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

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Brian M. Scott

Posts: 1,259
Registered: 12/6/04
Re: 2+2=4 ... How?
Posted: Sep 27, 2012 3:54 PM
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On Thu, 27 Sep 2012 01:40:34 -0400, wilson
<winslole@udayton.edu> wrote in
<news:op.wk94pwvp1hq4pq@leon-hp> 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: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>

> 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



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