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Re: 2+2=4 ... How?
Posted:
Sep 27, 2012 3:54 PM


On Thu, 27 Sep 2012 01:40:34 0400, wilson <winslole@udayton.edu> wrote in <news:op.wk94pwvp1hq4pq@leonhp> in alt.math.undergrad:
> 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:dca7ee3d94464fccb4fbc01fdbc685ea@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>
> A bit more complicated:
> First you need another axiom.
No, you don't.
> 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)
No extra axiom is used here. This is just a definition, made within the framework of the axioms.
[...]
Brian



