My Math Forum Is an Infinite Binary Sequence a Natural Number?

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July 13th, 2017, 02:28 PM   #11
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 Originally Posted by skipjack Your example showed that you defined the sequence as having a value equal to a sum with last term $a_n$
I think that was part of a badly formatted $a_n = 0,1$ meaning $a_n \in \{0,1\}$.

July 13th, 2017, 03:09 PM   #12
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 Originally Posted by zylo Convergence has nothing to do with it. Natural numbers don't converge. What does the sequence of natural numbers converge to?
Sure, the sequence does not converge in the usual meaning of convergence. But it can sometimes be very very useful to change the usual meaning of convergence to get good limits. Yes, it's a nasty piece of mathematical trickery, but it's very common and useful. This is what the 2-adics do.

July 13th, 2017, 04:38 PM   #13
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 Originally Posted by v8archie Grow up.
Archie, Haven't you and I rolled around in the dirt enough? Are you simply incapable of controlling yourself?

 July 13th, 2017, 06:17 PM #14 Senior Member   Joined: May 2016 From: USA Posts: 1,047 Thanks: 430 If I remember a long conversation with maschke about a year ago (on a zylo thread of course), it is sufficient to show that the possible number of infinite bit strings exceeds the number of natural numbers using the diagonal proof without any reference to the representation of real numbers. In other words, if infinity "exists," it "exists" in different flavors that are not commensurate, where commensurate means finding a means to put two sets into one-to-one correspondence. (That proof alone does not show that the set of real numbers exceeds the set of number of natural numbers, but it prevents a lot of confusion about the diagonal proof itself. Dealing with the number of real numbers then requires a separate proof.) It is perfectly reasonable to declare that there is no physical evidence for any kind of infinity nor any physical evidence of irrational numbers. Finitists have a respectable empirical basis for their position. Maschke and I disagreed (I can't remember why) about my premise that infinity certainly does exist as a mental construct, a creation of the human imagination. But if you insist on a demonstration that infinity is physically meaningful, then the natural numbers cannot be shown to be infinite. You can't have it both ways: either we agree to accept that infinity "exists" or, by denying that infinity "exists," to concede that the natural numbers are not infinite.
July 13th, 2017, 06:44 PM   #15
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 Originally Posted by Maschke Archie, Haven't you and I rolled around in the dirt enough? Are you simply incapable of controlling yourself?
Rather what I thought of your post. Hence the terse comment.

July 14th, 2017, 01:48 AM   #16
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Quote:
 Originally Posted by v8archie I think that was part of a badly formatted $a_n = 0,1$ meaning $a_n \in \{0,1\}$.
Okay, but then the sum zylo gave has an infinite number of terms. I'll assume for now that at least one term is non-zero (as in zylo's example). Without this assumption, the sum would presumably be zero if every term is zero, but zero isn't a natural number.

If there are an infinite number of non-zero terms, how do you define their sum, zylo, given that every natural number (such as 4 in the example) already corresponds to such a sum with a finite number of non-zero terms?

July 14th, 2017, 02:58 AM   #17
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 Originally Posted by Maschke Archie, Haven't you and I rolled around in the dirt enough? Are you simply incapable of controlling yourself?
Pass me that popcorn.

 July 14th, 2017, 08:17 AM #18 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100 https://en.wikipedia.org/wiki/P-adic_number p-adics: Abstract irrelevance. Std Trick 1. Subtraction is not defined for the natural numbers. I never claimed removing binary point from an infinite binary fraction was a mathematical operation. I simply said if an infinite binary sequence exists after a binary point, it exists without the binary point. Same trick over and over again: Joe saya the moon is made of green cheese implying Joe is a jerk. But Joe never said the moon was made of green cheese, proving what about the person who made the false attribution? Std Trick 2. Personally, I consider Tricks 1 and 2 as failure of the propounders to find anything wrong with OP. $\displaystyle All$ binary fractions are infinite, as are all decimal fractions. .840 is ambiguous. It is either exactly .840, ie, .840000000........(an infinite decimal fraction) or a number between .839500.... and .840500......., (infinite decimal fractions). EDIT: What defines a binary fraction: A unique infinite sequence of binary digits. The same unique infinite sequence of binary digits defines a natural number. Last edited by zylo; July 14th, 2017 at 08:31 AM.
July 14th, 2017, 08:42 AM   #19
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Quote:
Trick? Why would I use a trick?? For what

Quote:
 Subtraction is not defined for the natural numbers.
Well it is, it is a partial operation, but never mind. It's not essential.

Quote:
 Personally, I consider Tricks 1 and 2 as failure of the propounders to find anything wrong with OP.
Of course there is nothing wrong with the OP, you asked a question. You never made a definite statement.

Quote:
 $\displaystyle All$ binary fractions are infinite, as are all decimal fractions. .840 is ambiguous. It is either exactly .840, ie, .840000000........(an infinite decimal fraction) or a number between .839500.... and .840500......., (infinite decimal fractions).
But what does this have to do with the OP. In the OP, you seem to be talking about points going infinitely "to the left". Like ...11101101
You're not talking about decimal points.

But yes, I agree, .840 is simply shorthand for .84000000... But it is a useful and often used shorthand.

 July 14th, 2017, 09:53 AM #20 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100 The infinite binary (or decimal) sequence $\displaystyle a_{1}a_{2}a_{3}.......$ defines a unique natural number or unique binary fraction, depending on interpretation: Natural Number: $\displaystyle \sum a_{n}2^{n}$ which approaches a unique natural number as n -> infnity. Binary Fraction: $\displaystyle \sum a_{n}2^{-n}$ which approaches a unique fraction in [0,1) as n -> infinity. Note the unique association between natural number and binary fraction.

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