Understanding Numbers: Definition, Symbols, and Cardinality Explained

[C0nfused]

Hi everybody,
Is there any exact definition of numbers? I mean, what is a number? What are the symbols "0","1",2",...,"9" and what do they mean? Natural numbers are actually cardinals? And how do we define cardinals and ordinals? I keep asking about naturals because we started from them right?
NOTE: I have used numbers during my whole life. But although it seem that I have understood what they are and , of course use them almost mechanically, I haven't come across an exact definition (if there is any) of them. I have read some things about the Peano axioms but it wasn't very clear(it was a brief description of them) so I haven't found the answer yet

 

[matt grime]

The Natural numbers can be thought of as cardinals of finite sets, if you so wish. This is Peano arithmetic.

Simpler, in my view is to take the integers as the smallest ring without torsion. Ie a set with two operations + and *, with an additive identity we write as 0, and a mulitplicative one we write as 1. Then the other elements are 1+1, 1+1+1, 1+1+1+1, etc. the numerals 2,...,9 are then simply ways of representing these sums more elegantly. The naturals are then the positive part of this. You could start from naturals and say 1 is a symbol, and define addition etc in accordance with it.

Once you've got the integers you construct the rationals, and then you construct the reals from them.

I suppose even more formally, one starts with an axiom system for "naturals", and then the set cardinals are a model of the axioms, with union and product as the addition and multiplication.

Look up Peano Axioms or Peano Arithmetic.

 

[C0nfused]

I think cardinals are defined before defining naturals. Am i right? So is there any definition of cardinals? And what's their difference from ordinals?

 

[matt grime]

No, you are not correct. Natural numbers have been known and used for millenia, cardinals only for a fraction of that time. Really, there is no before or after; they are different. Cardinals are a product of set theory, the natural numbers do not depend on set theory.

Cardinals are a model of something that satisfies the abstract notion of "being the same as the natural numbers". In particular, not every result about the natural numbers can be deduced from Peano Arithmetic.

The model using sets is a good thing as it shows the natural numbers, which we usually do not bother to define since we all know what they are, can have a firm set theoretic foundation.

An oridinal is well ordered set, a cardinal isn't.

 

[C0nfused]

"The model using sets is a good thing as it shows the natural numbers, which we usually do not bother to define since we all know what they are..."

We know what natural numbers but we can't actually give a definition of them? I mean ,as you said, naturals existed before Peano. So it's all comes to our primitive need to create a means of counting and measuring . That's what 0,1,2,... actually are? And we have then taken naturals to a theoritical basis with algebra, and extended them to integers, rationals etc?

 

[matt grime]

We do not need to give a definition of them. At some point you have to stop referring back to simpler and simpler things, and say "this is the basic that we do not need to question". We intuitively know what numbers are, and we also know that there is a firm footing on which they can be put if we so desire. We really don't care about the foundations in ways that may make the beginning student, or the cynical outsider very surprised. 0 didn't come from a primitive need to count. Nor did place holder arithmetic (look at the romans). I know what one is, i know that if i add one to one i get something that i call two, i know how to repeat that. I know what the natural numbers are. They are the system that contains an element 1, and an associative binary operation +, and all the finite results of the binary operation applied to 1 to itself. Like I said in the first post. We can extend to * and a useful extenasion is a symbol 0 that is an identity etc. We can use them as a system for counting things (though not all things are countable like this), and cardinals give a good model of them. Things happen by DEFINITION.

 

[mathwonk]

to know what numbers are you need to ask what you intend to use them for.

if you mean to count finite sets, then you want the finite cardinal numbers, i.e. the non negative integers.

The number "five" for instance should be some object, real or imaginary, that can be assigned to each set with five elemens, and should distinguish such sets from sets with other numbers of elements.

One method would be to select a special set having five elements, such as the fingers on your right hand (assuming no accidents have occurred), and call that set the number "five". Then to decide whether any other set contained five elements, one would go and get this set, and try to match its elements bijectively with those of the given set.

One could imagine a choice of standard sets being made, one set for each number, and then storing these sets in the national bureau of standards in washington.


For example, the most basic objects in mathematics are usually considered to be sets, and the simplest set is the empty set ø, so one could call the number zero simply the empty set ø. then the number one is defined as the set {ø} containing precisely the empty set.

then the number "two" is the set containing all the previous numbers, i.e. the number two is the set { {ø}, ø }. Notice it has in fact two elements. Then each number n is the set containing the numbers 0,1,2,3,...,(n-1).


What we do in practice is rather assign a word to each set, or rather a procedure for generating such words from a few basic ones, and we memorize a way of reciting these words in order.

A set together with an ordering in which every non empty subset has a least element, represents not just a cardinal number but also an "ordinal" number.

