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July 6th, 2018, 07:47 AM   #41
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Originally Posted by zylo View Post
The answer is that in the limit of infinite decimal places there is no number between them though they remain distinct.
No they don't. The real number literally is the limit. You can't do mathematics if you ignore definitions.
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July 6th, 2018, 08:09 AM   #42
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Originally Posted by zylo View Post
2) Induction is the best nature gives you. Sure, you can imagine the moon is made of green cheese, or beyond the beyond.
Induction isn't "given by nature". You are avoiding responding to the assertion that you misused induction, and thereby used flawed reasoning.

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Maschke, Sorry, to me trans=finite is double=talk.
If you don't like that terminology, just read on a bit to find that Maschke was referring to oridnals or cardinals that aren't finite.

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There is no trans-finite
It's unreasonable to object to "trans-finite" as double-talk and then use the term. Induction isn't a limiting process, yet you readily state "in the limit" in a context where all you have is that something holds for each positive integer n. The wording "for all n" isn't "a little flaky", it's just an alternative wording. If you're unaware of a formal definition of "trans-finite" or "transfinite", that doesn't imply that such a definition isn't possible.

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. . . in the limit of infinite decimal places there is no number between them though they remain distinct.
Wouldn't that allow them to be treated as distinct representations of the same number, making it trivial that there isn't a number strictly between them?

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. . . one (you) might conceive that there are more points on a line than you can count with a number system.
What do you mean by "more points than you (or anyone) can count with a number system"? How would counting be done without a number system?
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July 6th, 2018, 09:57 AM   #43
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Induction isn't "given by nature". You are avoiding responding to the assertion that you misused induction, and thereby used flawed reasoning.
It's a natural, logical, way to refer to everything you can associate with a natural number.

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If you don't like that terminology, just read on a bit to find that Maschke was referring to oridnals or cardinals that aren't finite.
There is nothing beyond induction. That's the limit of a precise statement of "infinity."

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It's unreasonable to object to "trans-finite" as double-talk and then use the term. Induction isn't a limiting process, yet you readily state "in the limit" in a context where all you have is that something holds for each positive integer n. The wording "for all n" isn't "a little flaky", it's just an alternative wording. If you're unaware of a formal definition of "trans-finite" or "transfinite", that doesn't imply that such a definition isn't possible.
I agree, you can define anything, like trans-trans-finite, or beyond the beyond. Trans-finite only arose because Cantor didn't know how to count binary representations of the natural numbers.

Trans-finite is an imaginary concept which serves no purpose other than to throw a wrench into the gears while pretending to be logical, mathematical, and unassailable. None of the above. But it impresses the hell out of freshmen. I intentionally didn't say freshwomen- they're not that easily impressed.

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Wouldn't that allow them to be treated as distinct representations of the same number, making it trivial that there isn't a number strictly between them?
.33..... is never 1/3, but the difference approaches 0.

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What do you mean by "more points than you (or anyone) can count with a number system"? How would counting be done without a number system?
"more points may exist on a line than you (or anyone) can count with a number system"

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Originally Posted by v8archie View Post
No they don't. The real number literally is the limit. You can't do mathematics if you ignore definitis.
You are still confusing limits. There are no sums involved here, just limits of subdivisions of a line, corresponding to limits of decimal or binary or any radix places. Think of limit of a nest of closed intervals.

Decimal or any radix definition works fine for anything I can think of in analysis or topology (completeness, neighborhood, nearness of points, etc).
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July 6th, 2018, 10:32 AM   #44
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.33..... is never 1/3, but the difference approaches 0.
In the same way the difference between 2 and 3 approaches 1?
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July 6th, 2018, 10:35 AM   #45
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"more points may exist on a line than you (or anyone) can count with a number system"
Cool, so you believe the number line is something like the hyperreals or surreals. Or you believe that the number line is a purely geometric entity that can not be put into one-to-one correspondence with the reals or any field.

I'm fine with that, those are all respectable opinions in my book. But please don't refer to the number line as being the real numbers then.
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July 6th, 2018, 11:46 AM   #46
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Seems we haven't defined a line.

A line is all n-place decimals, or radixes, for all n.

It can be pictured physically as the points on a light ray between two points divided into 1/10^n segments, for all n. It follows that m(1/m) = 1 for all m, m=10^n, and $\displaystyle \infty 0$=1.

Edit.
If you like, as the number of sub-divisions becomes larger, the numbers (points on the line) become “real.” That’s about the best you can do without yoga.

Last edited by zylo; July 6th, 2018 at 12:36 PM.
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July 6th, 2018, 12:59 PM   #47
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Originally Posted by zylo View Post

A line is all n-place decimals, or radixes, for all n.
Your "line" consists only of terminating decimals. Not much a line. It's full of holes and doesn't even include all the rationals.
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July 6th, 2018, 01:43 PM   #48
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Originally Posted by Maschke View Post
Your "line" consists only of terminating decimals. Not much a line. It's full of holes and doesn't even include all the rationals.
You missed “for all n.” Granted, the interval [0,1/3] doesn’t exist (in decimal or binary notation) unless you interpret 1/3, or pi say, as endless decimals. It’s “for all n” that gives you the reals (the holes) and lets you use the shortcuts 1/3 or pi.

I know, I know, no matter what n is it’s a rational number. But that’s just semantics. As time goes by, your division construction fills up all the holes.

And where is the line you use as a yardstick for your numbers? It doesn’f exist except as a line of molecules, so you have to define it.
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July 6th, 2018, 01:57 PM   #49
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Originally Posted by zylo View Post
You missed “for all n.” Granted, the interval [0,1/3] doesn’t exist (in decimal or binary notation) unless you interpret 1/3, or pi say, as endless decimals. It’s “for all n” that gives you the reals (the holes) and lets you use the shortcuts 1/3 or pi.
By "for all n" I assume you mean for all natural numbers n. In that case I reiterate my remark. Your line consists only of the terminating rationals, which are few and far between in the scheme of things.

If you mean something else by "for all n" please say what you mean.



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Originally Posted by zylo View Post
I know, I know, no matter what n is it’s a rational number. But that’s just semantics. As time goes by, your division construction fills up all the holes.
You are contradicting yourself. Just semantics? Your definition of a line consists of only terminating rationals.

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And where is the line you use as a yardstick for your numbers? It doesn’f exist except as a line of molecules, so you have to define it.
Confusing math with physics.

The "real line" is defined as the set of real numbers.

As Micrm@ss noted, there are other philosophical conceptions of line. Are you referring to one of those alternate ideas?

Last edited by Maschke; July 6th, 2018 at 02:10 PM.
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July 6th, 2018, 02:59 PM   #50
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Originally Posted by Maschke View Post
By "for all n" I assume you mean for all natural numbers n. In that case I reiterate my remark. Your line consists only of the terminating rationals, which are few and far between in the scheme of things.

You are contradicting yourself. Just semantics? Your definition of a line consists of only terminating rationals.

The "real line" is defined as the set of real numbers.

As Micrm@ss noted, there are other philosophical conceptions of line. Are you referring to one of those alternate ideas?
I start walking. If every time I take a step I take another step, do I stop at some point or do I go on indefinitely. It's one or the other. And if I keep on going indefinitely, is there some place I will never reach, beyond the universe? That's mysticism. And if there is such a place, is there something beyond that? n steps is finite. n+1 steps if I take n steps isn't.

If it makes you happy, every time you divide the line 10^n times, divide it 10^(n+1) times.

And of course the question, what is your definition of a real number?

Some people see behind the beyond. Most people don't even see beyond the behind. Couldn't resist that old joke.
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