B What are the universally accepted fundamental definitions of finite and infinite?

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Erk

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I'm troubled by what I think the 'community' considers them to be, but I'm not sure if I'm correct. It appears as though finite is thought to have both an end and a beginning, but is it true that infinite (infinity) is thought to only have no end? Is this accurate? If so, then it would seem like they aren't polar opposites and infinity ends up being a composite of the two.
 
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I don’t know about the others, but I generally think of “infinite” as meaning “having cardinality ##\geq\aleph_0##. Equivalently, one could also characterize infinite sets as having subsets of size N for every natural number N.

Either way, there’s no need to discuss sets having beginnings or endings. Some infinite sets have beginnings and endings (e.g. the closed interval ##[0,1]##), or even an end but no beginning (the half-line ##(-\infty,0]##). So if I were to interpret your words very literally, I would find that your definition of infinity doesn’t quite work.
 

Erk

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I'm first and foremost wanting to understand math's definitions of them in a more fundamental fashion i.e. as nouns and not adjectives. That being said I suppose we should stick with finity and infinity?
 
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fresh_42

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I'm troubled by what I think the 'community' considers them to be, but I'm not sure if I'm correct. It appears as though finite is thought to have both an end and a beginning, but is it true that infinite (infinity) is thought to only have no end? Is this accurate? If so, then it would seem like they aren't polar opposites and infinity ends up being a composite of the two.
Finite is what has as many elements as ##\{\,1,2,\ldots,n\,\}## for some natural number ##n \in \mathbb{N}## or none. Infinite is what has at least as many elements as there are natural numbers, possibly more.
 

PeroK

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Whatever finite means, infinite means simply "not finite".
 
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I'm first and foremost wanting to understand math's definitions of them in a more fundamental fashion i.e. as nouns and not adjectives.
Your post asks about definitions of two adjectives: finite and infinite.
@fresh_42 defined "finite" in the context of sets in post #4. @PeroK's post provides a terse definition for the word "infinite."
Whatever finite means, infinite means simply "not finite".
That being said I suppose we should stick with finity and infinity?
To the best of my knowledge, "finity" is not a word used in English.
 
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To the best of my knowledge, "finity" is not a word used in English.
I though so too when I saw it, but I found this:
244061




The OP, by his own statement, was looking for nouns. Well, that's a noun, but I think it is a very poor choice of words, since I've NEVER heard it used in any context, scientific or otherwise.
 

PeroK

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I though so too when I saw it, but I found this:
View attachment 244061



The OP, by his own statement, was looking for nouns. Well, that's a noun, but I think it is a very poor choice of words, since I've NEVER heard it used in any context, scientific or otherwise.
Useful, nevertheless, if you are a Scrabble player.

PS although I just checked and it's not allowed. That's a pity.
 

Erk

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Whatever finite means, infinite means simply "not finite".
This is exactly what I'm getting at – what I'm trying to understand. I think it's fair to say that finite means a beginning and end. A measurable amount of anything. However, in math it appears as though infinite is not treated as its "not" or opposite. As I mentioned before, it's treated as a composite. A combination of finite and infinite but never its direct opposite.

I see a lot of hoops that math makes infinite or infinity jump through but nowhere do I see it rigorously treated as the opposite of finite or finity.
 

fresh_42

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This is exactly what I'm getting at – what I'm trying to understand. I think it's fair to say that finite means a beginning and end.
##\mathcal{F} =\{\,\text{ apple },\text{ orange },\text{ banana }\,\}## is finite, but I can't see neither beginning nor end.
A measurable amount of anything.
No.
Measurable is something different, and can be quite a complex lesson. It is something with volume. ##\mathcal{F}## has no volume. Countable is the correct adjective.
However, in math it appears as though infinite is not treated as its "not" or opposite.
You cannot put it out of context. As a cardinality it is the opposite of finite. In a way. But if something tends to infinity, then it is a point beyond all borders.
As I mentioned before, it's treated as a composite. A combination of finite and infinite ...
This is not true. It doesn't even make sense.
... but never its direct opposite.
Of course it is. Have a look at my definitions in post #4 and you will find that infinite is the opposite of finite.
I see a lot of hoops that math makes infinite or infinity jump through but nowhere do I see it rigorously treated as the opposite of finite or finity.
Then read this thread. I mean: Read it! We can explain it to you, but we cannot understand it for you.

This thread has run its course, so I'll close it.
 

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