On 7/18/2024 10:37 PM, Ross Finlayson wrote:
On 07/18/2024 02:51 PM, Jim Burns wrote:
[...]
Yet, there's a case for induction
that there's no case for induction,
which axiomless deduction usually
arrives at as insufficient.
Your (RF's) claim seems to be
| Axiomlessly,
| P(0) ∧ ∀j: P(j)⇒P(j+1)
| is insufficient to conclude
| ∀k: P(k)
I don't know of anyone who'd object to
that axiomless claim.
Axioms describe what we are talking about.
Axiomlessly,
what our claim is about could be anything.
For some things, induction is too strong a claim.
Induction is unjustified
without knowing what the claim is about.
Maybe it helps to think of the numbers
as ranging from zero to a large number,
then that it's infinite in the middle.
No, "large" is the opposite of helping,
because infinite is after all the finites.
Up.to.a.large.finite and after.all.finites
have different properties.
"Large" is misleading.
A finite is accessible.in.principle.
That includes numbers not accessible.in.fact,
such as Avogadroᴬᵛᵒᵍᵃᵈʳᵒ.
⎛ A number one step beyond an accessible number
⎜ is also an accessible number.
⎜
⎜ A number on the way to an accessible number
⎝ is also an accessible number.
That gives us some description of
the set ℕ of accessible.in.principle numbers.
(Axioms are description.)
⎛ ∀j ∈ ℕ: j+1 ≠ 0 ∧ j+1 ∈ ℕ
⎜
⎝ ∀k ∈ ℕ: k ≠ 0 ⇒ k-1 ∈ ℕ
Add to that that N is well ordered,
that is, each nonempty set holds a least element,
and that is a thorough description of
the familiar ℕ
( ∀A ⊆ ℕ: ∃j ∈ A: j ≤ᴬ A ⇐ A ≠ {}
From those three claims about ℕ, induction follows for ℕ
For whatever is outside ℕ, no claim is made.
For whatever is inside ℕ, we know that
it is accessible, and we can use its being accessible
to derive further information like
induction, unique prime factorization,
N fitting some proper subsets, and so on.
It is different, no matter what we call it,
no matter what other thing we call 'infinite'.
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