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wilson
Posts:
3
Registered:
9/27/12
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Re: 2+2=4 ... How?
Posted:
Sep 27, 2012 1:42 AM
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On Thu, 27 Sep 2012 01:40:34 -0400, wilson <winslole@udayton.edu> wrote:
> On Thu, 27 Sep 2012 01:18:28 -0400, Brian M. Scott <b.scott@csuohio.edu>
> wrote:
>
>> On Wed, 26 Sep 2012 21:21:54 -0700 (PDT), Madhur
>> <madhur.varshney@gmail.com> wrote in
>> <news:dca7ee3d-9446-4fcc-b4fb-c01fdbc685ea@googlegroups.com>
>> in alt.math.undergrad:
>>
>>> The natural numbers that we use are said to be derived
>>> from what so called Peano's Axioms.
>>
>> They *can* be; this is not the only possible formal
>> foundation for them.
>>
>>> While these axioms (listed below) give a method of
>>> building up counting numbers they do not define or
>>> construct basic arithmetic operations like addition,
>>> subtraction, multiplication, etc or basic comparisons
>>> like that of equality.
>>
>> Equality is assumed to be a known relation. The arithmetic
>> operations and the linear ordering on the natural numbers
>> are defined using the axioms. This is explained, albeit
>> briefly, in the Wikipedia article on the Peano axioms:
>>
>> <http://en.wikipedia.org/wiki/Peano_axioms?banner=none#Arithmetic>
>>
>> [...]
>>
>> Brian
>
> A bit more complicated:
>
> First you need another axiom. From the Wikipedia article:
>
> Addition is the function + : N × N ? N (written in the usual infix
> notation, mapping elements of N to other elements of N), defined
> recursively as:
>
> a + S(0) = a
> a + S(b) = S(a+b)
> Now can define 1 as S(0), 2 as SS()) and 4 as SSSS(0).
>
> the proof that 2 + 2 = 4 is then a matter of substituting the right
> thing in the right place.
sorry. my mistake. Define 2 as SS((0)).
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