Re: abundance of irrationals!)

aeo6 Tony Orlow wrote:
> Russell said:
> > I already responded once to this, but seems that the
> > Google dog ate it. Second try.
> >
> > Tony Orlow (aeo6) wrote:

[snip]

> > I didn't disagree that the branches (or paths, as I
> > called them) are infinitely long. Our disagreement
> > is this: you think this invalidates my earlier claim
> > (that each node in the tree is represented by a finite
> > path) and I do not.
> if a branch is infinitely long, how long is the string of bits that
specifies
> the node at the end (or finitely close to the end) of that branch? If
the node
> is infinitely far down a branch, it requires infinite bits, one for
each fork.

You assert that there *is* such a node, I can prove
that there isn't. Your only counter is that my axioms
are bad. But AFAICT you don't have any suggestion of
better axioms to use. Rather an impasse, don't you
think? It might be instructive to you, to try to come
up with a set of axioms that are acceptable to you.
I think it will be impossible, but go ahead and try.

> > I hope you can see that this is just the same old
> > disagreement in disguise, to wit, my claim that the
> > naturals are all finite (which you dispute) despite
> > the fact (which we agree on) that the set of all
> > naturals is infinite. To see the correspondence,
> > we of course only have to count along the nodes on
> > some infinite path of our choice -- each node along
> > the way gets a (finite!) number assigned to it by
> > this counting process. I claim we can walk along
> > and step on *all* of the nodes without ever stepping
> > on one that gets an infinite number assigned to it.
> > (Of course it would take forever, so I don't mean
> > this literally.) This is what you dispute.
> Yes, it's illogical to believe this. I can't emphasize this strongly
enough.
> You want to tell me you can walk an infinite number of constant
finite steps in
> a straight line, and still be a finite distance from your starting
point.

No, I don't want to tell you that. You have drawn
a false conclusion from what I said; the conclusion
is yours, not mine.

[snip]

> Well, I feel pretty resolved in my position, by allowing that the set
of
> naturals is infinite, and therefore requires infinitely long strings
of symbols
> to enumerate the elements,

Btw, there's a subtle trap here that has got you. Yes
you need a digital display (so to speak) with infinitely
many places, to be able to display every n. But, if you
allow *every* pattern to appear on that display, some of
those patterns do not represent natural numbers. Indeed,
a great many of them do not. The only patterns that
represent naturals are those with infinitely many blanks
appearing on the left. (Or zeroes, if you use that padding
convention.) I think you have failed to appreciate this
subtlety. If you try by brute force to redefine the
naturals such that any display will work, you end up with
something that does not have the properties we expect from
naturals, e.g. simple addition is not well defined.

[snip]

> > > You are new. I have proven that you cannot have an infinite set
of
> > natural
> > > numbers which are all finite, based on infinite series
> > (1+1+1+....=oo) and
> > > based on infomation science (S^L is the number of strings of
length L
> > made of S
> > > different symbols, one of which must be infinite for infinite
sets of
> >
> > > representations). It is patently impossbile to squeeze an
infinite
> > number of
> > > numbers with a finite difference between the values of each pair
of
> > them, into
> > > a finite range of values. Therefore, if the set of naturals is
> > infinite, then
> > > it must clearly contain infinite values, which must be infinite
> > strings of
> > > digits, given the defintion of the digital number system.
> >
> > I followed this with nodding assent all the way down to
> > the word "therefore"; but at that point I saw nothing
> > in what had preceded it to justify this word. All you
> > did, up to that point, was show (really, assert -- but
> > let that pass) the infinitude of the naturals. That
> > wasn't in dispute. Your sentence beginning "therefore"
> > puts the result of your previous reasoning into an "if"
> > clause and asserts that which was to be proved (the crux
> > of it, anyway) in the "then" clause -- but you don't tell
> > us anything at all to justify this if-then pairing. You
> > say "clearly", and that does convey your strength of
> > conviction that this disputed conditional stmt actually
> > holds -- but for us doubters, the word "clearly" by itself
> > is not adequate justification. Sorry, not a proof.
> So, you assert that S (number of symbols=digital number base) is
finite, and L
> (length of string=# digits) is finite, but S^L (the size of the set
of strings
> of length L constructed from a set of symbols of size S) is infinite?

No, I do not assert that. I told you in as simple
terms as I could find, what is wrong with your proof,
and you seem not to have understood anything I wrote.

Regarding S^L, what you wrote is true enough (if we
add some missing definitions) but it has nothing to
do with your sentence beginning "therefore", i.e. the
sentence that is in error. Although you *do* apply
the S^L stuff later in your argument to get your
infinite strings, this cannot resuscitate an argument
that is already dead. That's why I said nothing about
S^L in my previous post.

Had you *not* written your erroneous sentence, I might
have needed to go on and talk about blank (or zero)
padding on the left as per my other comments above.
This I think is the fatal flaw in the approach you
are trying to take; but you didn't even get that far.

.

Relevant Pages

  • Re: infinity
    ... The natural number, n, in the set of naturals, N, is finite iff the set ... If there is no finite upper bound to a set of finite strings, ... >> then the set has to be infinite. ...
    (sci.math)
  • Re: infinity
    ... > Cantor-infinite, and so the set of strings will also be ... Sure, Cantor-infinite, but not actually infinite. ... the Cantor-infinite set of finite naturals. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... uncountably infinite set of strings? ... representation of naturals conundrum, which is where I came in. ... A language is a set of strings constructed from a specific alphabet ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... Tony Orlow wrote: ... bit strings. ... How do you cull the set from uncountably infinite to a countably infinite set ... Naturals are natural, ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... and diagonal traversal does not cover all strings. ... Do the math, and stop playing bad logic games, and declaring nonexistent differences between the finite and infinite. ... Any such list is exponentially longer than it is wide in digits. ... If they are a larger set than the naturals, then that is a valid conclusion, perhaps, but to say they can't be enumerated like the naturals, is just wrong. ...
    (sci.math)