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Topic: 2+2=4 ... How?
Replies: 9   Last Post: Sep 29, 2012 10:40 AM

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wilson

Posts: 4
Registered: 9/27/12
Re: 2+2=4 ... How?
Posted: Sep 27, 2012 1:40 AM
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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.
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