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March 7th, 2018, 08:31 AM   #71
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Originally Posted by zylo View Post
Ref Post #68

A series of terms S1, S2, S3,.... either stops or it doesn't. There are no other options.
A sequence is a mathematical function, not a freight train. You seem confused on this point.

You also seem confused about the difference between a sequence and a series. The above is a sequence.
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March 7th, 2018, 08:47 AM   #72
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Zylo also doesn't understand the difference between "unbounded" and "infinite".

I don't understand how someone who is so obviously motivated to investigate mathematics can be comfortable being so ignorant about so many basic facts and definitions.

Last edited by v8archie; March 7th, 2018 at 08:59 AM.
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March 7th, 2018, 08:53 AM   #73
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I think I posted something to this effect last night ($x^2 \ge 0$). It's this implicit acceptance of the infinite (via the natural numbers) that might allow a construction of the reals. You just have to dress up the reasoning in language that avoids appealing to the infinite. The only real hurdle I can see is the need for infinite sequences. If you can make a rational-sounding argument that they are just sequences for which we can always get another element when we need it, the job will be done. The actual definition of the limit doesn't talk about infinity and only references it in passing with the "for all $n \gt N$" which is just another "every even number is divisible by 2" statement.
Since Goodstein has written a grad-level book about how to do analysis without infinity, I'm in no position to say you're wrong.

I would only say that when you write, "If you can make a rational-sounding argument that they are just sequences for which we can always get another element when we need it, the job will be done," you're only restating the problem, not outlining an approach to a solution. But perhaps you've described Goodstein's idea. I have no way to know
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March 7th, 2018, 09:07 AM   #74
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Originally Posted by Maschke View Post
you're only restating the problem, not outlining an approach to a solution
Yes, I did make some progress, but it's not maths so much as linguistics. As you pointed out, the starting point is rather contrived.

I certainly wouldn't suggest that I path to a watertight solution. It really would be restating the existing construction in finitist language.

As I say, though, it's interesting how little we appeal to the infinite to create the reals.
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March 7th, 2018, 09:45 AM   #75
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Quote:
Originally Posted by zylo View Post
Ref Post #68

A sequence of terms S1, S2, S3,.... either stops or it doesn't. There are no other options.
Corrected (U,B) per your suggestion. A sequence is a standard mathematical concept. So you can interpret it as a function, so what? Ever heard of the sequence 1,2,3,....? It either ends or it doesn't.

Quote:
Originally Posted by Maschke View Post
A sequence is a mathematical function, not a freight train. You seem confused on this point.

You also seem confused about the difference between a sequence and a series. The above is a sequence.
Thanks. By the way, I recall you were discussing ZFC. I had a question you might be able to answer:

ZFC Axiom of regularity
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March 7th, 2018, 09:46 AM   #76
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Originally Posted by zylo View Post
Corrected (U,B) per your suggestion. A sequence is a standard mathematical concept. So you can interpret it as a function, so what? Ever heard of the sequence 1,2,3,....? It either ends or it doesn't.
That sequence doesn't end. The only kind of sequence that ends is a finite one.
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March 7th, 2018, 09:52 AM   #77
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Originally Posted by Maschke View Post
That sequence doesn't end. The only kind of sequence that ends is a finite one.
I meant it as a generic sequence, which either ends or it doesn't, unless you have a third option.
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March 7th, 2018, 10:28 AM   #78
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Originally Posted by zylo View Post
I meant it as a generic sequence, which either ends or it doesn't, unless you have a third option.
You don't seem able to tell the difference, unless your unbounded/infinite sequence is a third option that terminates after an infinite number of terms. (That third option is nonsense, by the way - I thought I'd better say so to avoid confusing you.)
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March 8th, 2018, 02:15 AM   #79
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Originally Posted by zylo View Post
. . . If I specify for all n, Sn can't have a finite length for all n because the natural numbers are unbounded.
By that reasoning, if S_n is defined to be n for all n, and "finite length" is replaced by "finite", you would be asserting that though some values of n are finite, n isn't finite for every n. For what value of n wouldn't n be finite?
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