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

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From: David Libert on 10 Jul 2010 06:36

Transfer Principle (lwalke3(a)lausd.net) writes:
> On Jul 6, 12:23=A0pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
>> INFERENCE there is no oo
>
> Herc has joined the growing number of sci.math finitists.
>
> We already know that Han de Bruijn and R. Srinivasan are
> finitists as well. A few posters, including Archimedes
> Plutonium and WM, defy classification. AP believes that
> all numbers larger than 10^500 are infinite -- so
> technically speaking, he still believes in infinity, but
> his infinity is much less than standard infinity. (Note
> that both Herc and AP use Chapernowne's constant in their
> arguments regarding infinity.) WM, on the other hand,
> believes that some standard natural numbers don't really
> exist, although he has no upper bound on the largest
> natural number. It had been said that for that reason,
> one can't call WM an _ultra_finitist, although it's
> possible that WM is merely a finitist.
>
> In the theory ZF-Infinity, one can't prove that an
> infinite set exists (assuming consistency), and if we
> add an axiom such as ~Infinity, we can actually prove
> that no infinite set exists. (There's no need to point
> out yet again that ~Infinity asserts that there is no
> _inductive_ set as opposed to no _infinite_ set. In
> several other threads, a proof in ZF-Infinity+~Infinity
> that all sets are finite has been presented, and the
> proof uses Replacement Schema. So let's not travel down
> that road for the umpteenth time.)
>
> Srinivasan has also proposed another axiom, D=3D0, which
> implies that all sets are finite, as well as a logic,
> NAFL, which proves the same.
>
> 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.

These are closely related, and came up in discussions
in

[1] "How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)"
sci.logic 551 articles
also sci.math and others
http://groups.google.com/group/sci.logic/browse_thread/thread/4c784919ee892256/

I am in basic agreement with what was written about these axioms in [1], but I have
always wanted to note some technical points about it. I didn't get around to it
originally in [1], and since you mentioned it here i will do so now.

R. Srinivasan introduced that D=0 in

[2] R. Srinivasan
"How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)"
sci.logic,sci.math and others June 27, 2010
http://groups.google.com/group/sci.logic/msg/045183a90426396f

with:

>There is a simple way to sidestep this controversy. Suitably extend
>the language of ZF-Inf to admit the set D, where
>
>D = {x: An (x not in P_n(0))}
>
>Here 0 is the null set, n ranges over the non-negative integers and
>P_n(0) is the power set operation iterated n times on 0 with P_0(0) =
>P(0).
>
>Note that by definition, D does not include any hereditarily finite
>set, but it will contain every other set.
>
>Consider the theory F = ZF-Inf+{D=0} It is clear that F will only
>admit models with hereditarily finite sets. Use the theory F instead
>of ZF-Inf+~Inf in my post.

There were various points of discussion related to this in [1].

One question was defining the P_n(0) hierarchy. Some other discussion gave
a name to the compliment of this D, which in other posts I recall had also been
locally named D (different posts using the same name for compliementaey pairs)
and the point was raised that there may be in these ZF-Inf models no such set
as the second D.

You pointed out this last point could be handled instead as a definable proper
class, like L in ZF.

I think all that is ok. Handle the other D that way, and P_n(0) definitions
are not a major problem.

And in fact this D=0 axiom is itself closely related to ~Infinity axionm
anyway.

For me the D=0 or ~Infinity formulations are not a major point, things
can be translated back and forth. And the other issues raised above
are not big problems. And the claims made above about these, everything
hereditarily finite, or nom infinite sets, are ultimately ok.

But there is an additional technical point on the way to all that
that needs to be cleared up. Once it is the above becomes ok.

The definition above of D :

>D = {x: An (x not in P_n(0))}
>
>Here 0 is the null set, n ranges over the non-negative integers and
>P_n(0) is the power set operation iterated n times on 0 with P_0(0) =
>P(0).

depended on having n range over all natural numbers. So it needs to
have the collection of all natural numbers defined as at least a class.

~Infinity is just negating thge usual statement of the axiom of infinity,
so is no problem to state.

But discussion of D=0 and ~Infinity continued to discuss whether there
are any infinite sets, or to discuss hereditarily finite.

