Finite & Infinite: Univerally Accepted Definitions?

In summary, finite is defined as having a specific number of elements or none, while infinite is defined as having at least as many elements as there are natural numbers, and possibly more. While finite may imply a beginning and an end, this is not necessarily the case in mathematical definitions. Infinity is treated as the opposite of finite in terms of cardinality, but it can also be seen as a point beyond all borders. There may be confusion in the use of the term "infinity" as it is often used in different contexts, but in mathematics, it is a well-defined concept.
  • #1
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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  • #2
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.
 
  • #3
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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  • #4
Erk said:
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.
 
  • #5
Whatever finite means, infinite means simply "not finite".
 
  • #6
Erk said:
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."
PeroK said:
Whatever finite means, infinite means simply "not finite".

Erk said:
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.
 
  • #7
Mark44 said:
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.
 
  • #8
U
phinds said:
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.
 
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  • #9
PeroK said:
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.
 
  • #10
Erk said:
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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1. What is the difference between finite and infinite?

The main difference between finite and infinite is that finite refers to something that has a definite and measurable limit, while infinite refers to something that has no limit or is limitless.

2. How are finite and infinite used in scientific research?

Finite and infinite are used in scientific research as concepts to describe the boundaries and scope of a particular study. For example, a study may focus on a finite number of participants or an infinite number of variables.

3. Can something be both finite and infinite?

No, something cannot be both finite and infinite at the same time. These terms are mutually exclusive and refer to opposite concepts.

4. Are there any real-life examples of finite and infinite?

Yes, there are many real-life examples of finite and infinite. Some examples of finite things include the number of atoms in a specific molecule, the amount of time in a day, and the number of people in a room. Examples of infinite things include the number of possible combinations of a Rubik's cube, the number of stars in the universe, and the number of digits in pi.

5. Why are finite and infinite important in science?

Finite and infinite are important in science because they help us understand and describe the world around us. They provide a framework for defining and measuring the boundaries and limits of various phenomena, which is crucial for scientific research and understanding the natural world.

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