Re: infinity

stephen@xxxxxxxxxx said:
> 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?
All of them up to 100.
sum(x=0->100: 2^x)
>
> >>
> >>
> >> >>
> >> >> 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.
I didn't. You did.
>
> > 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.
That only proves there is no largest whole number. So what? Does the lack of a
largest finite mean that there can be no infinites? What is your point?
>
> 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.
The contradiction is not in the idea of a finite set but in naming the size F
as the largest finite.
>
> <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.
I mean larger than any finite, which apparently doesn't hold for you.
>
> >> 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?
I claim that there are a finite number of finite strings. I never claimed there
were F of them. You are again trying to shove the largest finite in my mouth,
and it's irrelevant. Whatever finite F you give me, S^F is finite, and sum(x=
0->F:S^x) is finite.
>
> 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.
Huyah huyah Ommmmm...... "Largest Finite". I never said 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?
Yeah, really not in anything I said. You concocted a contraidiction, but it was
irrelevant.
>
> 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)
Are you claiming there is a alrgest finite number? That would appear to be what
you are doing. Why would you do that?
>
> 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.
No, according to you that is the case. This has nothing to do with anything
that I have said. When did I make any such argument. You cannot deny that for
any finite L sum(x=0->L: S^x) is finite.
>
> > 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.
So, for which L does sum(x=0->L:S^x) become infinite?
>
> 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.
No kidding.
>
> >>
> >> >>
> >> >> 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.
I do not judge my thinking in terms of whether others will like it or agree. I
jusdge it in terms of the consistency of the conclusions it draws and how well
they mesh with reality and other thinking. I am obviously not the only one who
objects to the bizarre reasoning in this area, so I don't feel incredibly alone
anyway. But, even if I were the only one on the the planet who thought like I
did, I still would not see that as a areason to believe I am just wrong.
Everything has to start somewhere.
>
> Stephen
>

--
Smiles,

Tony
.

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