*Kronecker*

Concrete Nonsense

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Construction of natural numbers:

Meaning of ‘set’ and ‘is an element of’ are intuitively clear / assumed to be understood. Peano’s axioms assume the existence of a set N and a function f : N → N satisfying the axiom:

Whenever Φ belongs to N, then f(Φ) = {Φ} belongs to N.

Then, { N, f, Φ } is the model of natural numbers including zero. The claim becomes clearer when we define

0 := Φ

1 := {Φ}

2 := {Φ,{Φ}}

3 := {Φ,{Φ},{{Φ}}}

…

and so on.

The brace bracket { } is not to confuse, but to distinguish two distinct elements of N, e.g. to distinguish 2={horse, cow} from 3={apple, mango, banana}

It can be proved that f(a) = a+1 for all a. Similarly, a+f(b) = f(a+b) and thus, a+b becomes well defined.

(Perhaps now one can intuitively define product of a and b as addition of a to a, b-number of times)

A more interesting construction of integers from natural numbers is as shown:

Introduction- Until now, we know operations + and x in natural numbers. The operation -(minus) is unknown since it is not well defined for all ordered natural numbers (eg. 1-2 (=-1) is not a natural number).

Construction- Integers are the ordered pair (a,b) (see footnote) where a & b are natural numbers satisfying some relation among them.

We now have to define when two ordered pairs (a,b) and (c,d) are equal, and operations + & x in them. We do this as follows:

@ (a,b) = (c,d) if and only if a+d = b+c

@ (a,b) + (c,d) = (a+c,b+d)

@ (a,b).(c,d) = (ac+bd,ad+bc)

Claim- There is one-to-one bijection between this ordered pairs’ set and the set of integers that we have studied since school.

Proof-

Define

-1 := (0,1)

-b := (0,b)

Then, it is easy to check that

(a,b) = a+(-b)

Denote

a-b := a+(-b) (Note that -(minus) operation is now well-defined)

The addition and multiplication axioms can now be trivially proved.

Construction of rational numbers:

The division operation is until now, not defined, since division of an integer by integer may not be an integer. We construct rational numbers as follows:

Rational numbers constitutes the set of ordered pairs (a,b) such that a,b are integers, b≠0 and where equality, addition and multiplication are defined as shown:

@ (a,b) = (c,d) if and only if a.d = b.c

@ (a,b) . (c,d) = (a.c,b.d)

@ (a,b)+(c,d) = (ad+bc,bd)

(A smart reader will quickly recognize that (a,b) is the disguised form of a/b)

Define

1/a := (1,a)

Then, it is clear that

(a,b) = (a,1).(1,b)

Define

a/b := (a,b)

Construction of Real numbers:

Until now, we could construct upto countably infinite numbers. But Cantor showed in his elegant diagonal argument, that real numbers are uncountable. Roughly speaking, it means that natural numbers are as many as integers, which are as many as rationals. But reals are ‘more’. So, I am sorry to have disappointed you in that I cannot show an ordered pair of rationals which is ‘isomorphic’ to real numbers!

Interested readers may google for Dedkind cuts or Cantor’s method of construction of real numbers.

Construction of Complex numbers:

Finally, it is here…. Complex numbers. You don’t see them, but you can feel them. You have felt them since college days… Its time to construct them!

Fortunately, the ‘cardinality’ (i.e. number of elements of set) of Complex numbers is the same as the cardinality of Real numbers, so that I once again use my faithful ordered pair (a,b), this time, of real numbers to construct complex numbers as shown:

@ (a,b) = (c,d) if and only if a = c and b = d. (Finally, no ambiguity!)

@ (a,b) + (c,d) = (a+c,b+d)

@ (a,b).(c,d) = (ac-bd,ad+bc)

(Note that rationals and complex numbers are slightly different in that ac+bd is replaced by ac-bd. Nevertheless,)

It is trivial to prove (but prove) that:

(a,b) = (a,0) + (0,1).(b,0)

Define

i := (0,1)

a := (a,0)

Then, as expected, we have (trivial to prove)

i²=-1

a+ib = (a,b)

and thus we have complex numbers.

My advice to friends who do not believe in the root of -1 is to regard i as (0,1) and use only the above axioms for complex numbers.

God made the natural numbers. Everything else is the work of man.

–Leopold *Kronecker*

Further discussion:

@ Cantor’s diagonalization method is interesting and simple. Deserves to be in THE BOOK.

@ Google Was sind und was sollen die Zahlen.

@ Exhibit a bijective map from NxN into N (now, that each contains equal number of elements)

@ I have not defined subtraction for rationals and subtraction and division for complexes. Convince yourself that I don’t need to.

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January 30, 2009 at 02:31

Ameya Shashikant Khandekarcan a set have cardinality more than that of the set of Real Numbers?(or does God prepare hell for people who ask such questions [:)]?)

And are there infinities other than No and C ? if yes, which type of infinite sets have cardinalities other than those of naturals and reals?

January 30, 2009 at 23:13

Abhishek Parab@AMEYA- It is perfectly justifiable to ask whether there are greater infinities; observe that the power-set (i.e. set of all subsets) of a set has cardinality bigger than the set itself. For instance, a finite set of cardinality n has the cardinality of its Powerset as 2^n. As a matter of fact, it can be proved that the cardinality of the power-set of Natural numbers is equal to the cardinality of Real numbers, i.e. |2^N| = |R|

The power-set of the set of Real numbers is another larger infinity, the powerset of the powerset of the set of Reals is still bigger infinity….!!

The difficulty lies in visualizing such sets although their existence can be easily understood.

And there is no hell in mathematics, at least for those who enter at their own risk 😛

January 30, 2009 at 23:49

Ameya Shashikant Khandekarokk.. i was havin trouble visualising cardinalities more than that of real… so just as we progressed from naturals to real(in terms of cardinalities), we can go higher up from reals(to where, is hard to see) …so there are infinite types of infinities .. No<2^No(=N1)<2^N1(=N2)< …

thats great…

October 24, 2010 at 23:02

Sets | Gracious Living and the Two Meat Meal[…] sets were seen by luminaries like Bertrand Russell as being the salvation of math, since you could describe basically any mathematical structure in terms of sets. Nowadays, mathematicians tend to believe that the “guts” of math really aren’t […]

October 25, 2010 at 13:46

abhishekparabThough they say that Categories have taken over Sets as the backbone of mathematics, (and hence, Concrete Nonsense), I rephrase Hilbert,

No one shall expel us from the Paradise that Cantor has created!