finite
/fī′nīt″/
adjective
- Having bounds; limited. a finite list of choices; our finite fossil fuel reserves.
- Existing, persisting, or enduring for a limited time only; impermanent.
- Being neither infinite nor infinitesimal.
meaning - About the word 'finite' - English Language & Usage Stack Exchange
soft question - What do people mean by "finite"? - Mathematics Stack Exchange
Does "finite set" necessarily mean "there is a highest value"?
To you, does "finite" mean "not zero" or "not infinite"?
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something that is finite is something that has ends or limits, it's opposite is perhaps a slightly more common word, infinite, meaning without limits or ends, everlasting etc.
probably, certianly most things do have a finite lifetime, but it's a rather philosphical statement so either view could probably be supported. e.g. if everything in the medium of time ends what about time itself?
Finite = having bounds or limits, not infinite, [in maths] not zero.
Do I agree? Yes, because infinite is a logical concept that cannot be proved. As much as something might appear infinite, it's bounds may simply be beyond the range of your vision or comprehension.
Describing finite as not infinite is curious, but it comes about because the concept of infinity came first (from the latin word finitus), whereas finite came sometime later (15C so I've read)
If two people are arguing over whether or not “everything” is “finite”, then I'd say the difference between “cannot be put into one-to-one correspondence with a proper subset of itself” and “has a bijection to some set $\{1,\ldots,n\}$”, or any other definition of “finite”, is basically irrelevant. It's extremely unlikely they would suddenly agree if they meticulously chose a common definition, so why bother? (Consider an analogous situation with people arguing over the “existence” of “God”.)
Remember, even if a formal mathematical definition was assigned a particular natural-language name (like “finite” or “smooth”), it's only because it is felt to capture some aspect of that concept (at least according to the namer). I'd say that people arguing over whether “everything” is “finite” have a disagreement regarding the natural-language concept of “finite” itself; they don't need to have agreed about which mathematical approximation to this concept they like the most.
Well. This is a tough cookie to answer properly.
The reason is simple, though. Finite is one of those words which has a mathematical definition, but also a natural language definition and those are so close that we might confuse the two.
This is similar to what does a set mean. Is a set some predefined notion, is it an element of a model of $\sf ZF$, or $\sf Z$, or $\sf NF$ or $\sf KP$, or maybe an object in the category $\bf Set$. Do every set has a power set? Do every definable subset of a set is a set?
These are notions which are fuzzy, specifically because they are taken as somewhat of primitive notions in mathematics.
But suppose that you have happened to agree upon some notion of "set", and let's agree to stipulate that it satisfied some naive set theory which is close in flavor to $\sf ZF(C)$.
Now you have several options:
Claim that the natural numbers are not sets. They are urelements, or some atomic entities which satisfy the second-order axioms of $\sf PA$. Therefore the question what are the natural numbers is moot. And a set is finite if it can be mapped bijectively with a bounded set of natural numbers.
Define the finite ordinals, claim that the class of finite ordinals is "definable" (either as a set, or as a proper class if you want to reject the axiom of infinity). Then prove that the finite ordinals satisfy $\sf PA$, so they are worthy of being called "The Natural Numbers", and we are reduced to the previous case.
Use one of the many notions of finiteness which do not appeal to the natural numbers. These include, but not limited to, the following:
- Every self injection is a surjection.
- Every self surjection is a bijection.
- Every non-empty chain of subsets has a maximal element.
- Every non-empty collection of subsets has a maximal element.
Be forewarned, though, that apart of the last one, the axiom of choice is generally needed to prove that this is equivalent to the first suggested definition.
You may claim that the fact that there are definitions which are non-equivalent in the absence of the axiom of choice means that finiteness is not well-defined. And this is true. You can argue that you reject both the axiom of choice (and in fact, the axiom of countable choice), and the usual definition of finiteness. But you can also reject the axioms of induction in $\sf PA$ and claim that they are inconsistent, and you can reject the soundness of propositional calculus.
You can do all these things, but mathematics is a joint effort. If you are unwilling to agree on primitive notions like set, like finiteness, like natural number, then the problem lies in a deeper level than just this.