Re: abundance of irrationals!)

Randy Poe said:
>
> aeo6 Tony Orlow wrote:
> > Randy Poe said:
>
> > > "Usual order relationships"
> > >
> > > Two elements a and b are related by either "<",
> > > "=", or ">". That is, for any pair of elements we
> > > will say that exactly one of "a > b", "a < b" or
> > > "a = b" is true.
> > >
> > > > and tell me if that applies to your "set of all finite natural
> > > numbers".
> > >
> > > Yes, the finite natural numbers have a complete
> > > order relationship. Take any two and we can establish
> > > whether a<b, a>b or a=b, only one of these is true,
> > > and we know how to determine it from the base-10
> > > representation.
> > The same can be said of any two completely define infinite whole
> numbers.
>
> The same can be said of anything. You can always define
> an order relationship. You asked me to define "the usual
> order", I did.
>
> > I see nothing yet that implies any largest member,
>
> No, that's the definition. Most proofs have steps beyond
> the definition. You asked for a definition, so I gave that
> first.
>
> > but the largest infinite whole in
> > base 10 is 99999.....99999.0000....
>
> Tell me what's in those dots in the middle. Is it unique?
> Are there numbers like 9999...8...9999? Do they differ
> in the number of digits corresponding to the left
> ellipsis (...) and the right?
If you assume corresponding ellipses denote the same infinite substring, then
one can compare any two such numbers. 9999...8...9999 does not specify any one
exact value in the infinite set, but if the ellipses to the right of the 8
represent all 9's then the subsequent number is 9999...90...000, assuming the
ellipses to the left represent the same string. One has to assume in such cases
that corresponding ellipses are equal.
>
> > > If you like, I'll go into that too.
> > >
> > > > Axioms, please, and logic. No leaps.
>
> Did my best to comply, starting with the definition.
> Sorry that you took issue with my attempt to begin
> where you asked me to begin.
You didn't finish. Derive for me the fact that a finite set necessarily has a
finite number. You started. Where do you go next? Hmmmm.....
>
> > > Now consider a finite set. That means that the elements
> > > are in one-to-one correspondence with the natural numbers
> > > {1,2,3,...,n} for some finite number n. That is
> > > the definition of finite set.
> > That's one definition. Perhaps that should be called a "definite"
> set, with all
> > elements identified.
>
> Another starts from the bijective definition of infinite
> sets. Do you have another in mind?
Bounded, perhaps? All I know is that what we know about infinite series
contradicts the notion of an infinite set of distinct finite naturals. There is
no clear line between finite and infinite, value-wise.
>
> OK, so I'm starting to get the feeling that you're going
> to reject my argument on the basis that you have a
> meaning for "finite set" that includes what I would
> call "infinite set". In that case, I'm going to end
> up agreeing that not all Tony-finite sets have largest
> members.
>
> Do you agree that all "definite" sets have largest
> members?
Sure, by "definition". I have no problem with seeing that such sets exist and
are commonplace, and have largest members if they all have distinct quantities
associated with them.
>
> > You defined a finite set as one that has a last element. Is there
> > an axiom that states this?
>
> Two comments: I defined a finite set as one that has
> elements that can be labeled from 1 to n for some natural
> number n. There's no particular order implied, so
> there's no definite "last element". You can count
> them in any order you want.
As long as you go from 1 to n. Order is more important in many cases than set
theory admits.
>
