PROOF INFINITY DOES NOT EXIST! Not even ONE type [Logic]

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From: Transfer Principle on 13 Jul 2010 02:06

On Jul 11, 10:22 pm, "K_h" <KHol...(a)SX729.com> wrote:
> "Curt Welch" <c...(a)kcwc.com> wrote in message
> > some internal representation of separate "objects". Once you have one of
> > these machines, we get closer to the truth embodied in 2x7=14,. But
> > without that machine, that "truth" really doesn't exist at all.
> Nonsense. You just claim the truth embodied in 2x7=14 doesn't actually exist and
> that is plainly untrue. Marshall, thanks for the heads-up on Nam; it looks like
> Curt is another Nam.

And therefore, it should come as no surprise that I'm going
to defend Curt Welch, just as I defend Nam Nguyen.

> > "K_h" <KHol...(a)SX729.com> wrote:
> >> In regular arithmetic 4+5=9 is true but Curt was claiming
> >> that there is some tiny chance it could be wrong in regular arithmetic..
> >> Curt is obviously wrong there.

But there is some tiny chance that PA is inconsistent. Despite
K_h and Spight repeatedly calling those who even consider the
possibility that PA is inconsistent "loons" and "cranks," the
possibly remains that there's a chance, however tiny, that PA
is inconsistent.

> > and does exist, and all of math takes place in that fairy tale land. It's
> > highly useful and important to do math under that belief. But what's
> > invalid, is to assume the lies we use to do math, actually happen (or
> > exist) in the real world.
> Mathematical truth exists in the real world and those truths are not lies..
> Again, this is obvious.

But infinite objects aren't known to exist in the real world. So
I don't consider any statement concerning an infinite object to
be an absolute mathematical truth at all.

> > I can produce language that describes a reality where pink flying elephants
> > with no mass exist. But no one is going to get confused about whether the
> > reality I am talking about actually exists in our universe or not. It's
> > just a story I made up by taking things that do exist in our universe, and
> > combining them in a way that has never been seen, and which is highly
> > unlikely to ever be seen in our universe. That's how the idea of absolute
> > truth was created as well.
> No, there are absolute truths of the universe, for example conservation of
> electric charge.

There are absolute _physical_ truths, I admit, but not absolute
infinitary _mathematical_ truths.

> > But yet, somehow, many people get so engrossed in the stores we make up as
> > we talk the language of mathematics, they start to believe the world of
> > mathematics is not just a story, but that it actually exists. That it not
> > only exists, but that it "lives on" even after all the story tellers die
> > off. It is as if they believe the pink elephants exist and live on
> > forever, even after everyone that's heard the story has died off.
> No, because mathematical truth is discovered not invented. An alien on another
> planet must also discover the same prime numbers that humans have.

I am loath to bring up analogies about aliens due to their
association with JSH (Planet Contary), but since K_h brings
up aliens here, so do I.

Sure, if an alien defines "primes" the same way that we do,
then I admit that they'll discover the same primes. But an
alien need not have the same ideas about infinite objects
that we do.

I'll concede that _finitistic_ mathematical truth is
absolute and discovered. But _infinitary_ mathematical
truth is, to me, _invented_, with the inventors being
Cantor, Zermelo, Frankel -- in other words, those who
invented the _axioms_ postulating the existence of any
infinite objects. Without the axioms, there's no reason to
believe that any infinite object exists, no matter how much
K_h and Spight insist upon it.

Thus, this is where I draw the line between a defensible
poster and an indefensible poster. I won't defend those
who contradict the finitistic _theorems_ of PA, which do
include K_h's "4+5=9" and "2*7=14." But I will defend those
who don't necessarily accept the _infinitary_ proofs, which
include the _consistency_ of PA itself. (So I distinguish
between the _theorems_ of PA and the _consistency_ of PA.)

And so I defend Nguyen, Welch, and even finitists like Herc.

Yes, K_h and Spight are definitely biased by their own
notions of "absolute truth."

From: Transfer Principle on 13 Jul 2010 02:10

On Jul 10, 6:42 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> Herc's Axiom Of Pseudo Infinity (based on above equation AOF)
> There is a set, I, that includes all the natural numbers that could physically be computed
> (before the end of the computer sustainable Universe)

Out of curiosity, what does Herc consider to the the largest
number in I?

If Herc is like AP, then his upper bound might be somewhere
around 10^500 or so. But I suspect that Herc might be more
like WM (who influenced him greatly), and perhaps the set
I contains gaps -- so that googolplex is in I, but there
exist natural numbers less than googolplex but not in I.

