My Math Forum Cantor's Diagonal Argument. Infinity is Not a Number

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March 18th, 2016, 05:15 AM   #21
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Quote:
 Originally Posted by zylo In ZF, the "axiom of infinity" defines infinity as an inductive set. https://en.wikipedia.org/wiki/Axiom_of_infinity
The Axiom of Infinity does not define infinity. It asserts the existence of an infinite set from which we can extract the Natural numbers.

We then define $D=\{0,1,2,3,4,5,6,7,8,9\}$ and define some infinite sequences $S_n = \{a_{n,m} \in D \, | \, m \in \mathbb N\}$. And we define a set of sequences $S = \{S_n \, | \, n \in \mathbb N\}$.

Now each $S_n$ has a trivial bijection with the Natural numbers $m \mapsto a_{n,m}$ and $S$ also has a trivial bijection with the Natural numbers $n \mapsto S_n$.

Thus for any $k \in \mathbb N$ the element $a_{k,k}$ exists and we can define $T = \{b_k \in D \, | \, k \in \mathbb N, \, b_k \ne a_{k,k}\}$. $T$ is clearly a sequence and is not equal to any of the $S_k$ because $b_k \ne a_{k,k} \in S_k$. Thus $T \not \in S$ and Cantor is proved.

What do you not follow from the above? It really is very simple.

Last edited by skipjack; March 19th, 2016 at 04:26 AM.

March 21st, 2016, 08:14 PM   #22
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Quote:
 Originally Posted by zylo CANTOR'S DIAGONAL ARGUMENT: The set of all infinite binary sequences is uncountable. Let T be the set of all infinite binary sequences. Assume T is countable. Then all its elements can be enumerated: 1 0 0 1 1 0............... 0 0 1 0 1 1............... 0 1 0 1 0 0.............. ............................ Let s be the binary sequence consisting of the complemented underlined digits: s = 0 1 1................... s is different from every member of the list and s belongs to T. Contradiction. Therefore T isn't countable.
s doesn't exist because the list doesn't end.

Last edited by skipjack; March 21st, 2016 at 09:19 PM.

March 21st, 2016, 09:32 PM   #23
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Quote:
 Originally Posted by zylo s doesn't exist because the list doesn't end.
What exactly is the logical principle you're using? If I claim that the number 42 doesn't exist because the list of the natural numbers doesn't end, am I also correct? If I claim that $\pi$ doesn't exist because its value 3.14159... contains an endless sequence of digits, am I also correct?

March 21st, 2016, 10:12 PM   #24
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Quote:
 Originally Posted by skipjack What exactly is the logical principle you're using? If I claim that the number 42 doesn't exist because the list of the natural numbers doesn't end, am I also correct? If I claim that $\pi$ doesn't exist because its value 3.14159... contains an endless sequence of digits, am I also correct?
42 is a number.

pi exists as a procedure for calculating its digits. It doesn't exist as a sequence of digits for the same reason that infinity is not a number.

 March 21st, 2016, 10:42 PM #25 Global Moderator   Joined: Dec 2006 Posts: 18,956 Thanks: 1603 You haven't answered my first question. What exactly is the logical principle you are using? Why is it relevant that 42 is a number? Every natural number is a number. The list of natural numbers is endless. By your apparent reasoning, it would follow that 42 doesn't exist.
March 22nd, 2016, 05:36 AM   #26
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Quote:
 Originally Posted by skipjack You haven't answered my first question. What exactly is the logical principle you are using? Why is it relevant that 42 is a number? Every natural number is a number. The list of natural numbers is endless. By your apparent reasoning, it would follow that 42 doesn't exist.
It takes two digits to display (represent) 42. Two is a natural number.
It takes infinity digits to display (represent) pi. Infinity is not a natural number.

March 22nd, 2016, 05:58 AM   #27
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Quote:
 It takes infinity digits to display (represent) pi. Infinity is not a natural number.
Agreed, infinity is not a natural number.

However, it only takes two letters of the Roman alphabet or one of the Greek, to represent pi.

When you have met some more maths, you will discover there are many things in maths that cannot be adequately described using more primitive antecedents.

For instance, many functions are defined by differential equations, and there is no closed form solutions for some of these in terms of elementary polynomials, trig functions etc.
They are actually new functions.
For example Bessel functions, elliptic functions and many more.

You should rejoice that new numbers, new functions, new ideas makes maths all the more interesting.

Last edited by skipjack; March 22nd, 2016 at 12:49 PM.

March 22nd, 2016, 01:06 PM   #28
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Quote:
 Originally Posted by zylo It takes two digits to display (represent) 42. Two is a natural number. It takes infinity digits to display (represent) pi. Infinity is not a natural number.
You still haven't answered my question: What exactly is the logical principle you are using?

Instead, you make a statement about 42 and another about $\pi$. Those statements don't explain the logical principle that you were applying.

I used 42 as an example of a natural number, and your statements amount to observing that it's a natural number, not infinite - it remains the case that the list of natural numbers is endless and 42 is a natural number, so your apparent logic would imply that 42 doesn't exist.

 March 22nd, 2016, 01:27 PM #29 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,364 Thanks: 100 A finite sequence of natural numbers is defined: 42 An infinite sequence of natural numbers is undefined: pi=3.14159....... What is .......? The best you can do is n-digit approximation or induction. EDIT: 3.1416 gallons exist. pi gallons do not exist. Last edited by zylo; March 22nd, 2016 at 01:44 PM.
 March 22nd, 2016, 02:45 PM #30 Global Moderator   Joined: Dec 2006 Posts: 18,956 Thanks: 1603 It's acceptable to use representation or abbreviation. You refer to n-digit approximation as the best one can do, but that's just an observation about a particular representation. The mathematical constant commonly referred to as $\pi$ has mathematical existence and its exact value can be defined (as explained below). Your own use of the wording "n-digit approximation" implicitly accepts that some exact value that mathematically exists is being approximated, and that each digit of its representation can be precisely defined. Similarly, the sequence 1, 2, 3, 4, 5, 6, ... (continued without end) has mathematical existence, but one needs some method of abbreviation to refer to it. Even a single natural number requires some accepted way of representing it, such as its usual decimal representation. When you see "42", you know what it means. Another representation is "41.999999...", and both representations are of the same value that mathematically exists, but needs some method of representation so that one can refer to it. One can define $\pi$ as a circle's circumference divided by the circle's radius. Are you claiming that a circle's radius and circumference don't exist, or that if the radius exists the circumference doesn't? There's no logical principle that makes one representation "better" than another (as distinct from more convenient than another in particular circumstances), so you're still not explaining what logical principle you're applying that lets you deduce that s doesn't exist, but wouldn't also let you deduce that some arbitrarily chosen natural number doesn't exist.

 Tags argument, cantor, diagonal, infinity, number

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