Re: infinity

Virgil said:
>> Only for computer languages for which there is always some limit on
>> both S and L. For non-computer languages, there is no inherent
>> limit on either S or L, so no inherent limit on N. That TO limits
>> himself does not mean that everyone is similarly limited.
>>
Tony Orlow (aeo6) wrote:
>> If you limit your words to a finite length while using a finite
>> alphabet, then you have limited the language as a whole to a finite
>> maximum size.
>>
Virgil said:
>> See, TO is doing it again.
>>
Tony Orlow (aeo6) wrote:
> Actually I am exactly right, and you offer no alternative formula or
> interpretation.

Sure, given S symbols in L digits, you can represent exactly L^S
unique values. But why does this apply to natural numbers?

Why is there some fixed (presumably finite) upper limit on the
number of digits allowed for a natural number? Isthere some
law of numbers that has previously gone unnoticed?

It would help if you could specify what this limit L is. Something
more concrete than 'N' or ceil(log(N)), whatever those are.

>
Perhaps a little device might help. Consider Tr(x), a function
that takes the decimal fraction part of x and reverses its digits,
so that the 10^-1 digit of x becomes the 10^0 digit of Tr(x), the
10^-2 digit becomes the 10^1 digit, etc. For example, Tr(1/4) is
Tr(.25), which is 52. And Tr(6/25) = Tr(.24) = 42.

Now perhaps you can tell me how digits Tr(1/3) = ...333 has?
Or maybe even Tr(pi) = ...3823979853562951413? If you know L,
then perhaps can you tell me what the first (leftmost) digit of
Tr(pi) is? Is Tr(pi) greater or less than, say, Tr(sqrt(2))?

We cantankerous Cantorians will point out that Tr(x) is undefined
for all x that are not some multiple of 1/10^k for some finite k,
i.e., Tr(x) is undefined for values of x with nonterminating
decimal fractions, which produce infinite (divergent) values
for Tr(x). That should not stop you, of course, from declaring
that Tr(x) is well-defined for any x, since you posit the existence
of infinite integers.

.

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