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Re: 2+2=4 ... How?
Posted:
Sep 28, 2012 7:37 PM
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In article <op.wk94sq2i1hq4pq@leon-hp>, wilson <winslole@udayton.edu>
wrote:
> 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 Å~ N Å® 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)).
Also, a + 0 = a, instead of a + S(0) = a.
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