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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 8:03 PM

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,
it is consistent that there will be finite
sequences that are represented and that form
what is called a generic set. These sets
have a convergence property that permits the
construction of forcing models. Cohen used
this notion to derive generic reals.

So, with the current axiomatization of ZFC,
it is usually possible to take advantage of
countable ground models in such a way that one
can form a forcing model that includes all of
the original sets of the ground model and at
least one set not in the ground model.

This is one reason I do not agree with the
account of identity,

http://plato.stanford.edu/entries/identity-relative/#1

But, it is also why David Ullrich is correct if
I am interpreting your paraphrase faithfully.

You might look at the link:

http://plato.stanford.edu/entries/actualism/

Although it is not directly relevant to a
mathematical context, it reflects your sentiment
concerning the existence of objects that cannot
be actually realized (by formulas in this case).

http://plato.stanford.edu/entries/existence/#FreRusExiNotProInd

you will find that Frege and Russell did not view
existence as a first-order property. So, one finds
one's self in the awkward trade-off between a comfortable
to establish a fixed notion of identity.

Hence, formulas alone are not enough (in my non-standard
views of these things).

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