Re: infinity

Jonathan Hoyle said:
> >> That's because your countable infinities are actually unbounded finite sets.
>
> That is a contradiction in terms. If the number of elements is
> unbounded, it obviously can't be finite. (Obvious to those who use the
> understand the inferences of logic, anyway.)
The inferences of logic are dependent, not only on proper application of
logical operations, but on the use of proper assumptions. That is where we
differ.
>
> >> >> Just like you think that you can add 1 an infinite number
> >> >> of times and still get a finite, you are wrong.
> >>
> >> I never thought this, nor do I know of anyone here who claims it
> >> either. You have created what logicians call a "Straw Man" argument, a
> >> form of logical fallacy in which you have changed the opponent's
> >> argument to something you want to attack, when in fact your opponent
> >> never made such a claim.
> >
> > Wrong. The generation of the naturals involves an increment of 1 for
> > each natural number.
>
> Correct.
>
> > When you claim the Peano axioms generate an infinite set with an
> > infinite number of elements, one for each iteration, with an increment,
> > you are claiming exactly that.
>
> No, and here we find the Straw Man. We make no such claim. You're
> putting words in our mouths, presumably because you are unable to
> defend against what we really say. You create the Straw Man to defeat
> it because you can't defeat the real argument. I agree with you that
> the Straw Man is incorrect. But we are not claiming the Straw Man,
> only you are.
This is not a straw man. I notice you fail to state, if my characterization of
your position is incorrect, exactly how it differs from your actual position.
Let's see what you say next.
>
> >> You add 1 to a natural an infinite number of times, and yet claim
> >> it is finite
>
> The Straw Man again. We never add 1 to a natural an infinite number of
> times to get a number. We add 1 to the last natural number, and
> continue doing that an infinite number of times. You claim that it's
> the same thing, but that's your claim, not ours.

It is obviously the same. let's take 4 as an example. According to you, 4=3+1,
right? But 3=2+1, so 4=2+1+1, right? But 2=1+1, so 3=1+1+1, and 4=1+1+1+1. When
you have a set of nested increments, you are adding a value equal to the number
of levels of nesting. You want to make a distinction between 1+1+1+1.... and 1+
(1+(1+(1+(...(1)))).... What is this distinction that you feel is so salient?
It's not. You generate an infinite number of numbers, each 1 geater than the
last. If you generate N elements, the last is N-1 greater than the last. What
is your N? Is it infinite? Well, then, you have infinite values in the set.
Yours is the straw man argument.
>
> >> forgetting in your axiomatic haze, that induction rests on the
> >> notion of an INFINITE recursive chain of logical implication.
>
> It is in fact the axiomatic method that ensures that we are doing it
> correctly. Your lack of it is what causes your confusion and poor
> reasoning. What is surprising is that you somehow instinctively know
> that the rigorous logic of the axiomatic approach would quickly show
> the self-contradictions you have embraced, so you repel against it.
> Amusing.
The splinter you perceive in my eye is but a reflection of the log in yours.

The axiomatic method is what conveniently shields you from having to think
about what you are doing, so you can perform your actions mechanically. Axioms
are an important tool for logical analysis. One needs logical assertions taken,
for the sake of argument, as true, from which to draw inferences. However, when
one's conclusions are amiss, one needs to check their derivation, and if there
is no logical flaw, one needs to examine the underlying assumptions in their
starting assertions. Refusal to do this demonstrates a lack of understanding of
how logic works.
>
> >> I start with 0 and add 1, N times. I start with 1 and multiple by N.
> >> Which is larger? This is far from irrelevant. it's entirely germane.
>
> For finite N, they are the same. Still not seeing how it is relevant.
Because they are the same for all N, finite and infinite. Besides, even if it
were only true for finite numbers, wouldn't having a finite sum of 1's imply
that we have a finite number of elements? You are still not seeing. Try opening
your eyes, or get the log out of the way, anyway.
>
> >> >> And what does Robinson have to say about 0*oo?
> >>
> >> Robinson does not use such sloppy terminology. His hyper-integers
> >> and infinitesimals are rigorously defined and are not subject to the
> >> sloppiness of your arguments. In Non-Standard Analysis,
> >> hyper-integers and hyper-reals are are infinite with respect to the
> >> standard integers and standard reals, but 0 * x = 0 for all x, whether
> >> x is a standard real or a hyper-real.
> >
> >Are you sure that Robinson does not allow for a product of an infinitesimal
> >and an infinite to be a finite? That's too bad. Woe to the axiom-peddlers!
>
> How did you come to that conclusion from what I wrote??

You wrote: "0 * x = 0 for all x, whether x is a standard real or a hyper-
real." I take that as a no, it does not allow for the product of an
infinitesimal and a hyper-real to be finite. Did you misspeak? I certainly
didn't misread, did I? Is there a different interpretation of your statement?


> This is part
> of your problem, Tony. You need to *think* about what you read before
> emotionally reacting. I know that these ideas are threatening to you,
> but poor reasoning and Straw Man arguments is essentially why you think
> the way you do (and why no mathematician agrees with you).

The major part of my problem is that the majority of mathematicians spending
their time here are so steeped in what they have been taught that they can't
even think about the reflex nonsense that comes out of their mouth. You think
it is my problem when you write things you don't mean, or forget what you
meant, or don't know, or whatever caused you to say what I quoted, and then not
remember, even though it's right in front of your face? Sure, that causes me
some problems, but only when dealing with this kind of non-communication.
>
> Of course multiplying an infinitesimal with an infinite can sometimes
> give you a finite value. Let x be any infinitesimal. Let y = 1/x.
> Now y is infinite. x*y=1. Other times multiplying an infinitesimal
> with an infinite will get you another infinitesimal...sometimes it will
> get you another infinite.
Gee, why is that? Could there be different infinities and zeroes?

Hope that helps.
>
> Hope that helps,
>
> Jonathan Hoyle
>
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: infinity
    ... TO, as usual, tries to create a straw man to fight, since he cannot ... for the set Peano Naturals. ... >> nor is it infinite, or prime, or odd or even. ... > This is so Berkeley-esque it makes me kind of ill. ...
    (sci.math)
  • Re: infinity
    ... >> That's because your countable infinities are actually unbounded finite sets. ... The generation of the naturals involves an increment of 1 for ... > infinite number of elements, one for each iteration, with an increment, ... and here we find the Straw Man. ...
    (sci.math)
  • Re: infinity
    ... That is where we differ. ... we apply the word "infinite" to it. ... >> Yours is the straw man argument. ... If x is infinitesimal (non-zero), ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... the positive integers which I defined the other day. ... A finite real, then, may be defined as any finite natural, or any number between any two finite naturals on the real line, by subdivion of the unit interval. ... We can also construct a linear enumeration of the reals using powers as I suggested with the H-riffic numbers. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... you had said that the existence ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ...
    (sci.math)
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