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2+2=4 ... How?
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Madhur
Sep 27, 2012, 12:21:54 AM9/27/12
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The natural numbers that we use are said to be derived from what so called Peano's Axioms. 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. My doubt is how exactly we reach the conclusion: 2+2=4?
Peano’s Axioms of Natural numbers (N)
We assume that the set of all natural numbers has the following properties:
Axiom 1: 1 is a natural number. That is, our set is not empty; it contains an object called 1 (read ``one'').
Axiom 2: For each x there exists exactly one natural number, called the successor of x, which will be denoted by x'. x’ ≠ 1
Axiom 3: We always have . That is, there exists no number whose successor is 1. That is, there exists no number whose successor is 1.
Axiom 4: If x'=y' then x=y. That is, for any given number there exists either no number or exactly one number whose successor is the given number.
Axiom 5 (Axiom of Induction): Let there be given a set M of natural numbers, with the following properties:
I. 1 belongs to M.
II. If x belongs to M then so does x'.
Then M contains all the natural numbers.
Notice that there is no mention of such things as addition or multiplication. How are these to be defined?
Peano’s Axioms of Natural numbers (N)
We assume that the set of all natural numbers has the following properties:
Axiom 1: 1 is a natural number. That is, our set is not empty; it contains an object called 1 (read ``one'').
Axiom 2: For each x there exists exactly one natural number, called the successor of x, which will be denoted by x'. x’ ≠ 1
Axiom 3: We always have . That is, there exists no number whose successor is 1. That is, there exists no number whose successor is 1.
Axiom 4: If x'=y' then x=y. That is, for any given number there exists either no number or exactly one number whose successor is the given number.
Axiom 5 (Axiom of Induction): Let there be given a set M of natural numbers, with the following properties:
I. 1 belongs to M.
II. If x belongs to M then so does x'.
Then M contains all the natural numbers.
Notice that there is no mention of such things as addition or multiplication. How are these to be defined?
Brian M. Scott
Sep 27, 2012, 1:18:28 AM9/27/12
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On Wed, 26 Sep 2012 21:21:54 -0700 (PDT), Madhur
<madhur....@gmail.com> wrote in
<news:dca7ee3d-9446-4fcc...@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
<madhur....@gmail.com> wrote in
<news:dca7ee3d-9446-4fcc...@googlegroups.com>
in alt.math.undergrad:
> The natural numbers that we use are said to be derived
> from what so called Peano's Axioms.
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.
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
wilson
Sep 27, 2012, 1:40:34 AM9/27/12
to
On Thu, 27 Sep 2012 01:18:28 -0400, Brian M. Scott <b.s...@csuohio.edu>
wrote:
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.
--
Using Opera's revolutionary e-mail client: http://www.opera.com/mail/
wrote:
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.
--
Using Opera's revolutionary e-mail client: http://www.opera.com/mail/
wilson
Sep 27, 2012, 1:42:16 AM9/27/12
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Frederick Williams
Sep 27, 2012, 2:01:51 PM9/27/12
to
x + 1 = x
x + y' = (x + y)',
x*1 = x
x*y' = xy + x.
Also the natural numbers are defined as you'd expect: 2 = 1', 3 = 2',
etc.
If you ever come across The Number Systems: Foundations of Algebra and
Analysis by Solomon Feferman, you may find it a good read. It's
published by AMS, or Chelsea, or somebody.
--
Where are the songs of Summer?--With the sun,
Oping the dusky eyelids of the south,
Till shade and silence waken up as one,
And morning sings with a warm odorous mouth.
Michael Stemper
Sep 27, 2012, 2:15:38 PM9/27/12
to
In article <5064948F...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
>Madhur wrote:
>> The natural numbers that we use are said to be derived from what so called Peano's Axioms. 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. My doubt is how exactly we reach the conclusion: 2+2=4?
>>
>> Peano’s Axioms of Natural numbers (N)
>> We assume that the set of all natural numbers has the following properties:
>Madhur wrote:
>> The natural numbers that we use are said to be derived from what so called Peano's Axioms. 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. My doubt is how exactly we reach the conclusion: 2+2=4?
>>
>> Peano’s Axioms of Natural numbers (N)
>> We assume that the set of all natural numbers has the following properties:
>> Notice that there is no mention of such things as addition or multiplication. How are these to be defined?
>
>Recursively:
>
>x + 1 = x
Did you possibly mean
>
>Recursively:
>
>x + 1 = x
x + 1 = x'
--
Michael F. Stemper
#include <Standard_Disclaimer>
Life's too important to take seriously.
Frederick Williams
Sep 27, 2012, 3:09:33 PM9/27/12
to
Michael Stemper wrote:
>
> In article <5064948F...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
> >Madhur wrote:
>
> >> Notice that there is no mention of such things as addition or multiplication. How are these to be defined?
> >
> >Recursively:
> >
> >x + 1 = x
>
> Did you possibly mean
>
> x + 1 = x'
Oh yes! Thank you.
>
> In article <5064948F...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
> >Madhur wrote:
>
> >> Notice that there is no mention of such things as addition or multiplication. How are these to be defined?
> >
> >Recursively:
> >
> >x + 1 = x
>
> Did you possibly mean
>
> x + 1 = x'
Brian M. Scott
Sep 27, 2012, 3:54:33 PM9/27/12
to
On Thu, 27 Sep 2012 01:40:34 -0400, wilson
<wins...@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.s...@csuohio.edu>
> wrote:
>> On Wed, 26 Sep 2012 21:21:54 -0700 (PDT), Madhur
>> <madhur....@gmail.com> wrote in
>> <news:dca7ee3d-9446-4fcc...@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.
> 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)
made within the framework of the axioms.
[...]
Brian
Barb Knox
Sep 28, 2012, 7:37:21 PM9/28/12
to
In article <op.wk94sq2i1hq4pq@leon-hp>, wilson <wins...@udayton.edu>
> > Addition is the function + : N 癬 N 躓 N (written in the usual infix
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum videtur.
| BBB aa a r bbb |
-----------------------------
> > 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.
> > 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)).
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum videtur.
| BBB aa a r bbb |
-----------------------------
Frederick Williams
Sep 29, 2012, 10:40:07 AM9/29/12
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numbers begin at 1.
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