Re: infinity

Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> 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)

So when I plug in all the finite L, I get an infinite
sum of finite values, i.e.
sum (x>=0 : 2^x)
Note, I did not plug in a single value of L like you
kept insisting. There are an infinite number of finite values
for x.


>> >> 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.

Very mature response. Your entire S^L proof that there
are only a finite number of values for L is based entirely
on the assumption again that there are only a finite number
of values for L.

Explain again the relevance of S^L without assuming there
are only a finite number of finite values.

>>
>> > 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?

My point is that if I assume that there are only a finite number
of finite numbers that it leads to a contradiction.

>>
>> 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.

What does it matter what we name it? You claim there is
a finite number that is equal to the number of finite numbers.
It does not matter if we call it F, or X, or Tony or whatever.
You claim this number exists. If it exists, then that number
plus one exists, and that number plus one is also finite.

<snip>

>>
>> 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.

But you claim that the sum of all finite numbers, which clearly
should be larger than any individual finite number, is
a finite number. So no, I have know idea what you mean by
'infinitude'. After all, you claim that the sum
1+2+3+.....
is not larger than every finite number, and is in fact equal
to some finite number.

>>
>> >> 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.

You claim there are a finite number of strings. We can call
that finite number F, or X, or Tony, or whatever, if it
exists as you said it does.

You cannot claim that there are a finite number of finite numbers,
but then claim that we cannot call that number F.
You are just becoming increasingly dense in your effort
to not see the simple contradictions your ideas lead to.

>>
>> 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 are the one who keeps invoking 'Largest Finite'. All
your arguments implicitly rely on a largest finite.

Once again, tell me why
1 + 2 + 3 + ....
is a finite number?

<snip>

>>
>> 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?

I never mentioned anything about a largest finite number.
There are obviously numbers larger than F, such as F+1.
But according to you, F = 1 + 2 + 3 + ... + F + F+1 + F+2 + ...
because it equals the sum of all finite numbers, and it to
is a finite number.

You are the one who claims that
1 + 2 + 3 + 4 + ....
is a finite number. Accept the implications of your claim
or drop it. Stop trying to pretend that other people are
claiming these absurdities.

>>
>> 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.

Of course I cannot deny that. But you claim that
sum (x>=0 : S^x)
is finite. And you claim that
sum (x>=0 : x)
is finite. Why can't you talk about those claims? Why
do you always try to deflect the discussion to the irrelevant
bounded cases?

>>
>> > 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?

For none, which you have been repeatedly told. The
language is infinite the moment I define it. It does
not start out finite and 'become infinite'. Once
again you are making up ridiculous distractions that
have nothing to do with the argument.
>>
>> 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.

Well, you are the one who claimed that there was some
other way to construct an infinite language.


<snip>

>> > 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.

Yeah, and those cats might really be dogs.

Word definitions really cannot be wrong. Language is
a communal thing. If you insist on using words contrary
to the usage of the community, then you will just be
misunderstood. Insisting that the community's definitions
are 'wrong' is like saying that the French are wrong
because the do not speak English.

Stephen
.

Relevant Pages

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