Re: infinity

Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> stephen@xxxxxxxxxx said:
>> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>> > stephen@xxxxxxxxxx said:
>> >> I am not assuming that there is a longest word. A longest word
>> >> implies a largest natural number. You always deny that there
>> >> is a largest natural number, but once again you are using an
>> >> argument that depends on there being a largest natural number.
>> > I said "longest WORDS", not "longest word".
>>
>> There are no longest words either, so I am not assuming
>> anything about them. There is no maximum finite word length.
>>
>> >>
>> >> I have no idea what L is in your S^L. You are aware that there
>> >> is more than one string length, so picking a single L does not
>> >> make any sense. It almost makes sense if you think that L is
>> >> the maximum string length, i.e. the largest finite natural number.
>> >> Of course you also deny that there is a maximum string
>> >> length, so I have no idea what S^L is supposed to mean.
>> > Given any string length and alphabet, that is the maximum number of unique
>> > srings in the language.
>>
>> I asked what L is. L is not the maximum number of unique
>> strings in the language.
> You said you had no idea what S^L is, even though we have discussed this
> before. L is any string length as I said. What do you not understand?

Again, I do not understand what L is. Is L the length of any
string? Well, then lets look at the language of strings on
the alphabet {0,1} of length 100 or less. Is the size of this
language 2^55? 55 afterall is a length less than 100. Or is the size of
the language 2^23? What is the value of L I should plug into
2^L to determine the size of this language?

>>
>>
>> >>
>> >> You should have a summation:
>> >> S^0 + S^1 + S^2 + S^3 + .... S^k + ...
>> >> for all finite k. That is the number of finite length strings
>> >> over an alphabet with size S.
>> > Yes, and you are summing a finite number of terms, each of which is finite in
>> > value. Is the sum infinite? No.
>>
>> But I am not summing a finite number of terms. I am summing
>> all of the finite k. There are an infinite number of finite k.
>> You cannot assume there are only a finite number of finite
>> k when trying to prove that there are only a finite number
>> of finite k.

> Well, I have proven, at least to my own staisfaction, that you cannot have an
> infinite set of finite whole numbers, so when you say "finite k", I say you are
> counting a finite number of times and summing a finite number of terms, each of
> which is finite. You say I cannot assume that there are a finite number of
> finite naturals in trying to prove that there are a finite number of finite
> strings on a finite alphabet.

You cannot assume what you are trying to prove.

> Can you assume there are an infinite number of
> finite naturals, when trying to prove that there are an infinite number of
> finite strings using a finite alphabet?

No, but I do not need to. I already gave you three proofs
in another post. Here is the simplest of them again.

Suppose that the number of positive finite whole numbers is finite.
Lets call this number F. If F is finite, then F+1 is also
finite, and F+1 <> F. The set {1, 2, 3, .... F, F+1}
contains F+1 finite whole numbers. But this contradicts
the fact that there are only F positive finite whole numbers.
Therefore, there cannot be a finite number of positive finite
whole numbers.

Nowhere in that proof do I assume that are an infinite number
of finite naturals. In fact, I assume the opposite, and
derive a contradiction. If you do not agree with the above,
please point out the error.

<snip>

>>
>> >>
>> >> You claim that the above summation is finite. According to you
>> >>
>> >> F = sum S^k for all finite k
>> >>
>> >> is a finite number.
>> > That is correct.
>>
>> >> However if F is a finite number, then
>> >> there are strings of length F, and there are S^F strings
>> >> of length F, which is greater than F, the supposed number
>> >> of finite strings. That's a contradiction. Therefore
>> >> F cannot possibly be a finite number.
>> > Darryl just gave a similar "proof". When you say "for all finite k" then you
>> > are saying there is some upper bound to k.
>>
>> No, I am saying "for all finite k". There is no finite upper
>> bound for k. I noticed that you have conveniently ignored my proof
>> above.

