On 5/18/2024 4:11 PM, Ross Finlayson wrote:
On 05/18/2024 11:16 AM, Jim Burns wrote:
On 5/18/2024 12:09 PM, Ross Finlayson wrote:
One can contrive simple inductive arguments
that _nothing_ is so.
>
An example of such an argument
would be clarifying here.
Says nothing,
says nothing,
says nothing,
says nothing,
....
>
See,
just saying so doesn't make it so,
something that _goes_ has a _place_ to go.
Explain to me how that is an inductive argument.
----
Here is an inductive argument for n⁺¹≠n
BASE CASE.
⎛ Background for finite ordinals
⎝ k⁺¹≠0
for k=0
0⁺¹≠0
STEP CASE.
⎛ Background for finite ordinals
⎝ j≠k ⇒ j⁺¹≠k⁺¹
for j=n⁺¹ k=n
n⁺¹≠n ⇒ n⁺¹⁺¹≠n⁺¹
Therefore,
for each finite ordinal n, n⁺¹≠n
by induction.
----
In the context of finite ordinals,
that is complete, in that
we know by that argument that,
for each finite ordinal n, n⁺¹≠n
There exists a fuller argument which
details _how_ we know that,
the part which we don't often see,
because that part is essentially unchanged
from invocation to invocation of "induction"
Here is the fuller inductive argument for n⁺¹≠n
| Assume a finite.ordinal counter.example nₓ
| nₓ⁺¹=nₓ
|
| ⎛ Background for ordinals
| ⎜ The set of counter.examples
| ⎝ holds a first or is empty.
|
| There is a counter.example nₓ
| There is a first counter.example n₁
| n₁⁺¹=n₁
| k < n₁ ⇒ k⁺¹≠k
|
| ⎛ Background for finite.ordinals
| ⎜ Finite.ordinal n is 0 or
| ⎜ it can be decremented and
| ⎝ each before.ordinal can be decremented or is 0
|
| finite.ordinal nₓ
| n₁ < nₓ
| n₁=0 or n₁ can be decremented.
|
| 1.
| n₁=0
| 0⁺¹=0
| BASE CASE: 0⁺¹≠0
| Contradiction.
|
| 2.
| n₁ can be decremented.
| n₁⁻¹ < n₁
| (n₁⁻¹)⁺¹ ≠ n₁⁻¹
| STEP CASE: n⁺¹≠n ⇒ n⁺¹⁺¹≠n⁺¹
| n₁⁺¹≠n₁
| However,
| n₁⁺¹=n₁
| Contradiction.
|
| Contradiction or contradiction.
Therefore,
a finite.ordinal counter.example nₓ not.exists
For each finite.ordinal n: n⁺¹≠n
Completely.
----
Suppose we want a proof of some property P(n)
for the complete domain of finite.ordinals
We can swap out
proofs of 0⁺¹≠0 and of n⁺¹≠n ⇒ n⁺¹⁺¹≠n⁺¹
and swap in
proofs of P(0) and of P(n) ⇒ P(n⁺¹)
and we will have a correct proof with
a necessarily correct conclusion.
A proof.by.induction is a _general_ form
into which details can be inserted which
make it a correct proof of a _particular_ claim
not unlike a proof.by.contradiction,
in that respect.
What we often _call_ a proof.by.induction is
the details to be swapped into the fuller proof.
It's the _whole_ proof, seen and unseen,
which makes the conclusion invincibly complete.
See,
just saying so doesn't make it so,
something that _goes_ has a _place_ to go.
Yes,
just saying so doesn't _make_ it so.
Even
saying so within
a finite sequence of only not.first.false claims
doesn't _make_ it so.
However,
that allows us to _know_ that it's so.
Proof.by.induction is a telescope, not a ray gun.
V
Re: V