Re: infinity


Tony Orlow wrote:
> Randy Poe said:
> >
> > Tony Orlow wrote:
> > > Randy Poe said:
> > > > Your mapping is defined in such a way that there is no
> > > > x in X such that f(x) = X. This has nothing to do with
> > > > "largest finite". Your map, as you defined it, has the
> > > > property that no element x is in f(x). Therefore it's
> > > > pretty obvious without any references to "largest finite",
> > > > that there is no x in X such that f(x) = X.
> > > Not within the range of any given X, but in the overall infinite set, there is.
> >
> > Are you aware that if I claim I have a bijection from a set
> > A to a set B, then what I am claiming is that f takes
> > elements from set A and maps them to elements of set B?
> Sure, can you enumerate them all? Which even maps to the largest natural?

None does. There is no largest natural. You really are an idiot.

> > If f takes things that are not from set A and maps them
> > to set B, it is not a mapping from A to B.
> So, is twice the largest natural in the set of naturals?

No. Because there is no largest natural, there is no such thing as
twice it. You really are an idiot.

> > If you say "f is a not a one-to-one mapping from X to P(X)
> > but from an OVERALL set to P(X)", then f is not a one-
> > to-one mapping from X to P(X). It is a one-to-one mapping
> > from this larger set to P(X).
> If X is an infinite set, without end, and you start talking about a mapping to
> the entire set denoting every element individually, then you have identified a
> last element, and assigned a bit to it.This is the basis of your proof.

No you have not, for the umpteenth time. There is no last element. You
really are an idiot.

Got that?

Since this is abstract mathematics, not junior computing, we are not
assigning "bits" to anything.

> >
> > It's like saying you have an engine that runs on nothing
> > but water, and you prove it by filling your engine with
> > gasoline.
> No, it's like being asked, "How far can your car go?" and answering, "How much
> gas you got?". You folks seem to insist that I should be able to tell you how
> far the car can go without knowing how much gas I have to feed it. And yet,
> when it comes to power sets, suddenly the traqctor trailer doesn't have the gas
> to get all the way there. Motorcycles need gas too, and lesser bijections than
> the power set have value range considerations as well.

Because cranks can't understand proper arguments, it's terribly
tempting to assume they might understand analogies. But it never works
like that.

> > > > See how reasoning works? No "largest finite", in fact,
> > > > no reference to the properties of x at all. These could
> > > > be sets of mice or library books.
> > > But, in order to define any element mapping to the entire set, the last element
> > > must be identified
> >
> > Incorrect.
> >
> > For instance, consider this map from N to P(N).
> >
> > P(0) = {evens}
> > P(1) = {odds}
> > P(2) = {primes}
> > P(3) = N
> > P(k) = {multiples of k}, k=4,5, ...
> >
> > I did not have to "identify" a last element in order to find
> > an element (3) mapping to N.
> (sigh) No, and you didn't mention the last element specifically in your proof
> either, but if you want to talk about the element that maps to the entire set,
> that element's last bit corresponds to the last element.
> >
> > > and the set completed.
> >
> > All sets consist of their elements. Once you define membership
> > in the set, the set is completely defined. It's "complete".

> But an infinite enumeration has no last element.

Whoopee!! That's right, Tony. No last element. But we are still
considering the totality of all elements, none of them the last, with
no restrictions, no "value ranges", no "tenuous" or "variable"
not-the-last-element-but-let's-pretend.

> Any properties of the set
> which depend on this being identified will run into problems.

Absolutely they will. Sometimes I feel hope for you, because unlike
most cranks, you can occasionally produce a sequence of more than one
correct sentence.

> The best you can
> do is use the size/last element as a variable which tends to infinity, and
> derive properties that hold in the infinite case, inductively.

Oh dear. No, there is no "variable" blah blah blah inductively
anything. I'm beginning to see that the axiom of infinity may be hard
for some to swallow.


> > > Let me ask you this: When do you expect to "finish" this mapping?
> >
> > That's where this REASONING thing comes in. I don't have
> > to check each element one at a time, a process that will
> > never finish. I can REASON about all of the elements in
> > one step.
> You are asking about an element which maps to the entire set including it's
> last element, denoted with a bit in a specific place.

Three mistakes. "It's" is a misspelling. There is no last element. When
will you understand that if on lines 2, 4, 7, 11, 17, 23, 28, 33, and
40 of a post we say "There is no last element", on lines 46, 52, 57,
60, 63, 69, 72, 76, 88, 91, and 108 it should not be necessary to
repeat that there is no last element. There is no last element, you
moron?!!! Oh, and there are no "bits" either. This is mathematics, not
junior computing.

> What place is that, and
> why are there no bits after it?

Try answering this one yourself.

> > You really should try this deduction stuff, it's a real
> > time saver.
> Deduction has a partner, induction. The two go hand in hand, like yin and yang,
> the body and mind, female and male, plant and animal.
> >
> > > Are you going
> > > to go through all the numbers and find the set which corresponds to every
> > > single one?
> >
> > Once I've defined the general mapping rule, I use that
> > REASONING stuff to go through ALL OF THE NUMBERS at once.
> So, what happens to the top half of the naturals, in the mapping to the evens?

There is no "top half" of the naturals. The naturals do not end - it is
an ever-so-slightly surprising result (to an average intelligence) that
there is no point "half-way" to the end, because there is no end.

Incidentally, people have said that (e.g.) the fact that the set of
naturals and the set of evens have the same cardinality is at odds with
ordinary intuition. Well, it's not part of ordinary experience, but I
do not think it is at odds with common sense, actually. There's a joke
I've heard (not a mathematicians' joke either) in which some man is
offered three wishes by a pixie. We are supposed to giggle at his
dimness, because for the first wish he says:

"I'll have an inexhaustible bottle of Guinness, please."

And the pixie gives him this bottle, and sure enough, he drinks and
drinks and drinks, and the bottle is still full. Then for his second
wish he says:

"Begorrah(*), I think I'll have another one of those..."

(*) Yes I know.

Anyway, the point is that whereas a crate of real Guinness can be
improved on by having two crates of real Guinness, an inexhaustible
bottle of Guinness (which is fictional, but essentially an infinite
amount of Guinness) cannot be improved on by having two of them.

Brian Chandler
http://imaginatorium.org

.

Relevant Pages

  • Re: infinity
    ... >> Not within the range of any given X, but in the overall infinite set, there is. ... > elements from set A and maps them to elements of set B? ... it is not a mapping from A to B. ... far the car can go without knowing how much gas I have to feed it. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... You're starting to figure out that the reals and the naturals have ... You want to say that ...111 maps to N, ... The reason there's a natural mapping of ...1111 to N is that what we ... how to construct a well-ordering of an uncountable set, ...
    (sci.math)
  • Re: INFINITY Revisited
    ... such as the standard mapping of p/q to Natural numbers. ... Above in essence you said that the infinite ... > set naturals can "trivially" be placed in a one to one correspondence ... > numbers for a mapping between N the approximations of pi and pi, ...
    (sci.math)
  • Re: infinity ...
    ... >> to a mapping between *N and P. ... it was specifically a bijection between two sets of infinite binary ... >> from 0, both finite and infinite, in normal binary format, and on the other ... there are not enough naturals to map to every subset ...
    (sci.math)
  • Re: An uncountable countable set
    ... But, as in all infinite bijections, the formulaic ... "formulaic relationship", the explanation of which you seem to be avoiding. ... It's the mapping between set using a quantitative formula. ... naturals neN by the formula f=n-1. ...
    (sci.math)
Quantcast