5e7515@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
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On 3/12/2025 5:03 AM, WM wrote:
On 11.03.2025 23:51, Jim Burns wrote:
On 3/11/2025 5:28 PM, WM wrote:
On 11.03.2025 20:25, Jim Burns wrote:
On 3/11/2025 2:01 PM, WM wrote:

Zermelo defines Z to be an inductive set.
>
in order to ensure the existence of
an infinite or inductive set.
>
No.
>
Um aber die Existenz "unendlicher" Mengen zu sichern,
bedürfen wir noch des folgenden ... Axioms.
[Zermelo: Untersuchungen über die Grundlagen der Mengenlehre I, S. 266]
...which isn't
/ Um aber die Existenz "unendlicher" Mengen zu sichern,
\ bedürfen wir noch des folgenden ... Definition.

There is a definition and there is an axiom.
>
Where are they?
Please quote a definition
⎛ Menge Z, welche die Nullmenge als Element enthält und
⎜ so beschaffen ist, daß jedem ihrer Elemente a
⎝ ein weiteres Element der Form {a} entspricht
⎛ set Z, which contains the zero set as an element and
⎜ is such that each of its elements a
⎝ corresponds to another element of the form {a}

which is outside of the axiom.
Your advice to Zermelo would be to write
( Der Bereich enthält mindestens eine Z.
( The domain contains at least one Z.
The definition is used inside the axiom.
⎛ Der Bereich enthält mindestens eine Menge Z,
⎜ welche die Nullmenge als Element enthält und
⎜ so beschaffen ist, daß jedem ihrer Elemente a
⎝ ein weiteres Element der Form {a} entspricht
⎛ The domain contains at least one set Z,
⎜ which contains the zero set as an element and
⎜ is such that each of its elements a
⎝ corresponds to another element of the form {a}

They are different kinds of things.
>
And you are unable to understand the connection
between these two things.
A definition answers "What is Z ?"
An axiom answers "Does Z exist?"

Is it the language?
Or is it a general deficit?
>
The definition of Z states that
Z is inductive.
>
Please quote this definition which
is not the axiom.
"If it was a snake, it'd have bit me."

in order to ensure the existence of
an infinite or inductive set.
>
A definition doesn't ensure existence.
That's what an axiom does.
>
This axiom
Thank you.
Now you (WM) sound slightly less ignorant.
You don't deserve to sound less ignorant,
but it was really very grating.

This axiom ensures the sequence of numbers
by the same induction
"The same induction" is NOT
the induction which
⎛ proves that a subset is the whole.set
⎜ by proving that subset is inductive,
⎜ for a whole.set already known to be
⎝ its.own.only.inductive.subset.
"The same induction" is merely
something you decided that you'd call
'induction'.

which ensures the dark numbers:
...your darkᵂᴹ numbers about which
it cannot be said that they self.equal.

which ensures the dark numbers:
Since what follows is NOT
⎛ a proof that a subset is the whole.set
⎜ by proving that subset is inductive,
⎜ for a whole.set already known to be
⎝ its.own.only.inductive.subset,
how are your darkᵂᴹ numbers ensured?

{} ∈ Z₀, and for all x: if x ∈ Z₀ then {x} ∈ Z₀
ℕ \ F(1) = ℵo,
and if
ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n) = ℵo
then
ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n+1) = ℵo.

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