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Posts:
1,093
Registered:
12/4/12
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Re: Matheology § 203
Posted:
Feb 12, 2013 3:21 PM
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On 2/12/2013 11:21 AM, WM wrote:
> On 12 Feb., 17:59, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>> WM <mueck...@rz.fh-augsburg.de> writes:
>>> On 4 Feb., 13:35, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>>>> WM <mueck...@rz.fh-augsburg.de> writes:
>>>>> On 2 Feb., 02:56, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>>
>>>>>> "The logicist reduction of the concept of natural number met a
>>>>>> difficulty on this point, since the definition of ?natural number?
>>>>>> already given in the work of Frege and Dedekind is impredicative. More
>>>>>> recently, it has been argued by Michael Dummett, the author, and Edward
>>>>>> Nelson that more informal explanations of the concept of natural number
>>>>>> are impredicative as well. That has the consequence that impredicativity
>>>>>> is more pervasive in mathematics, and appears at lower levels, than the
>>>>>> earlier debates about the issue generally presupposed."
>>
>>>>> I do not agree with these authors on this point.
>>
>>>> So, on what grounds do you suppose that the notion
>>>> of natural number is predicative?
>>
>>> The notion of every finite initial segment is predicative because we
>>> need nothing but a number of 1's, that are counted by a number already
>>> defined, and add another 1.
>>
>> It's in the justification of the claim that induction yields a conclusion
>> that holds for *any* natural number where the impredicativity lies.
>
> Impredicativity is not a matter of quantity but of self-referencing
> definition. Further you seem to mix up every and all.
> I don't see any necessity to consider *all* natural numbers. I
> maintain, without running in danger to be contradicted: For every
> natural number that can be reached by induction, induction holds.
Speaking of self-referencing definitions...
Your observation that the natural numbers are
in one-to-one correspondence with the initial
segments is invoking the successor as a choice
function
|||({|,||,|||})=|||
It is one of the truly elegant observations
in your analysis. Sadly, you cannot see it
because of your religion.
It also suggests that all arithmetic
is impredicative.
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