August 1st, 2010, 01:25 AM  #1 
Senior Member Joined: Aug 2010 From: Germany Posts: 132 Thanks: 1  Cantorīs diagonal argument
To illustrate Cantorīs diagonal arument, we can also treat a restricted amount of numbers, e.g. the tenthousandths between 0 an 1. Since we are treating a restricted amount of numbers we will restrict our sequence to 4 numbers. To illustrate it: z1 = 0.a11 a12 a13 a14 z2 = 0.a21 a22 a23 a24 z3 = 0.a31 a32 a33 a34 z4 = 0.a41 a42 a43 a44 x = 0.x1 x2 x3 x4 If aii = 5, then xi = 4, otherwise xi = 5. using real values: z1 = 0.2327 z2 = 0.5249 z3 = 0.6670 z4 = 0.8221 x = 0.5555 Now we can say there is a new number x. But we can also see, that this number is only new, because we were restricted to list just 4 numbers, and not all the tenthousandths between 0 and 1 (0.0000, 0.0001 ... 0.9999). No matter which 4 numbers we choose, there will always be a new number x. If we are going to the limits now and treat an infinite amount of numbers between 0 and 1, the principle doesnīt change. We are always restricted to a finite sequence of numbers, even if the sequence is approaching infinity. And if we get a "new" number, then bound to the fact, that we are not able to list an infinite sequence of numbers. So, this diagonal argument doesnīt prove, that there is more real numbers than natural numbers. But this argument is basic to the set theory . 
August 1st, 2010, 06:36 AM  #2 
Senior Member Joined: Apr 2010 Posts: 451 Thanks: 1  Re: Cantorīs diagonal argument
If we could find some theorems ,axioms ,definitions which we could explicitly mention and base our arguments ,then Cantor's construction could be more convincing . But then again to say that Cantor's construction is wrong we should have laws and principals upon which we could base our arguments 
August 1st, 2010, 09:18 AM  #3  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Cantorīs diagonal argument Quote:
Quote:
http://us.metamath.org/mpeuni/canth2.html The axioms needed in this formalism are Propositional calculus: ax1 ax2 ax3 axmp Predicate calculus: ax5 ax6 ax7 axgen ax8 ax9 ax10 ax11 ax12 ax13 ax14 ax17 ZF: axext axrep axun axpow axreg Notice that the axiom of infinity ("the set Z of integers exists") isn't even needed!  
August 1st, 2010, 02:31 PM  #4  
Senior Member Joined: Aug 2010 From: Germany Posts: 132 Thanks: 1  Re: Cantorīs diagonal argument Quote:
z1 = 0.a11 a12 a13 a14 a15 z2 = 0.a21 a22 a23 a24 a25 z3 = 0.a31 a32 a33 a34 a35 z4 = 0.a41 a42 a43 a44 a45 z5 = 0.a51 a52 a53 a54 a55 The principle is still the same: The resulting number x is only new, because we were restricted to list just 5 numbers, and not all the hundredthousandths between 0 and 1 (0.00000, 0.00001 ... 0.99999). Feel free to enhance the list by another number. Thus we are no longer treating the hundredthousandths but the millionths. z1 = 0.a11 a12 a13 a14 a15 a16 z2 = 0.a21 a22 a23 a24 a25 a26 z3 = 0.a31 a32 a33 a34 a35 a36 z4 = 0.a41 a42 a43 a44 a45 a46 z5 = 0.a51 a52 a53 a54 a55 a56 z6 = 0.a61 a62 a63 a64 a65 a66 The principle is still the same: The resulting number x is only new, because we were restricted to list just 6 numbers, and not all the millionths between 0 and 1 (0.000000, 0.000001 ... 0.999999). Where do you want me to stop? At which stage do you think the principle changes? You can see, if you go on, the list is approaching infinity as well as the digits.  
August 1st, 2010, 03:32 PM  #5  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Cantorīs diagonal argument Quote:
 
August 1st, 2010, 03:51 PM  #6  
Senior Member Joined: Aug 2010 From: Germany Posts: 132 Thanks: 1  Re: Cantorīs diagonal argument Quote:
 
August 1st, 2010, 06:10 PM  #7  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Cantorīs diagonal argument Quote:
 
August 2nd, 2010, 12:21 AM  #8  
Senior Member Joined: Aug 2010 From: Germany Posts: 132 Thanks: 1  Re: Cantorīs diagonal argument Quote:
 
August 2nd, 2010, 05:10 AM  #9 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Cantorīs diagonal argument
Look, let's just make this simple (since I don't understand your philosophical objection). Do you think that Cantor's theorem is wrong? Then you should be able to make a list with all real numbers on it. Then we can apply Cantor's diagonalization to it and see what new number is produced. It's easy to list all the rational numbers, and you can even list all algebraic numbers if you like. But the proof shows that you can't list all real numbers (and hence can't even list the transcendentals). 
August 2nd, 2010, 05:24 AM  #10 
Math Team Joined: Apr 2010 Posts: 2,770 Thanks: 356  Re: Cantorīs diagonal argument
You can list all natural numbers: 0, 1, 2, 3 and so on. You will not forget any number between the first and the last one you have mentioned. You can't list all real numbers. If you would start (from 0) and would say a number larger than 0, for example , than one will say: you have forgotten This is not formal though.. it is to represent the idea. Hoempa 

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