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Topic: mathematical infinite as a matter of method
Replies: 25   Last Post: May 4, 2013 11:24 PM

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 fom Posts: 1,968 Registered: 12/4/12
Re: mathematical infinite as a matter of method
Posted: May 3, 2013 10:55 PM

On 5/3/2013 7:54 PM, Graham Cooper wrote:
> On May 4, 10:03 am, fom <fomJ...@nyms.net> wrote:
>> On 5/3/2013 5:03 PM, Graham Cooper wrote:
>>
>>
>>
>>
>>
>>
>>
>>
>>

>>> On May 3, 8:15 pm, fom <fomJ...@nyms.net> wrote:
>>>> On 5/3/2013 2:43 AM, Graham Cooper wrote:
>>
>>>>> Its not possible to test Equality by Extension in the inf. case.
>>
>>>> That is correct Herc.
>>
>>>> On the other hand, it is not possible to interpret
>>>> the universal quantifier as a universal statement if
>>>> it is interpreted as a course-of-values.

>>
>>>> Aristotle wrote this. It is ignored by a certain
>>>> contingent of the mathematical community who merely
>>>> argues on the basis of beliefs concerning infinity.

>>
>>>> You know well that any computer system balances
>>>> choices that affect performance. Relational databases
>>>> run faster on logic chips optimized for integral
>>>> arithmetic as opposed to floating point. The analogy
>>>> applies here.

>>
>>>> Brouwer had been clear concerning how the effectiveness
>>>> of working with finite sets differed from working
>>>> with infinite sets. But, the reason infinity enters
>>>> mathematics is because it is how the identity relation
>>>> is extended to convey the geometric completeness of a
>>>> line when used to represent the real number system.

>>
>>>> Infinity does not arise because of testability. It
>>>> arises because of the nature of the identity relation.

>>
>>> If there are more SETS in ZFC than FORMULA in ZFC
>>> (David C Ullrich)

>>
>>> ZFC FORMULA | ZFC SETS
>>
>>> 1 ___________ a i
>>> 2 ___________ b p q r
>>> 3 ___________ c j n
>>> 4 ___________ d s t k u v
>>> 5 ___________ e z w
>>> ...

>>
>>> THEN WHAT DO YOU MEAN BY ...
>>
>>> A SET OF ZFC ?
>>
>> I actually agree with you somewhat here.
>>
>> Nevertheless, if one restricts to countable
>> models, then it is clear that there must be
>> real numbers not represented. In particular,

>
>
> No, you're entitled to that view but hundreds of people say it is NOT
> clear.
>
> GIVEN AN INFINITE LIST
> YOU CAN CONSTRUCT A MISSING REAL
>
> is simply wrong! there is no turing machine that can do that and
> halt.
>
> ------------------------------------------------------
>
> Given there are oo choices for the 1st selected row of the list
> and only 10 digits to select from
> Every possible digit can appear at
>
> LIST_1_1
>
> by the Pigeon Hole Principle.
>
>
> +------------>
> | 0. 1 ..
> | ...
> |
> | 0. 2 ..
> |
> | ...
> |
> | 0. 3 ..
> |
> | ..
> |
> | 0. 4 ..
> |
> | ..
> |
> | 0. 5 ..
> |
> | ..
> |
> | 0. 6 ..
> |
> | ...
> |
> | 0. 7 ..
> |
> | ...
> |
> | 0. 8 ..
> |
> | ..
> |
> | 0.9 ..
> |
> | ..
> |
> | 0. 0 ..
> |
> |
> |
> |
> |
> V
>
>
> INFINITE CHOICES
> 10 OPTIONS
>
> By the P.H.P. the
>
> DIAG(1) = [ 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 ]
>
>
> NEGATING DIAG(1)
>
> 6 --> 4
>
> changes the Diagonal
> which gives you a different permutation.
>
>
> CHANGE PERMUTATION <=> CHANGE DIGIT ON DIAGAONAL
>
>
> --------------
>
>
> When you CHANGE THE DIGITS OF THE DIAGONAL
>
> 1 - at - a - time
>
> AT EVERY STEP OF THE ALGORITHM
>
> You have the SAME LIST (permuted)
>
>
> ----------------
>
>
> No one in their right mind would say:
>
> +--------->
> | 0.134...
> | 0.224...
> | 0.563...
> | ...
> v
>
> 0.123... is missing
>
> because:
>
> 0.5., is on the list CHANGE TO 1
> 0.X3... is on the list CHANGE TO 2
> 0.XX4... is on the list CHANGE TO 3
>
> but by sorting the list:
>
> +--------->
> | 0.563...
> | 0.134...
> | 0.224...
> | ...
> v
>
> YOU DO CLAIM 0.123... is missing!
>
> IT's just the DIAGONAL of list 1!
>
>
> -----------
>
> So you claim
>
> ALL ANTIDIAGONALS OF ALL PERMUTATIONS ARE MISSING!
>
> +--------->
> | 0. 1 [3] 4...
> | 0. 2 2 [4]...
> | 0. [5] 6 3...
> | ...
> v
>
> SO THAT PATH [5] [3] [4]
>
> CAN BE INVERTED!
>
> to 0.123...
>
> ----------------------------
>
> So you claim the DIAGONAL is CLEARLY ABSENT TOO!
>
> by the Pigeon Hole Principle!
>

I make none of these claims.

For the record, however, I do not use phrases
like "anti-diagonal". I speak of the "constructed
number" or "constructed representation". And, in
this, there is no changing of diagonal elements of
a given list.

And, you are quite correct that for each n, the
finite initial segment of the constructed number
will correspond with initial segments of listed
elements greater than n.

So, if I made the claims you attribute to me, then
one might argue that the "anti-diagonal" merely
arises from the combinatorics of transversal designs.

But, instead, it is a pinching argument that depends
on a completed infinity for the construction of
a counterexample to a claim that the real
numbers can be put in correspondence with the
natural numbers.

No claim... no argument.

No completed infinity... no counterexample.

In general, I have no quibble with people who do

Date Subject Author
4/21/13 fom
4/21/13 Virgil
5/2/13 Hercules ofZeus
5/2/13 fom
5/2/13 Virgil
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Brian Q. Hutchings
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 fom
5/4/13 Graham Cooper
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/3/13 Graham Cooper