On 3/12/2025 9:04 AM, WM wrote:
On 12.03.2025 11:54, Jim Burns wrote:
On 3/12/2025 5:03 AM, WM wrote:
On 11.03.2025 23:51, Jim Burns wrote:
There is a definition and there is an axiom.
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Where are they?
Please quote a definition
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⎛ 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}
That is just the induction.
A definition answers "What is Z ?"
An axiom answers "Does Z exist?"
That text answers the question "What is Z ?"
It is inside more text which answers "Does Z exist?"
⎛ Der Bereich enthält mindestens eine ...
⎛ The domain contains at least one ...
I can define whatever I want.
In order to show off my vast ability to define,
I occasionally define things as
flying rainbow sparkle ponies.
I have defined them.
You have watched me do it.
However,
that's NOT a claim of existence for
flying rainbow sparkle ponies.
I am afforded that ability to define because,
without existence (which I am NOT able to grant),
flying rainbow sparkle ponies mean so little.
Zermelo's Axiom of Inductivity grants the existence of
at least one set.which.could.be.Z
-- existence in the domain under discussion.
The axiom does not create an inductive set.
The axiom restricts
which domain might be under discussion
to domains holding a set.which.could.be.Z.
A change in axioms is a tectonic shift in
which domains might be under discussion.
It would be fair to say that one discussion ends,
and another begins.
I can define whatever I want, but
I can't get away with any axioms I choose.
They require agreement, where definitions don't.
This is why good axioms are boring,
to forestall disagreement.
Recall the system ST+F with which I plagued you.
The existence claims are for {} and X∪{y}
What could be more boring?
My hope was to get your (WM's) agreement and then
proceed _with you_ to less boring topics,
in a well.justified manner.
how are your darkᵂᴹ numbers ensured?
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Here:
Fron Zermelo's induction
{} ∈ Z₀, and for all x: if x ∈ Z₀ then {x} ∈ Z₀
Z₀ -- defined as the emptiest inductive set --
doesn't hold darkᵂᴹ {} and
doesn't hold any visibleᵂᴹ x and darkᵂᴹ {x}
The only inductive subset of Z₀ is Z₀
We know (above) that
the visibleᵂᴹ.number.subset {visibleᵂᴹ} ⊆ Z₀
is inductive.
There is only one subset of Z₀ it can be: Z₀
{visibleᵂᴹ} = Z₀
{darkᵂᴹ} = Z₀\{visibleᵂᴹ} = {}
Try again.
How are your darkᵂᴹ numbers ensured?
my induction is ensured
Your induction is not ensured by
your declaration that it's ensured.
Your induction is not ensured in the way that
⎛ 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.
is ensured.
How is your induction ensured?
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.