Re: universal quantification, because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

On 5/11/2024 5:44 PM, Ross Finlayson wrote:

On 05/11/2024 02:05 PM, Ross Finlayson wrote:

On 05/11/2024 12:24 PM, Jim Burns wrote:

[...]

[...]

>
It's like two inductive analysts were contradicting
each other. One says "base case, subsequent
case, case closed", and the other says "base case,
subsequent case, case not closed".
 You just pick one?

Pick the one in a finite sequence of
only not.first.false claims.
It is a true claim.
"Base case, subsequent case" is cisfinite induction.
Today, elsewhere, I gave an argument ==
a finite.sequence of only not.first.false.claims
for cisfinite induction.
Re: because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how
Date: Sat, 11 May 2024 14:49:46 -0400
But there is also transfinite induction, which also
is in a finite.sequence of only not.first.false claims,
claims about ordinals instead of naturals.
The sequence about ordinals is shorter,
and I don't think this is about the sequences themselves,
anyway, but about justification by the existence of
such sequences. I think the shorter will shed more light.
1.
Ordinals are well.ordered.
∃γ: p(γ) ⟹ ∃β: (p(β) ∧ ¬∃α<β:p(α))
2.
Transfinite.induction is true for ordinals
∀β: (̅p(β) ⇐ ∀α<β:̅p(α)) ⟹ ∀γ: ̅p(γ)
(2.) is not.first.false in that sequence
because
(2.) is merely a re.written version of (1.)
¬H ⟹ ¬C  for  H ⟹ C
∀¬  for  ¬∃
C ⇐ H  for  ¬(¬C ∧ H)
̅p  for ¬p
(1.) is not.first.false in that sequence
because
(1.) is not.false (there or anywhere)
If Q is first.false, then Q is false ==
If Q is not.false, then Q is not.first.false.
(1.) "Ordinals are well.ordered" is true
because _that's what "ordinal" means_
(1.) (2.) is a finite sequence of
only not.first.false claims.
Because (1.) (2.) is finite,
if (1.) (2.) held a false claim,
(1.) (2.) would hold a first.false claim.
(1.) (2.) doesn't hold a first.false claim
(1.) (2.) doesn't hold a false claim
(2.) is true.
Transfinite.induction is true for ordinals
∀β: (̅p(β) ⇐ ∀α<β:̅p(α)) ⟹ ∀γ: ̅p(γ)

It's exactly about "not.ultimately.untrue" that
describes how there are "inductive impasses"
that belie their finite inputs.

Please explain "not.ultimately.untrue" and
"inductive impasse". I would welcome examples.