From: Transfer Principle on 13 Jul 2010 03:29

On Jul 10, 3:36 am, ah...(a)FreeNet.Carleton.CA (David Libert) wrote:
> Transfer Principle (lwal...(a)lausd.net) writes:
> > I recommend that Herc read some of the other sci.math
> > finitists to see what they have to say. (Of course, he
> > already knows about WM.)
> You are writing above about the 2 axioms ~Infinity
> and D=0.

Thanks for the information. And so now we know that
Srinivasan's theory NBG-Infinity+D=0 will work, even if
we drop Replacement Schema and Regularity, as described
by Libert in this post.

> I posted in ZFC - Infinity transitive closures might not exist as a set:
> [4] David Libert "Axiom of infinity and the set of all hereditary finite sets"
> sci.logic Oct 3, 2007
> http://groups.google.com/group/sci.logic/msg/7593d4adf17732b7
> So if just expand the usual definition above as Russell is problems as for omega above.
> In
> [5] David Libert "Recursive cardinals"
> sci.logic, sci.math Jan 3, 2010
> http://groups.google.com/group/sci.logic/msg/02248254025cb4c8
> I posted how to defince TC as a class in ZF - Infinity.
> The same definition would work in Z - Infinity - R .

Ah, I remember that discussion about zuhair and his
"singleton towers" that work in ZF-Infinity. I
mentioned zuhair's theory in other threads, hoping
that maybe it might satisfy the desiderata of other
posters, but to no avail.

If Herc were merely a finitist, then he could choose
Srinivasan's theory (or possibly even zuhair's), but
in this thread he implies that he might either be a
ultrafinitist or have WM-like gaps in his set I of
natural numbers.

Still, Libert's post was interesting and helpful.

From: FredJeffries on 13 Jul 2010 09:29

On Jul 12, 2:24 pm, Dan Christensen <Dan_Christen...(a)sympatico.ca>
wrote:
> Correction
>
> On Jul 10, 7:22 pm, FredJeffries <fredjeffr...(a)gmail.com> wrote:
>
>
>
> > On Jul 9, 10:17 am, Dan Christensen <Dan_Christen...(a)sympatico.ca>
> > wrote:
>
> > > I can't imagine that you would be able to do very much using
> > > "finitist" methods. How do they handle such basic concepts as the
> > > square root of 2?
>
> > Terence Tao in "A computational perspective on set theory"http://terrytao.wordpress.com/2010/03/19/a-computational-perspective-...
>
> > in which he explores the question "what is the finitary analogue of
> > statements such as Cantors theorem or the Banach-Tarski paradox?"
>
> With your "countably infinite loops" (see link), it seems you are
> sneaking infinite sets in through the back door. You posit an
> algorithm that can complete an infinite, countable number of
> iterations (ranging over ALL the natural numbers) and arrive at some
> conclusion. Have such notions ever been successfully formalized
> without referring to the set of natural numbers as a whole?
>

I am not qualified to answer your question (if I even understand it)
and as I work in the real world as opposed to some theory I don't have
time to dig out the references now, but I believe that Ed Nelson and
Alexander Yessenin-Volpin (both regularly mentioned in these kinds of
threads) have tried to address that issue. See Nelson's Predicative
Arithmetic at
http://www.math.princeton.edu/~nelson/books/pa.pdf

My impression is that the answer to the "successfully" part of your
question is open.

From: |-|ercules on 13 Jul 2010 10:03

"Transfer Principle" <lwalke3(a)lausd.net> wrote
> On Jul 10, 6:42 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
>> Herc's Axiom Of Pseudo Infinity (based on above equation AOF)
>> There is a set, I, that includes all the natural numbers that could physically be computed
>> (before the end of the computer sustainable Universe)
>
> Out of curiosity, what does Herc consider to the the largest
> number in I?
>
> If Herc is like AP, then his upper bound might be somewhere
> around 10^500 or so. But I suspect that Herc might be more
> like WM (who influenced him greatly), and perhaps the set
> I contains gaps -- so that googolplex is in I, but there
> exist natural numbers less than googolplex but not in I.

I could redefine pseudo infinity to be any finite length = any computable length
with a program that terminates, if you can compute n then you can (theoretically)
compute n-1, so pseudo infinity might be the initial sequence of N up to the largest
physically computable number.

Let me clarify the argument.

let x = the length of some incremental sequence of natural numbers, starting at 1
let y = the last value of such sequence

As x->oo, y->oo
y = x

The limit of y does not exist
Therefore the limit of x does not exist

Quite Easy Done!

Herc