CANTOR DISPROOF <<<<<<<<<<<<<<<< [Math]

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From: |-|ercules on 29 Jun 2010 01:03

"Sylvia Else" <sylvia(a)not.here.invalid> wrote ...
> On 29/06/2010 1:57 PM, |-|ercules wrote:
>> "Sylvia Else" <sylvia(a)not.here.invalid> wrote
>>>> You haven't commented on the difference between induction over a single
>>>> data structure
>>>> and induction over numerous data structures.
>>>>
>>>
>>> I have questioned whether what you're doing is induction. P(n) ->
>>> P(n+1) is the result of an inductive proof, not the proof itself.
>>>
>>> Sylvia.
>>
>>
>>
>> Here's a story that tells of the difference that's more your level.
>>
>> You and I start a landscaping business Herc And Syl's Landscaping Ad
>> Infinitium
>>
>> I handle all the mowing, and you do the fencing.
>>
>> We get a call from Mr Fenceme and Mrs Mowme Blockheads.
>>
>> We drive to the property which appears to be divided into 2 blocks, both
>> infinite rectangular lawns.
>>
>> On one block, you start doing the fencing for Mr Fenceme, completing the
>> perimeters of larger and larger concentric
>> rectangular paddocks.
>>
>> I get to the mowing for Mrs Mowme, completing larger and larger
>> rectangular mown lawn areas, each building
>> upon the earlier smaller rectangular lawn area.
>>
>> Being a man, I'm well on my way to mowing the whole lawn.
>>
>> Because you're a woman, you never ever come close to fencing the entire
>> lawn! ;-)
>> Herc
>>
>
> Since neither of us makes the smallest dent in our respective infinite
> tasks, you cannot meaningfully say which of us is closer to completing it.

Wrong! the limit of mown lawn area as mowing time->oo is infinity.

However, infinitely many fence sizes all have finite perimeters.

>
> What has that to do with induction anyway?
>
> Sylvia.

You can prove a paint algorithm covers the screen using induction, just like mowing above,
just like proving a property for all natural numbers, or all digit positions.

Herc

From: herbzet on 29 Jun 2010 01:04

Sylvia Else wrote:

> But what I can do, if you like, is to killfile you.

Suit yourself, darling.

From: |-|ercules on 29 Jun 2010 01:09

"|-|ercules" <radgray123(a)yahoo.com> wrote
> You can prove a paint algorithm covers the screen using induction

i.e. flood fill

Herc

From: Sylvia Else on 29 Jun 2010 01:32

On 29/06/2010 3:09 PM, |-|ercules wrote:
> "|-|ercules" <radgray123(a)yahoo.com> wrote
>> You can prove a paint algorithm covers the screen using induction
>
> i.e. flood fill
>
> Herc

Last time I checked, screens were not infinite in size, and flood fill
is commenced using a finite sized dot.

Either way, the fact that some things can be proved by induction does
not imply that a particular proof is by induction.

Sylvia.

From: |-|ercules on 29 Jun 2010 01:46

"Sylvia Else" <sylvia(a)not.here.invalid> wrote
> On 29/06/2010 3:09 PM, |-|ercules wrote:
>> "|-|ercules" <radgray123(a)yahoo.com> wrote
>>> You can prove a paint algorithm covers the screen using induction
>>
>> i.e. flood fill
>>
>> Herc
>
> Last time I checked, screens were not infinite in size, and flood fill
> is commenced using a finite sized dot.

hardly relevant

>
> Either way, the fact that some things can be proved by induction does
> not imply that a particular proof is by induction.
>
> Sylvia.

Induction is a description of a method for proving a property is true for case N+1
given that case N is true. And showing a base step is true, usually N=1.

My and your examples fall under that class of descriptions.

So do the mowing and fencing algorithms.

Fencing is equivalent to your increasing sizes of lists.

Mowing is equivalent to my increasing widths of digits along an infinite digit sequence.

Herc