bonanova Posted June 12, 2013 Report Share Posted June 12, 2013 Cantor tells us that infinities come in "sizes" or cardinalities. Some infinities are "larger" than others. The smallest infinite set comprises the counting numbers, 1 2 3 4 5 ... and any other set that can be put in a 1-1 correspondence with them. Such infinities are called "countable." The rational numbers are also countable, but the real numbers are not. Occasionally someone will attempt to pair the counting number with the reals. One such scheme is shown below, and the argument goes as follows: The left column is the endless list of integers in numerical order The right column contains a decimal fraction formed by reversing the digits and placing a decimal point in front. Since the left hand list proceeds without limit it eventually contains every possible sequence of digits. Then the right hand list also catches every sequence of digits and thus represents the real numbers less than unity. Numbers greater than unity can be constructed by sets with a shift in the position of the decimal point. The union of two or more sets of equal cardinality has the same cardinality as its component sets. This correspondence is sufficient to prove the two sets have the same cardinality. The argument is obviously flawed, and the question is to expose the flaw. Here are the lists: Integers Decimal fractions 1 .1 2 .2 3 .3 4 .4 . . . . . . 10 .01 11 .11 12 .21 13 .31 . . . . . . 100 .001 101 .101 102 .201 103 .301 . . . . . . 1234 .4321 . . . . . . Quote Link to comment Share on other sites More sharing options...

0 James33 Posted June 12, 2013 Report Share Posted June 12, 2013 The left hand list only contains the set of terminating decimals, for example it does not contain sqrt(2)/2 Quote Link to comment Share on other sites More sharing options...

0 phil1882 Posted June 12, 2013 Report Share Posted June 12, 2013 (edited) i tend to agree with James. however let's try this construction. first, I'll use binary, so only 1 and 0. then I'll construct a transcendental number. let 0->01 and 1 -> 0 0 01 010 01001 01001010 repeat indefinitely, use this number as our 1. 1 -> 0.01001010.. we know the rational numbers are countably infinite, and doubling the rational numbers is not enough to change countability. so, i propose we rotate between the list of rational numbers, and xor the list rational numbers with this transcendental number. prove this doesn't go through every positive number. Edited June 12, 2013 by phil1882 Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted June 13, 2013 Author Report Share Posted June 13, 2013 The left hand list only contains the set of terminating decimals, for example it does not contain sqrt(2)/2 Right. Column 2 is only a subset of the rationals. Quote Link to comment Share on other sites More sharing options...

0 phil1882 Posted June 14, 2013 Report Share Posted June 14, 2013 does no one want to take on my challenge ???!!! :-) Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted June 14, 2013 Author Report Share Posted June 14, 2013 i tend to agree with James. however let's try this construction. first, I'll use binary, so only 1 and 0. then I'll construct a transcendental number. let 0->01 and 1 -> 0 0 01 010 01001 01001010 repeat indefinitely, use this number as our 1. 1 -> 0.01001010.. we know the rational numbers are countably infinite, and doubling the rational numbers is not enough to change countability. so, i propose we rotate between the list of rational numbers, and xor the list rational numbers with this transcendental number. prove this doesn't go through every positive number. Which rationals XOR with 0.01001010... to give e/3 and pi/4? Quote Link to comment Share on other sites More sharing options...

0 phil1882 Posted June 14, 2013 Report Share Posted June 14, 2013 i might not be able to find the exact coordinate since 0.01001010 is transcendental, and I'm not quite sure how to map one transcendental number to another, but i can give better and better coordinates depending on your approximation of e/3 and pi/4. what i can guarantee is that there are numbers that are as random as e/3 and pi/4. Quote Link to comment Share on other sites More sharing options...

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## bonanova

Cantor tells us that infinities come in "sizes" or cardinalities.

Some infinities are "larger" than others.

The smallest infinite set comprises the counting numbers, 1 2 3 4 5 ...

and any other set that can be put in a 1-1 correspondence with them.

Such infinities are called "countable."

The rational numbers are also countable, but the real numbers are not.

Occasionally someone will attempt to pair the counting number with the reals.

One such scheme is shown below, and the argument goes as follows:

The left column is the endless list of integers in numerical order

The right column contains a decimal fraction formed by reversing the digits and placing a decimal point in front.

Since the left hand list proceeds without limit it eventually contains every possible sequence of digits.

Then the right hand list also catches every sequence of digits and thus represents the real numbers less than unity.

Numbers greater than unity can be constructed by sets with a shift in the position of the decimal point.

The union of two or more sets of equal cardinality has the same cardinality as its component sets.

This correspondence is sufficient to prove the two sets have the same cardinality.

The argument is obviously flawed, and the question is to expose the flaw.

Here are the lists:

Integers Decimal fractions

1 .1

2 .2

3 .3

4 .4

. .

. .

. .

10 .01

11 .11

12 .21

13 .31

. .

. .

. .

100 .001

101 .101

102 .201

103 .301

. .

. .

. .

1234 .4321

. .

. .

. .

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