Re: infinitely many nn's = infinite nn's?

In sci.logic, Phil
<toob-headman@xxxxxxxxxxxxx>
wrote
on Sat, 31 Mar 2007 21:38:18 GMT
<460ED4CC.1050606@xxxxxxxxxxxxx>:
The Ghost In The Machine wrote:

[snip]

So, FN is an *infinite* set of *finite* natural numbers, and reality
seems none-the-worse for it.

One can also deduce this from Peano's Axioms, which are:

[1] There exists a 0 in N.
[2] For any n in N there exists a successor successor(n) also in N.
[3] 0 is not a successor of any n in N.
[4] For any a and b in N, if successor(a) = successor(b) then a = b.
[5] If P(0) and (for all n in N) (P(n) => P(n+1)) then one can
deduce that (for all n in N) (P(n)).

Well, god hasn't personally told me otherwise, and as I said to Barb,
options #1 or #2 MIGHT be correct, meaning that the premises underlying
"there are infinitely many numbers," and the premises underlying "all of
which are finite," might be either #1, the same, or #2, different, but
compatible. However, so far, every attempt on my part to get an answer
ends up with option #3, the premises are different, and incompatible.

These are axioms and assumed correct. They may not match your axioms.
If so, please identify your axioms.

I obviously didn't make myself clear: these axioms do not explicitly
state whether they should be used with potential, or actual, infinity.

Is there an infinity somehere in the axioms, then? If so, I don't see
it.

In any event, N is provably finite from these axioms, if what you're
stating is true.

I
don't use different axioms, I am merely AWARE that mathematicians use
more than is explicitly stated in these axioms when deriving the
properties of the natural numbers. My question is whether we can or
cannot use potential infinity for some conclusions, and actual infinity
for other conclusions, without creating an inconsistent, contradictory
whole (or mess).


Let's look at Axiom [2]:

For any n in N there exists a successor successor(n) also in N.

Suppose you generate successive numbers, or cross successive halfway
points between you and the door. Will you generate INFINITELY many
numbers?


No. Not physically possible. The concept of "rollover" is very well
known, both regarding car odometers (which at some point added an extra
digit) and computer-represented integers.

I think this is just silly. There is no rollover function in the
premises behind either numbers, or number generation. The limitations of
electronic calculators are irrelevant to this discussion.


Will you leave the room? It DEPENDS on whether or not you allow
the "use" of actual infinity.


One cannot use infinity any more than one can use a
moon that one has never been to. At best, infinity is a
symbolic construct, much like numbers. Or can you catch a
3 at one's local zoo? How about finding a 5 in the wild?
A 1 might be lurking right behind you...or not.

Another silly comment. Infinity is used in mathematics all the time,

Mostly erroneously.

and
we have PHYSICAL analogies even in the real world, where we do, in fact,
cross infinitely many halfway points between us and the door every time
we leave the room.

No can do. If one assumes an atom is a halfway point, there are only so
many atoms. Each foot contains about 3 billion atoms, give or take.

There are uncountably infinite mathematical points, yes, but if N is finite,
R is finite as well. Stands to reason.

The fairly recent introduction of Robinson's
nonstandard analysis has even brought infinitesimals, along with
infinite numbers, back into mathematics, and on firm ground. Look it up
in Wikipedia.


If you produce numbers or cross halfway
points at a rate that is NOT infinite, for a period of time that is NOT
infinite, then the answer to both questions is no. Of course, in
mathematics, we do not have to actually "produce" numbers, but even in
pure definition, we have the POWER to define things using EITHER
potential infinity, or actual infinity, so the use of "generators" is
actually a VERY good idea, from the viewpoint of both avoiding and
detecting errors. And the screams by shithead notwithstanding, the mere
fact that these terms are not consciously stated does NOT mean that they
are not being used, or don't exist. When children cover their eyes and
say "I can't see you so you're not here," we ARE, in fact, still here.

I must emphasize that I'm not blaming YOU here, or accusing you of
anything, but you ARE a human being, so these facts and potential
oversights MUST apply to you as much as they do to the rest of us.


Not necessarily. I speak English. Those in China will probably be
speaking a Chinese dialect.

Another irrelevant comment. I obviously was speaking about
characteristics of thought, and the kinds of errors that humans can
make, which really do apply to all of us, even if certain individuals or
groups make certain errors more or less frequently than others.