> Second, I can't imagine what you mean by "axiom" in
> addition to the definition. You're asking if there's
> an axiom that says the word "finite" means "has certain
> properties" implies that "finite" means something has
> those properties? What?
You said finite sets by definition have largest elements, and you offered a
definition that doesn't mention anything about largest members, so all I can
imagine is that you must have an axiom that states this, or be applying some
axiom to that definition to derive that fact as a theorem. Is either one the
case, and if not, then how can you simply declare such a "fact"? Are you not
just being argumentative?
>
> "Finite set" is a shorthand for "can be put into one-one
> correspondence with {1,2,...,n} for some n." What axiom
> is missing? I'm just setting up a synonym.
Yes, finite means it stops. There is an end. Does that end need to be larger
than the middles? If a set is a finite ring, does it have a largest element?
>
> > > When it stops, M has the property that M >= a_i for
> > > i={1,2,3,...,n}. The proof of this is a finite
> > > induction on iteration number.
> > So, counting stops?
>
> Counting to a finite natural number stops. That
> is part of the definition of finite natural
> number.
Fine. Does counting the naturals as a whole ever stop? No? What does that tell
you?
>
>
> > > P.S. I was looking for your set cardinality definition,
> > > but haven't found it. Please repost and I'll comment.
> > >
> > >
> > >It was the one I posted at 1:19 yesterday, 5/10. I don't think you
> > >looked very hard,
>
> I scanned about 100 posts. That's a lot of words.
>
> > or maybe didn't get to the bottom of the post. Here's what I wrote:
>
> OK.
>
> > The size of a set of real numbers defined using a function f(n) on
> the naturals
> > from x to y,
>
> What does this mean?
>
> Suppose x = {1.3, 2.6, 5.7} and y = {0.5, 0.4, sqrt(37)}.
>
> What is "f(n) on the naturals from x to y"?
> Can you give an example? Normally f:x->y, a
> mapping from x to y, takes elements of x and
> produces elements of y. You are talking about
> three sets: x, y and N. I don't know what you have
> in mind.
Those don't look like the kind of sets I have defined this method as
addressing, now do they? I don't see a formula defining them, do you? If I say
I can prove something about red apples, do you complain that it doesn't apply
to your green bananas? Pay attention to the scope of the proof and reread.
>
> > is defined as the largest whole number less than the value of the
> > complementary function g(n) at y,
>
> What does "complementary function g(n) at y" mean?
I should use the term "inverse function" as it turns out, but I defined it
clearly: g(x) is the complement (read 'inverse') of f(x) iff g(f(x))=f(g(x))=x.
>
> > minus the largest whole number less than the
> > value of the complementary function at x-1.
>
> Again I can't figure this out. Example?\
I gave examples. Are you daft? Try reading the entire thing, and then
responding.
>
> > The complementary function of f(n)
> > is defined as the function g(n) for which f(g(n))=g(f(n))=n.
>
> OK, we normally call that the inverse. But what sort of things
> are f(n)? Are they in x? y?
What? Whatever you are asking is probably answered further down, since you
can't seem to read an entire sentence without objecting at every conjunction.
>
> > Let's look at the set {2,3,4.....}. We map this to the naturals s.t.
> > f(n)=n+1, the complement of which is g(n)=n-1.
>
> So f(n) takes elements of N and produces elements of x. There
> is no y.
Yes a function from the naturals to the reals, as I said. If you don't
understand those words, it is a function that takes a natural number as a
parameter, and produces a real number result (which may of course be a
natural).

