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What is the opposite of "infinitesimal"?

Infinitesimal means a value smaller than the minimum possible measurement. For example, if we have a ruler with 1 centimeter intervals, then any length measured shorter than 1 cm is infinitesimal for that measuring instrument - smaller than the smallest possible measurement, but still greater than zero.

The opposite of infinitesimal means a value larger than the maximum possible measurement. Analogously, if we have a 100 centimeter ruler, then any length measured longer than 100 cm is that opposite term.

"Infinite" is not the opposite of infinitesimal, although that term is commonly (mis)used. Infinity is an abstract concept for something that never ends, not merely a value above measuring capability. For example, the question "how long is this road?" may have the answer "the opposite of infinitesimal" if we lack the capacity to travel along a straight road in its entirety and measure its length. On the other hand, if the road is short enough to see in its entirety but loops in on itself, we could say that its length is "infinite".

A term I came up with for this is "supermaximum"; above the maximum possible measured value. I would like to know if an official term exists.

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Where do you get that definition for "infinitesimal"? Anyway; immeasureable: "too large, extensive, or extreme to measure" –  Peter Shor Feb 27 '13 at 2:03
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Infinitesimal can be used as a noun for a mathematical concept, and then its opposite is indeed an infinity. Even when it is used as an adjective it can have this mathematical meaning. I'm not sure where your definition is from, I don't see it anywhere. –  HodofHod Feb 27 '13 at 2:13
    
My definition of infinitesimal is pretty much from Wikipedia. –  Core Xii Feb 27 '13 at 2:18
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If I had a 100 cm ruler, I think I could figure out how to measure longer distances with it. –  Canis Lupus Feb 27 '13 at 2:25
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If you consult Wiktionary instead, you will see that you are using an informal usage of the word, and that technically, infinitesimal is actually the opposite of infinite. –  HodofHod Feb 27 '13 at 2:29

6 Answers 6

up vote 11 down vote accepted

I believe that infinite is the opposite of infinitesimal.

Infinite : extending beyond, lying beyond, or being greater than any preassigned finite value however large

Infinitesimal : taking on values arbitrarily close to but greater than zero

So while infinitesimal talks about values arbitrarily smaller than any finite value but greater than zero, infinite talks about values that are arbitrarily larger than any finite value however large.

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Hm, perhaps the question is actually more philosophical than linguistic. –  Core Xii Feb 27 '13 at 4:10
    
But mathematically speaking, that definitions relates a real number (strictly positive, still being a member of the numeric axis) to an entity not in the set of reals. I'm not saying it's incorrect here. I'm just curious if one could view e.g. identity matrix of rang 17 as an opposite (or in any other way related to) a triangle. The relation stretches across the definition sets which usually sets off a warning beep in my head. –  Konrad Viltersten Jul 13 at 16:07

The source of the term "infinitesimal" is 17th century Latin "infinitesimus", which refers to an "infinite-th term in a series". It was introduced by the mathematicians Nicolaus Mercator and Gottfried Wilhelm Leibniz (see this article). If one has a sequence of terms getting smaller and smaller, the "infinite-th term" is taken to be infinitesimal. In this sense, infinitesimal can be taken to be the opposite of "finite", referring to the values of the ordinary terms of the sequence (before one gets to the "infinite-th term"). However, Leibniz also thought of infinitesimals as "inassignable" quantities, related to ordinary assignable quantities by means of a pair of heuristic principles: law of continuity and law of homogeneity. In this sense, the opposite would be "assignable".

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Skipping past the confusion regarding your definition of 'infinitesimal' and 'opposite', the term I'd use to mean 'bigger than any number you care to mention' would be arbitrarily large.

Granted I study mathematics so my usage is informed by that, but it seems like it might get at what you want to say.

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Even though I don't have the response you are looking for, I can tell you that physicists use the term finite as the opposite of infinitesimal. Of course, this has a lot of problems of its own if one thinks about it. But it is indeed part of the theoretical physicist's jargon.

You are asking the wrong crowd. You are asking a non-mathematical crowd about mathematical jargon.

Indeed, if 0 < e<<1, then we call e infinitesimal. That's a definition. Hence if d = 1 then d is not infinitesimal and of course 1 is not infinity.

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Measureless or immeasurable. Synonyms of immense.

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Oops. I didn't see Josh's post. –  Canis Lupus Feb 27 '13 at 2:22

The simple term I would use is immeasurable.

adjective
too large, extensive, or extreme to measure:
immeasurable suffering

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"Immeasurable" encompasses both infinitesimal and its opposite term. "Immeasurable" does not communicate whether the value is too small or too large to be measured; merely that it cannot be measured. –  Core Xii Feb 27 '13 at 2:15
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But without measure is typically only used when conveying extremely large –  Jim Feb 27 '13 at 2:23
    
Although something can be immeasurably small, every dictionary (and common use) suggests that the meaning defaults to being too large. –  Josh Feb 27 '13 at 2:34
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@Core Xii: The word immeasurable cannot mean infinitesimal. Maybe you're confusing it with unmeasureable. –  Peter Shor Feb 27 '13 at 14:51
    
I'd argue that "immeasurable" is more useful when used as an adverb, i.e. paired with another adjective, such as "immeasurably more" or "immeasurably greater." –  MarkHu Sep 17 '13 at 19:01

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