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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: 1,289
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
<> wrote in
<news:op.wk94pwvp1hq4pq@leon-hp> in alt.math.undergrad:

> On Thu, 27 Sep 2012 01:18:28 -0400, Brian M. Scott <>
> wrote:

>> On Wed, 26 Sep 2012 21:21:54 -0700 (PDT), Madhur
>> <> wrote in
>> <>
>> 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:

>> <>

> 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.



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