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James4 comments on Artificial Addition - Less Wrong

36 Post author: Eliezer_Yudkowsky 20 November 2007 07:58AM

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Comment author: James4 22 November 2007 12:34:00AM -2 points [-]

I read a book on the philosophy of set theory -- and I get lost right at the point where classical infinite thought was replaced by modern infinite thought. IIRC the problem was paradoxes based on infinite recursion (Zeno et. all) and finding mathematical foundations to satisfy calculus limits. Then something about Cantor, cardinality and some hand wavy 'infinite sets are real!'.

1.999... is just an infinite set summation of finite numbers 1 + 0.9 + 0.09 + ...

Now, how an infinite process on an infinite set can equal an integer is a problem I still grapple with. Classical theory said that this was nonsense since one would never finish the summation (if one were to begin). I tend to agree and I suppose one could say I see infinity as a verb and not a noun.

I suggest anyone who believes 1.999... === 2 really looks into what that means. The root of the argument isn't "What is the number between 1.999... and 2?" but rather "Can we say that 1.999... is a sensible theoretical concept?"

Comment author: taryneast 30 December 2010 12:15:17PM *  -2 points [-]

1.999... does not equal 2 - it just tends towards 2

For all practical purposes, you could substitute one for the other.

But in theory, you know that 1.9999... is always just below 2, even though it creeps ever closer.

If we ever found a way to magickally "reach infinity" they would finally meet... and be "equal".

Edit: The numbers are always going to be slightly different in a finite-space, but equate to the same thing when you allow infinities. ie mathematically, in the limit, they equate to the same value, but in any finite representation, they are different.

Further Edit: According to mathematical convention, the notation "1.999..." does refer to the limit. therefore, "1.999..." strictly refers to 2 (not to any finite case that is slightly less than two).

Comment author: DanielLC 01 November 2011 03:45:37AM *  1 point [-]

Classical theory said that this was nonsense since one would never finish the summation (if one were to begin).

It was nonsense in classical theory. Infinite sum has its own separate definition.

I tend to agree and I suppose one could say I see infinity as a verb and not a noun.

There are times in modern mathematics that infinite numbers are used. This is not one of them.

I doubt I'm the best at explaining what limits are, so I won't bother. I may be able to tell you what they aren't. They give results similar to the intuitive idea of infinite numbers, but they don't do it in the most intuitively obvious way. They don't use infinite numbers. They use a certain property that at most one number will have in relation to a sequence. In the case of 1, 1.9, 1.99, ..., this number is two. In the case of 1, 0, 1, 0, ..., there is no such number, so the series is said not to converge.

... "Can we say that 1.999... is a sensible theoretical concept?"

No. The question is "Can we make a sensible theoretical way to interpret the numeral 1.999..., that approximately matches our intuitions?" It wasn't easy, but we managed it.