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Topic: Infinity: The Story So Far
Replies: 16   Last Post: Mar 7, 2014 8:31 AM

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Peter Percival

Posts: 1,302
Registered: 10/25/10
Re: Infinity: The Story So Far
Posted: Mar 7, 2014 8:28 AM
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Virgil wrote:
> In article <lf9qbf$so$1@news.albasani.net>,
> Peter Percival <peterxpercival@hotmail.com> wrote:
>

>> Virgil wrote:
>>> In article <lf83m8$tqh$1@news.albasani.net>,
>>> Peter Percival <peterxpercival@hotmail.com> wrote:
>>>

>>>> Virgil wrote:
>>>>> In article <4cfd9bb3-5708-4ed7-9285-e2735a4fcd85@googlegroups.com>,
>>>>> mueckenh@rz.fh-augsburg.de wrote:
>>>>>

>>>>>> But he claimes that the Peano-axioms supply the natural numbers of formal
>>>>>> mathematics. So the natural numbers of formal mathematics are:
>>>>>> 1, -10, 100, -1000, ...

>>>>>
>>>>> Only in wierd places like WM's wild weird world of WMytheology.
>>>>> Elsewhere, the successor operation is denoted by adding one.

>>>>
>>>> I don't think that is correct.

>>>
>>>
>>> In abstract induction it need not be, but in the standard set of
>>> naturals, which is what WM is arguing about, it most assuredly is.

>>
>> But what are the naturals defined to be? If it's "things satisfying
>> Peano's axioms" then 1, -10, 100, -1000, ... (with the appropriate S) do
>> the job.

>
> The natural are defined to be the members of a Peano set, but only when
> the definitions of addition and multiplication of natural numbers are
> suitably ( and inductively ) defined, and whether the ur-element is
> interpreted as 0 or 1.
>
> Starting with 0
> ? A x1 in N, 0 ? S(x1)
> ? A x1,x2 in N, S(x1) = S(x2) => x1 = x2
> ? A x1 in N, x1 + 0 = x1
> ? A x1,x2 in N, x1 + S(x2) = S(x1 + x2)
> ? A x1 in N, x1 * 0 = 0
> ? A x1,x2 in N, x1 * S(x2) = (x1 *x2) + x1
>
> Starting with 1
> ? A x1 in N, 1 ? S(x1)
> ? A x1,x2 in N, S(x1) = S(x2) => x1 = x2
> ? A x1 in N, x1 + 1 = S(x1)
> ? A x1,x2 in N, x1 + S(x2) = x1 + x2 + 1
> ? A x1in N. x1 * 1 = x1
> ? A x1,x2 in N. x1 * S(x2) = (x1 *x2) + x1


Does anything exclude S(x) from meaning "x multiplied by -10"?
Perhaps there is a problem equating the informal "multiplied by" with
the formal *.

--
Madam Life's a piece in bloom,
Death goes dogging everywhere:
She's the tenant of the room,
He's the ruffian on the stair.



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