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June 24th, 2016, 10:32 AM   #41
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Quote:
 Originally Posted by v8archie What you have written (in that post) is mostly correct but they are not definitions. It's not ambiguous. The natural numbers are finite (that's a property that every one of them holds). The set of natural numbers is infinite (that's a property of the set, not of its elements which are natural numbers.
Zylo's posts repeatedly make the same error of confusing the attributes of the members of a set with the number of members of the set.

He thinks it is a contradiction to say that each member of the set of natural numbers is finite, but the number of members of the set is infinite. Bizarre.

Last edited by skipjack; June 25th, 2016 at 08:37 PM.

 June 24th, 2016, 12:38 PM #42 Senior Member   Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 367 Thanks: 68 But he indicated that to say, "The natural numbers collectively are finite" is false. Isn't that another way of saying that the number of elements in the set of natural numbers is infinite? Perhaps to be clear, it would have been better for him to have said, "To say, 'The natural numbers collectively are finite in number' is false."
June 24th, 2016, 02:05 PM   #43
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Quote:
 Originally Posted by Timios But he indicated that to say, "The natural numbers collectively are finite" is false.
He is deliberately using language that can be misunderstood in order to later make false claims about countability and Cantor's diagonal argument.

This is why he highlights the two possible meanings. At some point he is going to reach the sentence with one meaning and then interpret it using the other.

Or perhaps I'm doing him a disservice and he really is incapable of distinguishing between a set and its elements despite having had the difference highlighted to him many times.

June 24th, 2016, 10:28 PM   #44
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Quote:
 Originally Posted by zylo Finite: A natural number Infinite: All the natural numbers
Below is a question for you, zylo.

Is it incorrect to state that a "countably infinite sequence of digits is a natural number"?

 June 25th, 2016, 06:17 AM #45 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,010 Thanks: 83 n binary digits define 2^n natural numbers for every natural number n. Cantor's proof fails. The reals are countable. Last edited by skipjack; June 25th, 2016 at 08:49 PM.
 June 25th, 2016, 06:46 AM #46 Math Team   Joined: Dec 2013 From: Colombia Posts: 6,700 Thanks: 2177 Math Focus: Mainly analysis and algebra Only for natural numbers $n$ and thus only for finite sequences. Cantor doesn't talk about finite sequences. Even if the diagonal argument were flawed, there are other proofs that the reals are uncountable, so you are completely wrong. You are obviously incapable of rational thought on this subject. You have done exactly as I predicted in using the artificial "ambiguity" that you set up in exactly the way I predicted that you would.
June 25th, 2016, 06:56 AM   #47
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Quote:
 Originally Posted by zylo n binary digits define 2^n natural numbers for every natural number n.
This statement is true. Unfortunately, you do not know what it means!

For any finite number n, n binary digits define 2^n natural numbers. You are immediately extending that to a (countably) infinite set of digits without justifying the jump.

Last edited by skipjack; June 25th, 2016 at 08:44 PM.

June 25th, 2016, 05:33 PM   #48
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Quote:
 Originally Posted by zylo n binary digits define 2^n natural numbers for every natural number n. Cantors proof fails. The reals are countable.
n binary digits define 2^n natural numbers for every natural number n.[/U][/B]

And this is totally irrelevant to real numbers.

I feel a singularity in the derp field approaching.

Question: how many binary digits does it take to define all natural numbers?

Let's say it is p. So how many binary digits does it take to define $2^p + 10.$

EDIT: And what is magical about binary digits anyway?

Last edited by skipjack; June 25th, 2016 at 08:45 PM.

 June 25th, 2016, 08:47 PM #49 Global Moderator   Joined: Dec 2006 Posts: 16,944 Thanks: 1255 derp?
 June 25th, 2016, 08:52 PM #50 Newbie     Joined: Jun 2016 From: ha noi Posts: 2 Thanks: 2 Same applies for binary digits. Thanks from manus

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