Mathematical definition of number?

  • #26
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I would agree with HoI's reply. Numbers, as mathematical expressions and statements in general, are independent of notation (be it symbols, numeral systems, or different human languages). The statement encapsulated by the statement in decimal numerals: 1+1=2 is still true in binary: 1+1=10.

You can define numbers starting with any base, but individual definitions are needed for each digit that are more basic than the notion of a set. It doesn't matter which number system you use because they all equivalent notations of the same thing.
 
  • #27
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You can define numbers starting with any base, but individual definitions are needed for each digit that are more basic than the notion of a set. It doesn't matter which number system you use because they all equivalent notations of the same thing.

Why do you believe that digits are more basic than sets?
 
  • #28
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You can define numbers starting with any base, but individual definitions are needed for each digit that are more basic than the notion of a set. It doesn't matter which number system you use because they all equivalent notations of the same thing.

Why do you believe that digits are more basic than sets?

lolgarithms said:
(But then we cannot define our notations without using another notation. We can say that "the symbol 2 at n places left of the decimal is defined as the number two times ten to the (n-1)th power", but it is a definition in English, just another notation.)
Plus, how do you define "two" using only sets and ordering? -> Then how do you define divition and rationals?
then,yeah, i would agree that numbers are fundamental in some respect.
 
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  • #29
HallsofIvy
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Plus, how do you define "two" using only sets and ordering? -> Then how do you define divition and rationals?
then,yeah, i would agree that numbers are fundamental in some respect.
The standard method is this: The number "0" is the empty set: {}. The number "1" is the set whose only member is the empty set: { {} }. The number "2" is the set whose only members are 0 and 1. In general, the "successor" to a number, n, is the set containing the set "n" and all of its members.

Once you have done that, you can show that these "numbers" satisfy Peano's axioms and prove all the properties of the non-negative integers from that. After defining the non-negative integers, you look at the set of pairs of non-negative integers and define the integers to be equivalence classes of pairs of non-negative integers with equivalence relation "(a, b) is equivalent to (c, d) if and only if a+ d= b+ c". We define the addition of two integers by: if x and y are integers (equivalence classes of pairs of non-negative integers) choose a 'representative' from each class, that is (a,b) from x, (c,d) form y. x+ y is the equivalence class containing the pair (a+c, b+d). Of course, you would have to prove that choosing different pairs from the same two equivalence classes you would wind up with the same equivalence class. Similarly, xy would be defined as the equivalence class containing (ac+bd, ad+ bc). If, for some (a,b) in equivalence class x, [itex]a\ge b[/itex], you can show that every pair has first member larger than the second member, and, in fact, "first number minus second" number is always the samd so we can associate that equivalence class with the non-negative number a-b. If, for some (a,b), b> a, that is true for all pairs in the class and we associate that with the number -(b-a), the additive inverse of b- a.

To define rational numbers, consider the set of pairs (a, b) where a is an integer and b is a positive integer and use the equivalence relation (a, b) is equivalent to (c, d) if and only if ad= bc. The rational number "m/n" would be the equivalence class containing (m, n).

There are a number of different ways of defining the real numbers. For example, consider the set of all Cauchy sequences of rational numbers and use the equivalence class [itex]\left{a_n\right}[/itex] is equivalent to [itex]\left{b_n\right}[/itex] if and only if the sequence [itex]\left{a_n- b_n\right}[/itex] converges to 0. Another way is to use increasing sequences having an upper bound instead of Cauchy sequences and the same equivalence.
For example, the equivalence class containing the sequence {3, 3.1, 3.14, 3.141, 3.141, 3.1415, ...} corresponds to the number "[itex]\pi[/itex]". A completely different way is to define the real numbers to be "Dedekind cuts", sets of rational numbers satifying
(1) A cut is non-empty: there exist at least one rational number in the set
(2) A cut is not all rational: there exist at least one ration number not in the set
(3) There is no largest number in the set
(4) If b is in the set and a< b, then b is also in the set.
For example, the set of all rational numbers, whose square is less than 2, together with all negative rational numbers, is a cut and corresponds to the irrational number "[itex]\sqrt{2}[/itex]".

Does that satisfy you? These and proofs of all properties of numbers from them should be "five finger exercises" for any mathematician. But I still assert that the concept of "number" itself is an "undefined term". These are all definitions giving specific "instances" of "number".
 
  • #30
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a number that is in the naturals is simply defined as the set (containing members from zero up to n-1) itself, and not as the set's cardinality?
 
  • #31
HallsofIvy
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Yes, because you can't define "cardinality" until after you have the natural numbers.
 

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