Perhaps the easiest way to see the implicit use of potential infinity is
through Axiom [5], mathematical induction, or as it is sometimes called,
finite induction:

If P(0) and (for all n in N) (P(n) => P(n+1)) then one can deduce
that (for all n in N) (P(n)).


This is a meta-axiom, and IMO very problematic. However, it is one of
Peano's axioms.

Out of curiosity, what do you find problematic about it? I obviously
believe that it must be used with potential infinity, so I am open to
the idea that it has problems.

The problem with the fifth axiom is (or should be) obvious.
The other axioms talk about numbers, but the fifth talks
about properties relating to numbers and quantifiers.
That makes it a level-1 rather than level-0 axiom, at least;
it might even be a level-2, but I'm not that familiar with
Russel's Orthodoxy.

However, one should be able to replace it with alternative axioms,
much like the parallel postulate.



A property of induction that most are reluctant to state is that it is
true for ANY property, not just a chosen few properties (if this were
not true, then it would be useless, unless we had another axiom that
stated which properties fall under induction). I think the reluctance is
due to the vastness of the claim; obviously we cannot verify it by
testing EVERY property, so people take the stand they normally would use
in a DEBATE, as opposed to a SCIENTIFIC discussion, and avoid such
all-encompassing, sweeping claims. But either induction REALLY DOES
apply to ALL properties, or it's useless, so we really have no choice
but to accept its true power.


Correct.


Since 0 is finite, and given a finite n, n+1 is also finite, therefore
for ALL n in N, n is finite.


And thus you see the problem.

Ummm ... if you say so? Maybe the following lists some of those problems?

Not all numbers are finite;

Are you referring to numbers other than those generated using finite
induction?

it depends on
how one defines "number". It also depends on how one defines "finite";
all symbolic representations are in fact finite by necessity. The
tilted-8 is little more than a few strokes on a page, but represents a
concept neither comprehensible nor truly expressible.

This is not a problem if one is clear on one's axioms and deductive
rules.

So, what is made clear, or what problems are eliminated, if you are
clear on your axioms and deductive rules? I'm not trying to be difficult
here, but if you have some point, I truly am missing it.

No doubt...however, it's quite clear that N is effectively finite.



However, if we generate numbers by adding
marks to a set, as in

{|}
{||}
{|||}

etc., then if we add INFINITELY many marks -- which we can do in a
finite period of time, if we add them at an infinite rate -- then we
will have an INFINITE NUMBER.


Now you're verging on the ridiculous. How does one add marks at an
infinite rate?

How does one cross the infinitely many halfway points between oneself
and the door?

The same way Zeno did.

Or how does one move at 1 meter per second, when each
meter contains infinitely many points? The infinite, and the use of the
infinite, although difficult, really is a normal part of mathematics,
and our mathematical descriptions of the world.

Then N is finite, as you've said. It can't be infinite.

It is a concept which we
must accept, difficulties notwithstanding.

Not at all. Infinity should never be accepted without strong, provable
rigor.

Mathematically speaking, I
can, in fact, do just about anything at an infinite rate, but if you
prefer, we can settle for instantaneously defining a set as having
infinitely many marks in it. The "add marks at an infinite rate" is
merely a tool of understanding, something to make you focus on the
difference between potential and actual infinity, and to notice that
actual infinity is inherently incompatible with induction.

And the fact that you can do anything at all shows evidence that
infinity is a meaningless concept.



Now, by the current definitions of natural
number, this will not be a natural number, or to use terms from
nonstandard analysis, we will have a nonstandard integer of the
"unlimited" type, but it will be a number, and it will be infinite. Now
you see why induction is sometimes called "FINITE induction," because it
is valid iff you use POTENTIAL INFINITY, and NOT actual infinity! Note
that induction does not state that there is some specific, finite number
that we can go up to, after which we can quit and be assured that the
property is true for all n in N, so it does NOT assume the existence of
a specific, finite upper bound, but at the same time, it CANNOT be used
with actual infinity. This is an extremely important point; although
induction CANNOT use a specific, finite upper bound, this lack of an
upper bound does NOT equal actual infinity. Instead, it equals EITHER
actual or potential infinity, and we can choose either option.

So, induction PRESUMES potential infinity,


The cardinality of N in the standard axiom set is greater than any element of N.

That is indeed the current belief, but I fail to see what that fact --
assuming it's true -- has to do with this discussion about induction.

So the cardinality of N is in fact equal to a member of N? Which member, then?



and with it, we can prove
that for all n in N, n is finite.


Which is highly circular.