When we say f(x)=*&^(&^, that is the same as y=*&^(&^. f(x)=y, when you are
graphing functions. Where did you learn math?
>
> Is f(n) taken by axiom to be a bijection? How do you know
> it always exists?
Bijections are irrelevant to what I am doing. In my world there is a 1-1
correspondence between the mention of bijections and the resulting objections.
So please leave them at the door with your muddy shoes. This has been a public
service announcement. Thank you.

To answer your question in real words, I think you are asking whether the
output of the function is always a natural. It need not be. The set is a subset
of the reals defined by a function on the naturals. For every natural there is
a real value in the set.
>
> > We take the entire range, up to N (aleph_0 if you must),
>
> Well, I can't call it N, since N is a set and not a number.
It is both. We're not in Cardinality Land anymore, Randy. N is the set, and
also the set size for shorthand. In English we learn to glean a lot from
context. Does |N| make it clearer? The bars seem superfluous if we're talking
about the size of the set. Alephs suck anyway. Who needs a new special symbol,
except to make things more mysterious?
>
> > so if the entire set of naturals is N, this set is N-1.
>
> Do you mean the cardinality of this set is aleph_0 - 1?
> Because N is a set, and N-1 is a subtraction of a number
> from a set.
>
> Is aleph_0 - 1 < aleph_0?
Yes, correct. A proper subset is always smaller than the superset. You may
consider that an axiom until I figure out whether it is derivable from more
basic axioms.
>
> > If we try
> > a finite range, say x=3,y=10, we get g(10)-g(2)=8 members of the set
> in that
> > range. Not convinced? Let's try another.
>
> No, I'm convinced that a plausible order of "bigness"
> can be constructed this way to compare some subsets of the
> naturals to the set of naturals. I'm not convinced
> you can compare all sets this way, nor am I convinced
> that you can even compare all subsets of the naturals
> to each other.
I can order all subsets of the reals that are defined using an invertible
function on the naturals in this way. You will notice that using this method,
there is no function, like 2^N, as Cantor suggested, that states the
relationship between the naturals and reals. That is, there is no finitely long
function on the naturals which completely fills the real line with values.
>
> Is the set of odd numbers bigger or smaller than the
> set of even numbers?
The same. Half of all whole numbers. You see (crank alert, yeah yeah, I know)
infinity is odd, which balances zero's even. There is a reason out there,
somewhere waiting to be found, that infinity is prime, but it is still eluding
me. I hope that gave you a big laugh. I hope more that it actually made sense
to you, but I doubt it.
>
> How big is the set of rationals on [0,1]? Here's an
> f(n): f(n) = 1 - 1/n. Using that, what do you get
> for your "bigness" of the set of numbers {1-1/n}?
> How about the "bigness" of the set of all rationals
> 1-1/n?
This method doesn't address the rationals, or any other dense ordering of the
reals. Dense orderings cannot be defined by finite functions on the naturals.
So let's move on to the pertinent part, where yo define your set as a function
of a single variable in N.

F(n)=1-1/n, g(n)=1/(1-n). Let's look first at finite subsets to make sure this
works. The range of the function is from zero to 1, so we must use values in
taht range to define a subset. We can see that there is no element under 1/2 in
the set, so we need to pick numbers between 1/2 and 1. Let's say 0.5 and 0.7.
Floor(g(0.7))-floor(g(0.5))= 3-2=1. The only element of that form in the range
is 1/2; the next is 3/4, outside the range. Let's extend the range to 0.8. Now
we have 5-2=3. We have now included not only 3/4, byt 4/5=0.8 as well. Seems to
work! So, in applying the method to the entire range from 0 to 1. g(1)=1/0 and
g(0)=1/1, so we have 1/0-1/1. Ah, but you say 1/0 is undefined? Well guess
what? 1/0 is DEFINED in this system to be exactly equivalent to the unit
infinity, N. Therefore, we have N-1 as the size of the set. This jibes pretty
well with the fact that we can draw a 1-1 correspondence between the naturals
and the terms 1-1/n, with both in quantitative order. It stands to reason they
would be the same size set. The difference of 1 is slightly troubling, but not
terribly. errors of 1 are the most common programming mistake and easily fixed.
> What about sets for which no easy functional relationship
> exists?
What does Cantor say? Can you draw a bijection? If we can mathematically define
an asymptotic limit to the size of the set, as with primes, then we can make
statements using those facts. This method cannot be applied to every single
set, but it certainly can be applied to the sets as I've defined them, and
produces interesting results that cardinality denies.
>
> > Do you still think bijections are the most robust possible
> generalization from
> > finite to infinite sets?
>
> Yes.
>
> > Do you still think I am a crank?
>
> I think you have some naive misconceptions that you
> are fond of, and think that mathematics needs to be torn
> down rather than yourself being educated. You can put
> any label for that you like, but apply that description
> to somebody storming the halls of any field, and ask
> yourself how you would describe such a person.
Kind of like Einstein and Newtonian physics? This is a relativistic approach. I
am not tearing down all of mathematics, only the rotten bug-ridden part that
needs to be rebuilt. The rest of the mansion is in good condition, except the
paths from room to room are too long and sometimes inaccessible. It's this
particular outbuilding, this tool shed, that is ramshackle.

You are the particle and I am the wave. You are the deducer, I induce. Without
the axioms that induction creates, your deductions have nothing to work with.
Without research into facts, induction has nothing. Induction ensures that our
deductions have basis in fact. If this is naive, then what is sophisticated?
The idea that truth can be contradictory or have nothing to do with reality?
Perhaps that is mature, in the sense that any senile old man is mature.
>
> - Randy
>
>

--
Smiles,

Tony
.

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