My Math Forum Cantor's Diagonal Argument and Infinity

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March 19th, 2018, 06:49 PM   #11
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Quote:
 Originally Posted by Micrm@ss The ordinal $\omega +1$
Well to be fair, a sequence by definition has order type $\omega$, yes? $1, 2, 4, 5, 6, \dots, 3$ is not a sequence. By my understanding, anyway.

March 19th, 2018, 06:51 PM   #12
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Quote:
 Originally Posted by zylo If infinite means endless, you can never get to the end of CDA.
This might have merit if Cantor had described an algorithm. He didn't. He simply defined an infinite sequence. If you accept that the natural numbers can be defined inductively and taken as a set - a precondition of Cantor's work - you must also accept his definition of a particular infinite sequence.

His definition is no different to defining a sequence $(s_n)$ as $s_n= n \pmod{2}$ or a function $f(x)=x^2$. The sequence is well-defined for every $n$ and requires no iterative process to produce any element you like. Similarly for the function. It's defined perfectly well for all $x$ with no need for any iterative process.

March 19th, 2018, 07:07 PM   #13
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Quote:
 Originally Posted by Maschke Well to be fair, a sequence by definition has order type $\omega$, yes? $1, 2, 4, 5, 6, \dots, 3$ is not a sequence. By my understanding, anyway.
True. I might still call it a transfinite sequence though. But while you are right, the original post didn't say anything about sequences or binary digits, it just said infinity is endless, which I showed is wrong. A sequence is almost by definition endless, true.

 March 19th, 2018, 07:33 PM #14 Global Moderator   Joined: Dec 2006 Posts: 20,288 Thanks: 1968 The notation 1, 2, 4, 5, 6, . . . , 3 leaves it unclear what numbers are implied by the dots.
March 19th, 2018, 07:45 PM   #15
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Quote:
 Originally Posted by skipjack The notation 1, 2, 4, 5, 6, . . . , 3 leaves it unclear what numbers are implied by the dots.
That's an example of $\omega + 1$. All the naturals are there and $3$ is stuck at the end. By definition it's not a sequence. It's an example of an infinite order type -- a well-order in fact -- with a smallest and largest element. I'm not sure how else I'd notate it.

March 19th, 2018, 08:09 PM   #16
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Quote:
 Originally Posted by zylo If infinity means endless, you can never get to the end of CDA.
Then it's a good thing this isn't what infinity means! Even better than Cantor's argument never mentions infinity nor is it required. Cantor proves that if you provide a countable list of reals (which may be infinite but isn't required), then you have missed at least one. Nobody has to "get to the end" of anything or "go" anywhere.

By all means though, please feel free to redefine every definition you don't like, replace it with a definition you do like, and then use these bad definitions to argue against proven theorems as if you are still talking about the same objects.

It's better than the alternative, which is to stop making all these ridiculous threads and open a gd book.

Last edited by skipjack; March 20th, 2018 at 05:15 AM.

 March 19th, 2018, 08:33 PM #17 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,599 Thanks: 2587 Math Focus: Mainly analysis and algebra Let's face it: if the OP were even close to being a valid criticism of Cantor, it would be astonishing, not to say utterly negligent of all mathematicians over the last couple of hundred years, that it hadn't been raised before. Even the tiniest grain of self-awareness and critical thinking would have killed this thread before it even got started.
March 20th, 2018, 07:08 AM   #18
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Quote:
 Originally Posted by zylo Infinity, infinite, and endless are synonymous. But if it makes you happy: If infinite means endless, you can never get to the end of CDA.
Infinite, adjective: (1) extending indefinitely, endless.
Infinity, noun: (1a) the quality of being infinite.
While related, these are not quite the same thing.
Axiom of Infinity: THERE EXISTS a set $\mathbb N$ that:
1. Contains the number 1, and
2. If it contains the number n, it also contains the number n+1.
While it is an abstraction, so is the existence of all other mathematical objects. This one's existence is no more abstract than the concept of a point ("that which has no extent") in geometry. It escapes you because you keep trying to treat it as something it is not.

+++++

You can choose not to accept this Axiom in your preferred Mathematics if you want to. But then, you cannot discuss whether Cantor's proposition ("there exists an infinite set $\mathbb M$ which cannot be put into a bijection with $\mathbb N$") is, or is not, valid. If you deny the existence of $\mathbb N$, of course it can't be put into a bijection. Even with itself.

So it seems you do accept it, at some subconscious level, since you insist on discussing it and asserting properties for it. Clearly, the set itself is infinite by the definition given above. BUT IT IS ENDLESS. Cantor's great insight was to not try to describe what you find at the end it does not possess, like you are doing, but to accept it as a whole and describe how its "the quality of being infinite/endless" fits in a valid Mathematics.

You seem to think "infinity" describes either a member of $\mathbb N$, or a property approached in the endless sequence. There is no number in $\mathbb N$, as identified by step #2 of the Axiom, that can be called "endless," or associated with the quality of being endless. People who talk about limits are describing them in terms of properties of the members of the set. This seems to be your misunderstanding - you want "infinity" to mean the end of what is defined to be an endless iteration. It isn't.

So get this straight: Cantor does not apply the word "infinity" to a natural number, or to anything that even looks like a natural number. It is applied to a property of the set $\mathbb N$ itself called "cardinality." That property describes how sets can be compared to one another. For finite sets, the proerty can be likened to a natural number in the set. But we aren't dealing with a finite set. We are dealing with one that is, by definition, "endless" or "infinite."

Since the set DOES EXIST, and can be compared to other sets, it needs the property "cardinality." But the only quality that can be applied is "endless" or "infinite," so it is a kind of "infinity" by the definitions you oversimplified. And it is a property of the set, not any member of it.

March 20th, 2018, 08:04 AM   #19
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Quote:
 Originally Posted by zylo If infinity means endless, you can never get to the end of CDA.
I assumed respondents would be familiar with Cantors infinite sequences {wiki),

This sequence is either finite, 1001101010011, or infinite 1001101010011................. (endless).

Endless means you can't get to the end, ie, you can't complete CDA. I assume all irrelevant replies (such as parts of speech-I said synomous) are a confirmation of this.

March 20th, 2018, 08:15 AM   #20
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Quote:
 Originally Posted by zylo Endless means you can't get to the end, ie, you can't complete CDA. I assume all irrelevant replies (such as parts of speech-I said synomous) are a confirmation of this.
As usual you are deliberately ignoring all the useful information that has been posted to help you correct your misunderstanding.

There's nobody as stupid as someone who refuses to learn.

 Tags argument, cantor, cantors, diagonal, infinity

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