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Topic: Infinity: The Story So Far
Replies: 16   Last Post: Mar 2, 2014 4:04 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Infinity: The Story So Far
Posted: Feb 28, 2014 12:49 PM
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On 2/28/2014 10:36 AM, Ralf Bader wrote:
> On Fri, 28 Feb 2014 07:01:25 -0600, fom wrote:
>

>> On 2/28/2014 2:01 AM, mueckenh@rz.fh-augsburg.de wrote:
>>> On Thursday, 27 February 2014 22:52:54 UTC+1, fom wrote:
>>>

>>>>> The question was whether Peano defines the natural numbers. He fails.
>>>>
>>>>
>>>>
>>>> Why do you say that?
>>>>
>>>>

>>> I say that because it is widely assumed that Peano defined the natural
>>> numbers. People assume the natural numbers and find that Peano is
>>> rigth. But the other way dos not work. Assuming the Peano axioms does
>>> not yield N. I emphasize this because it has been hundred years taught
>>> falsely.

>>>>
>>>>
>>>> It would be correct to say that he assumes the
>>>>
>>>> natural numbers and places identity criteria
>>>>
>>>> onto his denotations to restrict their
>>>>
>>>> interpretations. In that sense he does not
>>>>
>>>> define them. But I do not think this is what
>>>>
>>>> you mean.

>>>
>>> It is precisely what I mean. And I am in particular happy that there
>>> are some like you who have not yet been completely perverted by the
>>> study of mathematics but can understand that some topics have been
>>> taught wrong.
>>>
>>>

>> Ok. Yes, Peano's actual axioms are different from the
>> recharacterizations according to the needs of computer systems or the
>> first-order axioms.
>>
>> Thanks.

>
> There is exactly one natural numbers structure as defined by Peano's
> axioms, up to isomorphism, with second order logic. And more than "up to
> isomorphism" doesn't make any sense. Mückenheim is obviously unable to
> grasp this, especially the clause "up to isomorphism". I am unable to
> grasp how one can take a person like Mückenheim serious in these matters.
> Mückenheim's idiotic babble has nothing to do with the non-uniqueness in
> first order logic, because a non-standard model of first order Penao
> arithmetic is a beast totally diofferent from Mückenheim's idiotic
> garbage.
>


True.

But, why would you expect a person without
training to understand that in the same way
that you do?

And I am in full agreement with what WM said
here. It fits the facts of what the axioms
as given by Peano say. He assumes the natural
numbers and does not define them.

It is your statement "up to isomorphism"
that reflects modern notions of structure.

But, here's the rub. Most of the grand
philosophy that underlies your statement
has rejected Frege's notions of sense and
reference. So, while your symbols may be
mappable "up to isomorphism", your philosophy
says that those symbols have no denotations.

So, can you even say what you are talking
about?

Or are you just expecting me to fill in the
blanks?

Words become meaningless because everybody
wants to look smart by fucking his neighbor
up the ass. And, the more intelligent among
us seem to be worse than most: "Oh, that's
just words" followed by "This is the right
way."

I am critical of WM for that exact
approach to the agenda he pursues.





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