> That was a proof? All it is is a rehashed statement that there is no largest
> finite integer. Sure, finite F can always be incremented, since finite k can
> always be incremented. That lack of a largest element, or longest string,
> doesn't prove infinitude of the set, as far as I'm concerned, so that doesn't
> prove anything to me.

Until you define what you mean by 'infinitude' no one will know
what you are talking about.

>> What about the strings of length F? Which of the following
>> do you disagree with:
>> 1. F is a finite number
>> 2. There are strings of length F
>> 3. There are S^F strings of length F
>> 4. S^F > F
>> 5. There are more than F finite strings.
> This is exactly the same as the idea that there is no largest finite. SO
> WHAT????

You claim that there are only F finite strings. But there
are clearly S^F strings with the finite length F. If
S^F is greater than F, how can you claim there are only
F strings?

Do you really not see the contradiction? How can there
be only F finite strings, if there exist finite strings with length
F, and S^F is greater than F? Just answer that.


>> You apparently disagree with 5, because you insist
>> there are only F finite strings. You definitely agree
>> with 1, and you seem pretty certain about 3 and 4. So
>> I guess 2 is the one you must disagree with, but
>> I do not see how you can claim that the set of all
>> finite strings does not include strings of length F
>> if F is a finite number.

> This is all based on trying to pinpoint that "largest finite F", which is a
> waste of all our time. It doesn't exist. What is your longest finite string, or
> largest finite natural? It doesn't exist. The nonexistence of a largest finite
> is irrelevant. There is no F s.t. no finite language can be larger. SO WHAT???

So what???? Do you really not see the contradiciton?

Here is another one for you. Let
F = sum of all finite k > 0
i.e.
F = 1 + 2 + 3 + 4 + ....

You of course claim that F is finite. If that is the
case, then F will appear on both sides of the above equation.

F = F + (sum of all finite k>0 and k<>F)

Given that F is finite, we can safely subtract it from
both sides of the equation, giving us:

0 = (sum of all finite k>0 and k<>F)

So according to you, the sum of all finite k>0 and not
equal to F equals 0. So either 1+2+3+ ... adds up to 0,
or there are no finite numbers greater than 0 other than F,
or some other equally bizarre case. I know you will
say 'So what?', but I find it hard to believe you really
cannot see the contradiction.

> I don't know what restrictions you put on the meaning of "language", but a
> language is simply a set of strings constructed from a set of symbols.

I have no idea what you are talking about. A language is a set of
strings. I have never said otherwise.


>>
>> L = a*(ba+ab)*b
>>
>> is a language containing an infinite number of strings
>> (all of which are finite). I do not create this language
>> a string at a time. I define that language with some
>> finite structure, such as a regular expression, and I am done.
> That is one way to construct a language. BTW your language isn't infinite
> unless you allow the strings to become infinite.

My language is infinite, even though all the strings are finite.
I can even prove it. Of course you are using your weird
private definition of 'infinite', but I do not care about
your definition.

And that is the only way to "construct" an infinite language.
It does not have to be a regular expression, but it has
to be defined with some finite structure. You cannot
list all the elements of an infinite language.

>>
>> >>
>> >> The set of all finite length strings over the alphabet {0,1}
>> >> is an infinite set. There is no longest string in this
>> >> set, and there is no L to plug into your S^L formula.
>> > If the string length cannot be infinite, then the language cannot be infinite.
>>
>> So you keep saying without proof. That may be your definition
>> of infinite, but your definition of infinite does not
>> include unending sets that any reasonable person would call
>> infinite.
> Apprently, I have a non-standard understanding of infinity, but nothing I have
> heard here convinces me that I am in the least bit wrong.

You do have a non-standard understanding of infinity. If
you would actually share your definition of 'infinity' perhaps
someone could make sense of what you are talking about, but
I doubt you will as that will probably just reveal more
contradictions. If the fact that noone else in the world
agrees with you does not convince you that you are in the
least bit wrong, then I suppose nothing will. Hey, if you
decide that the word 'cat' should really mean 'dog', then
you probably would not think you were in the least bit wrong either.

Stephen
.

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