Notice the sets above used to define cardinals are well ordered by "elementhood", hence also represent ordinal numbers. I.e. one can say in this model of ordinal numbers that a given ordinal number is less than another, if it belongs to the other one, as an element.

One can simply define an ordinal number as a set which is well ordered by the relation of elementhood. i.e. an ordinal is less than another if it is an element of the other, and a set is an ordinal number if the binary relation "is an element of" defines a well ordering on the set.

Russell the logician, defined cardinal numbers as the equivalence classes of all sets under bijection. I.e. the cardinjal number of a given set is simply the class of all other sets which are bijectively equaivalent to it. How handy this is for calculating, is readily visible. But then he was not a mathematician.


I live mostly by Hausdorff's maxim that we do not care what numbers are, only how they behave.

 

[mathwonk]

to distinguish ordinals from caridnals, notice that the set of all positive integers, although having a fixed cardinality, can be well ordered in many ways. E.g. we could require all the even numbers to be greater than all the odd numbers. then we have a well ordering of the set of positive integers, in which there exists an infinite number of elements all less than 2, namely all the odd numbers. this represents a different ordinal number than does the usual ordering of the natural numbers.

 

[mathwonk]

so the natural numbers in their usual ordering represent the smallest infinite ordinal.

then one wants the negative numbers, which one defines in terms of the non negative integers by introducin a sign to go with each positive integer, and calling the combination negative. then one introduces rational numbers as usual, as equaivelence classes of pairs n/m under cross mulotiplication.

then one introduces real numbers as say infinite decimals, modulo the usual rule that one ending in all 9's equals another one ending in all 0's.


if one want to solve algebraic equations, one then creates complex numbers,etc...

 

[robert Ihnot]

Mathwonk: Russell the logician, ...But then he was not a mathematician.

Perhaps this is a little off the subject, but I am not sure Russell should be described as "Not a mathematician." My professor of mathematical logic certainly thought he was a mathematician. Here is one of Russell's quotes:

At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness. Bertrand Russell

See: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Russell.html

Also, I see he is listed under "Mathematicians born in Wales":
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Russell.html

 

[C0nfused]

Thanks for all your answers. They were really helpful

 

[mathwonk]

Robert, a mathematician is not to me someone who enjoys studying mathematics, but someone who also produces mathematics. Have you ever heard of Russell's theorem, or a problem that Russell solved, or any mathematical idea that Russell introduced? Your professor of mathematical logic may also have not been a mathematician; mine was not in my opinion, although he graduated summa in math as an undergraduate.

Also some of the authors of the textbooks people study on calculus and other topics, are by people who are only barely mathematicians. You may of course disagree, but now you can understand what I meant. By analogy, a grammarian is not necessarily a poet. He studies the structure of language, but may not attempt to create anything beautiful using that language.

For example, it may be that Euclid was not a mathematician, but a textbook writer, recording other people's ideas. By all accounts, Eudoxus was a mathematician, and also Archimedes, however.

But we digress.

As to your last argument, I post here for future reference:

"Mathematicians born in North Carolina"
1. Michael Jordan


(I have heard he was a math major in college, or at least math minor.)

Well after perusing some web pages, I have notioced that some logicians occur among mathematics departments as professors. so I guess i am just narrow minded.

 

[robert Ihnot]

My mathematical logic professor was a mathematician by his own reckoning, and was paid by the math department. "A working mathematician" was his term for someone who published frequently.

My Logic Professor had utter distain for Philosophy, which he felt spent its time on useless questions like, "What is the difference between the Morning and the Evening Star?" As for Newton, he refused to consider him a Mathematician because, "He was only interested in Physics." (That is another type of definition, simply to employ Mathematics does not a Mathematician make.)

However, on the strength of his Mathematics and urging of Barrow, Newton replace Barrow at Cambridge, who went on to become the Chaplain to the King.

Flexible people besides Russell come to mind: Alan Turing, who's website declares that he was in Computer Science, Mathematics, Philosophy, and Code Breaking.

 

[mathwonk]

to say Newton was not a mathematician already disqualifies your subject from serious consideration on the topic, in my mind.

inventing calculus seems a reasonable criterion for admission to the club. obviouisly your logic professor had strong, but fairly unique, opinions.

for him to say that he considered himself a mathematician but not Newton, is also a rather interesting remark.

 

[CrankFan]

"Mathematicians born in North Carolina"
1. Michael Jordan

Haha, are you serious? That appears to be true... at least according to one website which says that he was a mathematics major until his Junior year.

I never would have guessed it.

It looks like David Robinson was also was a mathematics major, and he stuck with it long enough to get a BS.

 

[mathwonk]

kind of humbling isn't it, to think that Michael Jordan and David Robinson were among other things, possibly also better mathematicians than us.

by the way, this is simply more evidence that i am never wrong. and i never lie to you.