I understand there is a symbol for infinite. Is there one for finite?
I searched and found there is none. How is finite represented symbolically?
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9$\begingroup$ In writing, you'll probably be better off being clear and using words, not symbols. $\endgroup$ – lhf Jun 18 '12 at 12:57
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1$\begingroup$ Possible duplicate of this question? $\endgroup$ – Zev Chonoles Jun 18 '12 at 13:16
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$\begingroup$ @lhf: do you mean to say that symbols will be more handy than words when talking? $\endgroup$ – tomasz Dec 8 '12 at 1:18
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I have never seen a notation for 'finite,' but what I do very often see is denoting something finite as simply being less than infinity. For example, $|A| < \infty$, or $[G:H] < \infty$.
Small thing I'd like to add: Of course something like $[G:H] < \infty$ isn't technically meaningful, but it certainly gets the point across and in my experience at least seems to be pretty standard.
answered Jun 18 '12 at 12:52
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1$\begingroup$ What's not technically meaningful about it? It's a comparison between two cardinalities. $\endgroup$ – Qiaochu Yuan Jun 18 '12 at 14:21
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$\begingroup$ @QiaochuYuan: True, it does work in that case if we let $\infty$ represent some transfinite cardinal number. I should have used the second example there, and in fact I will edit it. (and sorry for the late reply) $\endgroup$ – Alex Petzke Jun 20 '12 at 15:14
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1$\begingroup$ I don't see what's not technically meaningful about $[G : H] < \infty$ either. You are committing a fairly small abuse of notation in identifying a finite cardinal with a natural number. One can make perfect sense of the poset $\{ 1, 2, ... \infty \}$. $\endgroup$ – Qiaochu Yuan Jun 20 '12 at 15:45
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$\begingroup$ One can write $|[G:H]|<\infty$. The notation $|[G:H]|$ means the number of members of $[G:H]$. ${}\qquad{}$ $\endgroup$ – Michael Hardy Dec 7 '15 at 19:37
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How about using $$\not\infty$$
answered Jun 19 '12 at 5:11
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1$\begingroup$ I had the same idea, but didn't know \not. $\endgroup$ – draks ... Jun 19 '12 at 10:23
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2$\begingroup$ You might want to consider using "\!" a few times to write $\not\!\!\infty$. Just typographical concerns ;) $\endgroup$ – Alex Nelson Jun 20 '12 at 15:48
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$\begingroup$ Does anyone know if the unicode symbol ⧞ = 29DE (infinity negated with vertical bar) was meant to be used in situations like this? $\endgroup$ – Kedar Mhaswade Dec 27 '15 at 22:46
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I'm guessing you mean the symbol $\infty$, for a non-specific non-finite cardinality. In this case, in the same way you would say $|X|=\infty$ to mean "the set $X$ has infinitely many elements", I would write $|X|<\infty$ to mean "the set $X$ has finitely many elements".
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$\begingroup$ He doesn't say he is working with cardinals. When we have a function $f$ with (according to its definition) values in the extended real numbers $[-\infty,+\infty]$, and we want to emphasize that a certain value $f(x)$ is finite, we may write $|f(x)| < \infty$. When we have a series $\sum a_n$ and we want to say it converges absolutely, we may write $\sum |a_n|<\infty$. $\endgroup$ – GEdgar Jun 18 '12 at 13:45
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$\begingroup$ Sure, I used cardinals as an example. (This is why I like Alex's answer better, because he gives two examples). $\endgroup$ – mdp Jun 18 '12 at 13:48
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Writing instead of symbolizing is a very bad idea but some rare cases. The reasons are:
- Doesnt exist an universal language more than one that is symbolic
- Symbols have visual meaning, words not... it is slow reading (and writing) instead of just knowing
- Symbols carry more meanings than just words, this is exactly the meaning of "symbol"
For this I think is a good idea use the symbol, as noted before, $\not\infty$.
This is too long for a comment so I write an answer.
EDIT:
A more convincing, and objective argument for my apology: nobody write number or basic operands with words... we don't write $\text{"two more two"}$ when we are doing mathematics... we write $"2 + 2"$.
When we are using words in advanced mathematics this is because doesn't exist a good symbolic way to express complex things (by now), but this doesn't mean that is the "best" or "correct" way to do.
Just compare words, mathematical notation of the roman empire before the use of Indian numbers, with the actual notation. The evolution tends to destroy words in favor of the extremely more efficient pure symbol... but what is needed is a good symbology, if this doesn't exist of course the best is just write words and describe things with natural language.
