• alan1000
    200
    1. An infinity is a number which is larger than any natural number

    This definition is still surprisingly popular, even (I regret to say) among mathematicians, although it was exploded by Cantor more than a century ago.

    It is easy to prove that an infinity is larger than some particular natural number; it does not follow that it must, in principle, be larger than ANY natural number, because the natural numbers are themselves extendable to infinity. Indeed, the set of the natural numbers is the definitive example of what mathematicians call a 'countable' or 'aleph-null' infinity. In an aleph-null infinity, every member can be counted off with a natural number; therefore, there is necessarily some natural number which is equal to the infinity, though of course we can never know what it is, because (paradoxically) we can never reach it by counting.

    It is possible to prove that an infinity may be smaller than some natural number. (For the sake of brevity, in what follows, I do not distinguish between a set and its cardinality).

    (1) Consider the infinite set of the natural numbers, beginning from 0. We will call this set A. Now consider the infinite set of the natural numbers which are greater than or equal to 10. Call this set B.

    (2) In the arithmetic of the countable infinities, A = B; but: A - B = 10.

    (3) In other words, if we pair the elements of infinite set A one-to-one with those of infinite set B, we have 10 numbers left over.

    (4) Therefore, an infinity may be smaller than some natural number.

    We can never assign a specific value to "some natural number"; but every aleph-null infinity is equal to or smaller than some natural number. There other kinds of infinity which are definitely larger, but that is another subject.

    ---

    2. An infinity is any non-finite number

    A finite number is a number whose value can be expressed with perfect exactitude. It is finished, complete, precise, and nothing more can be added to it.

    If we can identify one non-finite number which is not an infinity, this definition will fail. Is there such a number? As it happens, there are infinitely many. The infinitesimal, as defined by the symbol 'h', is not a finite number. In contexts where 'h' is important, this may imply that 0 also is not a finite number; however, considered as a natural number, 0 is finite. Any number which entails an infinite series of non-trivial decimal places, like pi or any of the irrationals, is a non-finite number; it is not bounded and its true or final value is unknowable. Infinitely many fractions between 0 and 1 are non-finite in this sense.

    The definition, then, is incorrect and seriously misleading. All infinities are non-finite, but only some non-finite numbers are infinities. The confusion arises from the careless assumption that "infinite" and "non-finite" are synonymous. If they were really synonymous, the definition would be vacuous and non-informative instead of merely incorrect.


    ---

    Correct Definition:

    3. An infinity is the cardinality of a set which contains one or more proper subsets equinumerous with itself

    A 'proper subset' is a collection of some of the elements of a set, but not all of them, and not including anything taken from any other set; it's really just a part of a set. "Equinumerous" means, having the same cardinality or numeric value. So the definition says that if a part of the set turns out to have the same number of members as the whole set, even though it is only supposed to include some of them, then the parent set must be an infinity.

    In classical arithmetic, the part is always less than the whole. If we cut a cake into portions, every portion is necessarily something smaller than a whole cake. In the arithmetic of infinity, however, at least one portion ('proper subset') always persists in being just as large as the original cake, no matter where or how often we cut it. (Sounds almost like a children's fairy tale, doesn't it!)

    Suppose we have N quantity of coins in a jar, and we take out five coins. Then we check the number of coins remaining in the jar. If there are N-5 coins, we will know that N is a finite number and, if we keep taking coins out, eventually the jar will be empty. On the other hand, if we find that the jar still contains N coins, and never contains less than N coins, however often we take some out, then N must be an infinity.

    Set theory is not as clearly defined as some other branches of mathematics, so there is some room for dispute, but this essentially defines infinity in terms of what it IS, rather than what it isn't, and for that reason, it succeeds where the other two popular definitions fail. Infinity can only be correctly defined in terms of this simple, special logical property, which is unique to itself. The definition makes no stipulations about finitude or absolute magnitude relative to any other kind of number; the criteria to which the other two definitions appeal are thus seen to be irrelevant and useless, and have no place in mathematics.

    Moral of the story: believe only 33.3% of what you read in Wikipedia!
  • BlueBanana
    873
    (2) In the arithmetic of the countable infinities, A = B; but: A - B = 10.alan1000

    Incorrect. That only holds for natural numbers. When A or B equals infinity, that's not true.

    (3) In other words, if we pair the elements of infinite set A one-to-one with those of infinite set B, we have 10 numbers left over.alan1000

    Does not follow.

    (4) Therefore, an infinity may be smaller than some natural number.alan1000

    Does not follow.
  • alan1000
    200
    Arguments?
  • alan1000
    200
    Thanks for your reply, BlueBanana. You understand, I guess, that the arithmetic of the countable infinities does not follow all of the same rules as classical arithmetic? For example, that infinity plus any number equals infinity, but infinity minus infinity may equal any number between 0 and infinity, depending on how the infinities were defined in the first place?
  • BlueBanana
    873

    3) You could assign every number n from the set A to number n+10 in the set B, or you could assign that to n+11 and have a number left over from B.
    4) Would not follow even if 3) was true so I'll just leave this to the burden of proof.
  • alan1000
    200
    I'm not entirely sure I follow.

    The numbers >=10 from set A can be mapped exactly onto the corresponding numbers in set B. Both sets are infinite, but when set B is exhausted, the numbers 0-9 in set A are left over with nowhere to go. These numbers must therefore be extra to the infinity of set B.

    This is dealt with by Russell in one of the later chapters of "Mathematical Philosophy". I don't have the book ready to hand so I'm afraid I can't give you any more precise reference than that; but it's a golden text - any mathematician who has not read it is not a complete mathematician.
  • sime
    1.1k
    The fact that infinite cardinals do not represent our intuitions of the relative sizes of countably infinite sets shouldn't feel problematic. After all an infinite cardinal is merely one suggested measure of size. You're still free to think of A and B as having different sizes and could even create your own formal definition of size satisfying this purpose.
  • BlueBanana
    873
    but when set B is exhaustedalan1000

    It won't be, because it's infinite.
  • sime
    1.1k
    It won't be, because it's infinite.BlueBanana

    I used to think that way, but that is to dogmatically assume that mathematics must serve practically intelligible purposes, as opposed to purely fictive or aesthetic purposes.

    Yes, the aesthetic reasons for defining finitely unconstructable hierarchies of infinitely large numbers are entirely subjective, practically meaningless and involve a degree of self-deception i.e. are what many irreligious non-platonists would call "bullshit", yet for some reason it is still fun.

    For some reason, convincing oneself that it is possible to exhaust countable infinity say via a non-standard construction of the integers, can produce pleasant feelings of expansiveness.
  • MindForged
    731
    A lot of this is just wrong but I'll focus on one thing because @BlueBanana is on the rest.

    (3) In other words, if we pair the elements of infinite set A one-to-one with those of infinite set B, we have 10 numbers left over.

    This is incorrect. You are confusing lacking particular unique elements (the numbers 1-10) as changing the cardinality. It doesn't. Both sets you specified have the same cardinality so there will be nothing left, they can be put into a one-to-one correspondence with each other.

    The set of even numbers lacks all the odd numbers, and yet that set can still be put into a one-to-one correspondence with the set of natural numbers which has both the even and odd numbers. I personally have never met a mathematician who (when speaking to a mathematically literate audience) define infinity as anything other than:

    "A set whose members can be put into a one-to-one correspondence with a proper subset of itself."

    Obviously that will be not understood by the lay audience. And further, infinity is a concept in common parlance too, it is not just a mathematical concept so the definition won't necessarily match. Perhaps I am misunderstanding you or your point; I am rather low on sleep at the moment, heh.
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