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

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 Frederick Williams Posts: 2,164 Registered: 10/4/10
Re: 2+2=4 ... How?
Posted: Sep 27, 2012 2:01 PM

>
> 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?
>
> Peanoâs 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'. xâ â 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?

Recursively:

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

--
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.

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