So all the discussion rests on the definitons of natural number, finite
and heritarily finite.

Nothing was specified about these definitions, so the default would be
presumably the usual defintions.

I claim the usual definitions must be reworked in the context of
ZF-Inf.

With that the above becomes ok, but a complete account should mention this point.

Namely the usual definition I am used to of finite is having cardinality a
member of omega.

(And no: we don't define omega as the set of all finite ordinals :) ).

If we take the usual Russell definite description expansion of definitions,
this definition amounts to saying

x is finite <-> there exists y such that y satisifies the defintion of omega and
x member y.

So on existential exansion of omega in ZF - Infinity + ~Infinity, nothing satisifies the
definition of omega so the definition always vacuously fails and everything is actually
infinite by this definition.

Or idf you take instead univeral reading of the definite description, everything is vacuously
finite in ZF - Infinity + ~Infinity, seems closer to what we wanted but this definition
being vacious doesn't readily prove the familar properties of finite.

So instead we must provide an alternative definition of finiteness.

Similarly to define natural number has similar issues.

I will discuss the pitfalls of doing this in weak fragments of ZF, and solutions.

Above I spoke about the difficulties created by dropping Infinity.

I will also discuss issues with AC and regularity R dropped.

So ultimately below I will speak of solutions that work well even in ZF - Infinity - R.

I will also discuss some points about dropping replacement, back to Z.

One possibility would be Dedekind finiteness. See my next reference for details. But
the proof that Dedekind finiteness supports proofs by ordinary mathematical induction uses
AC (there are counter models of ZF if ZF is consistent).

I seek definitions of finiteness that make it work reasoinably in weak theories. The definitions
I will give below over the theories in question will then support the discussion above about
D=0 and ~Infinity, namely allow the defintion of the P_n(0), and show the claims about
no infinite sets, ie with these defintitions of infinite (not finite as finite to be
defined).

One defintion of finiteness that gets induction and function defintions by ordinary recursion
was discussed in

[3] David Libert
"How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)"
sci.logic,sci.math and others June 27, 2010
http://groups.google.com/group/sci.logic/msg/7011b0c9510d6764

[3] discusses some points about the definitions and references an older article that gives it.

The definition will give induction and recusrion in Z - Infinity - R .

So that theory would at least get the discussion started, to work from those definitions.

However to get from ~Infinity to every set is finite, you need replacement. This is the
result Transfer Principle mentioned above, which had been discussed several times in [1]
as Transfer Pirncile noted.

Other people had mentioned this point, that replacement is needed.

My older article referenced in [3], also gave this proof as well as contructing a
counterexample model of Z + ~Replacement, showing replacement really is needed as claimed.

That is handling ~Infinity from the [3] definition of finiteness.

What about D=0 ? That definition depended on the definition of natural number.

So we can define natural number as a finite von Neumann ordinal, using that definition
of finiteness.

But in this weak theory Z - Infinity - R there are some complicatiions about defining
von Neumann ordinals. They can be overcome, but some definitions need to be restated.

The usual ZF definition of von Neumann ordinal is transitive and hereditarily transitive.

The usual defition for property P that x is heriditarily P is every member of TC(x)
has property P, TC being transitive closure.

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 .

I remarked in [5] we could also define TC as having a finite sequence of members below the
top, except how to define finite. So an alternative would be to define it this way, using
[3] 's definition of finiteness.

With either of these, we can redefine heritarily transitive, and so redefine von Neumann
ordinal.

In the special case of von Neumann ordinals, since they are also defined to be transitive
they do have a transitive closure, namely themselves. So you could also define them as
being transtive and every member is transitive. Or you could just return to the usual
definition of transitive closure, because the definition expands properly by Russell in this
special case. I still think this is worth pointing out, when the general definition can fail.

Anyway, in all these ways we can restate a formulation of von Neumann ordinal.

But we are not done yet. The restatement so far would work in Z - Infinity.

But when we drop regularity this defintion allows for ordinals to be not linearly
ordered.

So instead for Z - Inifinity - R, define von Neumann ordinal to be the above
reworkings of transtive and hewreditarily transitive and also well-ordered by epsilon.

With this definition we get a reasonable recasting of von Neumann ordinals in
Z - Infinity - R.

