Infinate Numbers, Its Crazy.

[DaneH]

Heres my question, if there are an infinate amount of numbers, then there has to be names for all of these numbers, right? So does that mean no matter what sound you make with your mouth, even if its jibberish, its a word for a number??

[Finagle]

Hmmm. Two quick misspelled questions. I can't decide if the poster is a troll or just another product of the MTV generation.

But oh well. The primary misconception is that you actually have to have a name for each number. Heck, why? We don't go naming every atom in the universe, and there's probably a finite number of those.

But you don't have to make up new names for very large numbers because they can be expressed as multiples of existing numbers, plus some additive quantity with a known name. Or just use some form of exponential/scientific notation and be done with it. Someone with a better math theory background than me can say whether there are infinitely large numbers such that it would take an infinite long expression to describe it.

[Konrad]

Actually, this question isn't that stupid. You can say that each number has a name as long we assume that 1 000 000 000 = 1 thousand million and so on. That way each name is a multiple of other names.

So, yes there are an infinate number of names BUT that doesn't mean that any sound you make is one of those names.

After all, there are an infinate amount of even numbers, but not any number you pick is an even number.

[pluto]

I'm willing to concede that there may be a name for every number. The name may be extremely long for extremely large numbers but it wouldn't be infinite because no single number is infinite.

However, that doesn't mean that every sound we make corresponds to the name of a number. There are also an infinite number of possible utterances, only some of which will correspond to the name of a number. The existence of two infinite sets doesn't imply that the sets are equivalent or even that they overlap at all.


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Let's see. I lathered and I rinsed. But did I repeat?

[pluto]

That Konrad is a smart fellow. I can tell because he thinks like me.

I do, however, spell better than he does.

It's spelled infinite.

[bantmof]

For just the briefest of moments I thought I was in the "What's the deal with LSD" thread.

--
peas on earth

[mega the roo]

in elementary school we had this computer that would read everything you'd type. so we'd just sit there and type a one, then hold down the zero for a couple of minutes. it came up with some pretty interesting things, i wish i remembered some of them.

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"human beings, vegetables, or cosmic dust; we all dance to a mysterious tune, intoned in the distance by an invisible piper." - albert einstein

[dlv]

What is a finite ordinal number, like 2 or 5?

My favorite construction is to take the equivalence classes of finite sets (two sets are equivalent if they have the same cardinality) and say, e.g., that 2 is the class of sets with cardinality 2, 5 is the class of sets with cardinality 5, etc.

We can extend this construction to cardinalities of infinite sets. For example, there are infinitely many integer numbers (no matter how many we list, we can always add one more). George Kantor denoted the infinite number of integers (the cardinality of the set of integers) by Hebrew letter Aleph with a subscript 0. We mathematicians pronounce it Aleph-nought. It looks sort of like X0, except it's Aleph instead of X.

Kantor also showed that the infinite number of real numbers (like 1/2, pi, sqrt(2), etc) is greater than the infinite number of integers. He denoted this infinite number by X1 (pronounced Aleph-One). (You would probably understand his proof, but find it boring.) In a similar way, one can construct larger and larger infinities, called X2, X3, etc.

[KeithB]

Actually, there is an interesting variation on the "long, boring" diagonal proof in this thread:

Take the integer numbers: 1,2,3...

Now take all the *names* for the numbers:
one, two, three...

Find a *sound* that is not a name:
knot-won

Since there are more sounds than names, the number of sounds is greater than the number of integers.

[Earl Snake-Hips Tucker]

Here's a great riddle that shows how you can get some flaky results when you start dealing with "infinite" numbers:

How many numerals contain the numeral "3?"

Answer: 100%

Amazing.

[Cabbage]

He denoted this infinite number by X1


Actually, he denoted it by c (that's the way it's usually denoted anyway, don't know if he actually was the first to do it), and also showed that c = 2X0.

The idea that c = X1 is known as the Continuum Hypothesis. Unfortunately, it can neither be proven or disproven.

[dlv]

Oops! You're right. I stand corrected.

[DrMatrix]

KeithB You must be careful when using infinite numbers like that.
If I map 1 -> a2, 2 -> a3, ... n -> a(n+1) then a1 has no match, but you cannot conclude that { a(n) } is larger than { n }.

There is a paradox involving naming numbers. What is "the smallest number that cannot be named in less than 100 syllables"? Oops.

Finally just to add to what Cabbage said:
The idea that 2 to the Xn equals X(n+1) is the Generalized Continuum Hypothesis. GCH, like CH cannot be proven nor disproven.

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Virtually yours,

DrMatrix

[C K Dexter Haven]

I'm a little confused by the OP... Do you think numbers come with God-given names, so if you mumble some phrase like "plbbbbt", that's the name of a number?

The names of numbers are the names we give them. In English (and in most modern languages) there is a naming convention for how to assign a name to a number.

Yes, it is true, a few special numbers like "google" and "google-plex" were invented (mostly for the fun of it), but that does not mean that "plbbbbt" is also a number.

There are an infinite number of numbers, but there are only a finite number of words (let's say, for the purpose of illustration, that a "word" has fewer than 100 letters from the English alphabet)... so there are only a finite number of words -- very many, to be sure, but still a finite number.

[Revtim]

There is in fact an infinite number of (possible) words.

Since there is an infinite amount of numbers, you can create a new word for each number (each new word being a concatenation of previously-defined words, as others have pointed out). Of course, you may feel that it isn't a "new" word, being built of previously-existing words.

Getting back to the OP, any random noise you make will only be a number if it happens to follow the rules for building number-names.

[Revtim]

CKDextHavn, why should words have limits in length?

[pluto]

I get your point DrMatrix, re the difference between countably and uncountablyl infinite, but we can still name any number we come up with. In that sense every number has a name.


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Let's see. I lathered and I rinsed. But did I repeat?

[handy]

One thing about the names for numbers is that the word gets longer the bigger the number.
[Usually, haven't checked an infinite number yet].

[DrMatrix]

I think Revtim answered the question. What I was trying to say in my previous post was that I question the premise:[T]here [have] to be names for all of these numbers, right?I meant that if by numbers you mean reals you will not be able to find names for all the numbers. You cannot even name all the reals between zero and one. Cantor's famous diagonal proof shows this.

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Virtually yours,

DrMatrix

[DrMatrix]

If by numbers you mean real numbers (instead of just the integers) then even if we put no upper limit on the size of names for numbers, you cannot have a name for every number. The set of (finite) strings of letters is countably infinite, but the set of reals is uncountable.

Revtim If I may answer for CKDextHavn[W]hy should words have limits in length?Answer: So we don't have to scroll left and right when reading posts on this message board.

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Virtually yours,

DrMatrix