numbers as sets?

[alan2here]

An article I read in focus magazine stated that numbers could be represented only using sets as proposed by Von Neumann.

 

The concept as stated by the article is that "The number 1 is defined simply as the set containing just one thing: the Empty set. The next number, 2, is then the set containing 2 things: the Empty set, plus the set representing the number 1 - and so on, each successive number being built out of the previous set, plus the empty set."

 

Therefore

 

1 = {}

2 = {{}, {}}

3 = {{}, {{}, {}}}

 

and so where [math]s_{n}[/math] is the set that represents the number n and n > 0

[math]s_{n}[/math] = {{}, [math]s_{n-1}[/math]}

 

Is this correct and how is this abstraction helpful compared with for example?

 

3 = {{}, {}, {}}

or

3 = {1, 1, 1}

or

3 = 3

[Mr Skeptic]

The idea works OK for integers, but for real numbers not so much. Think of it this way: one is an arbitrary definition, our starting number. If you look up the numbers in the dictionary, it says it's the number that comes after the previous number. Ie, 2=1+1, 3=2+1, 4=3+1. This can be gotten from the successor function. So in this definition, each number is the sum of the previous number and 1, and in this case 1={}.

Edited April 12, 2010 by Mr Skeptic

[the tree]

Sets don't contain duplicates, so {1,1,1} = {1} and { {},{},{} }= { {} }.

 

Correctly,

0 = {} the empty set

1 = { {} } the set containing the empty set

2 = { {} , {{}} }

3 = { {} , {{},{{}}} } et cetera

 

The point is that you only need the definition of a set, the successor function as defined and one axiom 'there exists an empty set' to get a definition for all the natural numbers.

 

Once you've got a solid definition like that, it's easier to define addition, multiplication etcetera.

 

 

The idea works OK for integers, but for real numbers not so much.

Well, you can eventually define the reals working this way, it just takes a very long time and isn't worth it. Edited April 12, 2010 by the tree
Consecutive posts merged.

[khaled]

here is a way to make this theory fits for real numbers,

 

i just thought of it,

 

n > 0

 

0 = ..{ }

1 = { { } }

2 = { { }, {{}} }

3 = { { }, {{},{{}}} }

4 = { { }, {{},{{},{{}}}} }

 

guess you notice that there is always a single empty set at the beginning of the global set

 

so, we can use it to define precision as follows,

note: i used ,{} to distinguish 0 from 1 ...

 

0.0 = { }

0.1 = { ,{} }

0.2 = { ,{},{} }

0.03 = { ,{ {},{},{} } }

0.004 = { ,{{ {},{},{},{} }}

 

example:

 

3.1235 = { { {}, { {},{}, { {},{},{}, { {},{},{},{},{} }}}}, {{},{{}}} }

................. DECIMAL ..................... INTEGER ....

 

so degree of precision is specified in levels of { }'s into the single set

 

PS, i should get a certification for this, you know

[alan2here]

I still don't quite get why turning numbers into a fractal is useful. Why not use binary representation to define numbers formally?

 

A variation of this almost seems useful in certain OO programming context where 3 = {{}{}{}} would probably have benefit in terms if abstraction, but would be very inefficient.

 

A fraction, or a floating point like implementation seems like a good way of representing a real number.

Edited April 22, 2010 by alan2here

[khaled]

but in the end, this solution does not solve the main problem

 

saving space .. no,

 

0 = __

1 = {}

2 = {}{}

3 = {}{}{}

4 = {}{}{}{}

5 = {}{}{}{}{}

6 = {}{}{}{}{}{}

7 = {}{}{}{}{}{}{}

8 = {}{}{}{}{}{}{}{}

9 = {}{}{}{}{}{}{}{}{}

 

i think i should waste time on more beneficiary ...

[alan2here]

I don't think you can beet [math]101_{2}[/math] = [math]5_{10}[/math] for storing integers in binary from a space saving point of view.

Edited April 22, 2010 by alan2here

[the tree]

Of course the point here is nothing to do with saving space or efficiency, the whole idea obviously gloriously inefficient and unnecessary. Formality at this level really exists for it's own sake.

 

If you give a separate definition for each number then you are claiming "there exists a '1'", "there exists a '2'", "there exists a '3'" etc and you need infinite axioms just to get an infinite amount of natural numbers - which just wont do when trying to create a well defined system.

 

With the Von Neumann system all you need is "there exists an empty set" and "every set has a containing set".