With this, we want to define a natural number to be a finite von Neumann ordinal.

Use the [3] definition of finiteness.

With these reworkings, we do get proofs in Z - Infinity - R that D=0 gives every
set is finite and is hereditarily finite, using those definitions of finite and
hereditarily finite.

Zuhair, in the same thread as [5], posted other defintions of finiteness for
von Neumann ordinals:

[6] Zuhair "Recursive cardinals"
sci.logic, sci.math Jan 8, 2010
http://groups.google.com/group/sci.logic/msg/65ef21f30e4578de

Defining von Neumann ordinal as I just said, we can also use these definitions,
to replace [3] 's definition of finiteness.

Another definition of finite von Neumann ordinal that works reasonably
in Z - Infinity - R is von Neumann ordinal as defined above which is not
a limit ordinal and has no members which are limit ordinals.

With any of these Z - Infinity - R and D=0 give the claims every
set is finite and is hereditarily finite. We can use these alternative
definitions of natural number for the D definition. To define finite
use [3], or use had cardinality a natrual number the current definition
of natural number. Abd hereditarily finite in the conclusion defined as
above in the discussion of von Neumann ordinal.

So with these roeworked defintions, I think the claims above are ok.

--
David Libert ah170(a)FreeNet.Carleton.CA

From: hagman on 10 Jul 2010 06:53

On 9 Jul., 02:43, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> A REVISED PROOF OF THE NON-EXISTENCE OF INFINITY
>
> C10 = 0.12345678910111213141516...
>
> x = the number of digits in the expansion of C10
> y = the number of consecutive digits of PI in C10
>
> As x->oo, y->oo
> x = oo
>
> Assume the limit exists.
> y=oo
> Contradiction (for each finite starting digit of PI in C10 there is a finite ending digit)
> Limit doesn't exist.

There is a great difference between
lim_{n->oo} f(n) = oo
and
E n in NN: f(n) = oo
The former simply states that for all m in NN there is a k in NN such
that for all n>k we have f(n) > m.
OTOH, the latter states that there are n where f(n) has a value that
isn't even a number.

>
> y cannot reach infinity
> therefore x cannot reach infinity

For finite subsequences, there is clearly a maximum value of y that is
attained (at least once).
For the infinite sequence, the supremum of pi subsequence lengths need
not be attained.

>
> x = the number of digits in the expansion of C10
> x =/= oo
>
>
>
> > INFERENCE there is no oo

Non sequitur

>
> > Herc
> > --
> > Conan do we REALLY have to hear the lamentations of the women?
>
>

From: |-|ercules on 10 Jul 2010 07:07

"hagman" <google(a)von-eitzen.de> wrote
> For finite subsequences, there is clearly a maximum value of y that is
> attained (at least once).
> For the infinite sequence, the supremum of pi subsequence lengths need
> not be attained.
>

I was thinking along those lines (so to speak) today but it would imply a bizarre conclusion.

As x->oo, y->larger subsequence lengths

Here's a simpler version:

H1= 0.101101110111101111101111110...

apologies if I borrowed someone's constant!

So you're saying if H1 had an infinite amount of digits in it's expansion, the number of consecutive 1's
would be (a non attainable non number?)

Herc

From: Wolf K on 10 Jul 2010 09:47

On 09/07/2010 15:45, K_h wrote:
[...]
> [These] question[s] [don't] need to be answered in order to know that these truths
> always exist. How does anything exist? How does the galaxy exist? People don't
> need to have those answers to know that they exist.
[...]

In your posts, the term "exist" is vague and unclear. It also shifts
meanings. Exactly which meaning of "exist" do you have in mind? Eg, the
four uses of the term in the snippet I quoted are all different, and
three of them are mutually exclusive (insofar as one can decode the
probable intensions of the terms). The fourth asks about the meaning of
the term itself.

FWIW, I think Plato's ontological questions are so ill-formed that his
answers aren't even wrong.

cheers,
wolf k.

From: George Greene on 10 Jul 2010 11:31

On Jul 9, 6:38 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> Perhaps you don't understand the proof, it only contradicts a well used axiom, not a well established fact.

AND WHICH axiom is that??
If you can answer that one question (which if course you can't),
we MIGHT get somewhere.