No, given the axiom of induction -- INCLUDING its necessary use of
potential infinity -- we can in fact deduce that for all n in N, n is
finite.

I'm not interested in arbitrary elements of N. I'm interested in N's
cardinality. How many elements are in N?

I think you are using the term "circular" as a claim of bad
reasoning,

That is correct.

but without pointing out where this circular reasoning
exists, assuming it does. Circular reasoning implies that a premise is
accidentally used to "prove" itself;

In some cases it's quite deliberate. We assume that for any n, there
exists n+1, both in N; therefore, N is infinite. The premises are
axioms but need not be true.

we think we've added a deduction
from existing premises, when in fact, all we've really done is to add
another premise. I didn't "prove" that induction must use potential
infinity here (although I tried to do so elsewhere).

It matters little. Induction is effectively infinite, as it requires a
quantifier.

In other words, one is proving stuff about ALL members of N.

I merely pointed
out that induction + potential infinity = finite natural numbers, which
is what everyone currently believes is correct (I should mention that I
believe that having a subset of the nonstandard unlimited integers, in
which every non-negative integer is finite, is a VERY good idea).

OK. So N is a finite subset of the nonstandard unlimited integers?



Note that this statement DOES mean
that no n is infinite, that there are NO infinite natural numbers under
the premise of potential infinity! So, what else does induction prove?


Induction proves nothing; it is *assumed*. It is the fifth axiom in
Peano's set. It's a bit like the geometric axioms of Euclid; the
parallel axiom led to many strange deductive paths (and ultimately at
least two additional geometries).

Now wait a minute! Both induction AND the parallel postulate can be used
to prove all sorts of things! Yes, they must usually (maybe always?) be
used with other postulates/axioms, but they certainly can be used to
prove things. Perhaps you would have been happier if I had said, "So,
what else CAN induction BE USED TO prove?"

So how does one prove that card(N) is finite?


Well, the number of numbers from 0 to 0 is finite, and if the number of
numbers from 0 to n is finite, then the number of numbers from 0 to n+1
is finite, so for all n in N, the number of numbers from 0 to n is
FINITE. There are NO infinite sets of numbers under the premise of
potential infinity! Not one. N is finite, not infinite.


Interesting logic. So, the number of all even numbers is finite?

Under finite induction, yes.

Is this the same as Peano's fifth axiom, then?


How
about the cardinality of the set of all squares?

Right again.

The number of all
powers of 2 might be finite, too.

The number of ANY subset of the natural numbers,

Including the natural numbers, which is of course a subset of itself.

under finite induction,
is finite. By the way, this is sort of said in nonstandard analysis,
when they say that ANY infinite set has nonstandard elements. I say
"sort of," because elsewhere, it seems that they claim something else
(R. Srinivasan has said something about this in other posts in this
thread, although I'm not certain I followed everything he said).

And yes, I am well aware that mathematics has infinite sets of natural
numbers,

No it does not. Math has *representations* of sets. In short, one
describes these entities with notations such as

QSQ = {x^2: x in Q}

which basically is an attempt at a decision procedure.
In this case one can attempt to translate it into an
algorithm for determining whether something is in a set:

inQSQ(a) { if (a in Q and sqrt(a) is rational then return true else
return false; }

or one can even attempt enumeration:

enumQSQ() { for(q in enumQ()) { return q^2; } }

where one might write

enumQ()
{
return 0;
for(s in N)
{
for(n=1;n<s; n++)
{
if(gcd(n,s-n)==1)
{
return n/(s-n);
return -n/(s-n);
}
}
}
}

or just explicitly write it out, where people might attempt to
infer what one has written:

QSQ = {0, 1, 1/4, 4, 1/9, 9, 1/16, 4/9, 9/4, 16, ... }

but absent more intelligence by whoever's trying to figure out QSQ, the
next element might as well be "3".

All of these representations are by necessity finite.

(The notation, BTW, is slightly unusual. However, coroutines have
effectively fallen out of favor in the computing world, although one
might today use some sort of pushback list.)

and I like and accept the idea, but I see no way for those sets
to consists of exclusively finite numbers, or for induction to fully
apply to such sets. But then, you said yourself that you have issues
with induction.

I do, but they are not unresolvable.



If we try to say, well, there are infinitely many n's, because induction
only applies to finite subsets of N, then why not say there are infinite
n's, because induction only applies to finite subsets of N? What does
induction prove, if it does NOT prove some property for the entire set N?

and by the elementary observation that, for any finite
subset F of a totally ordered set (which N clearly is),
then a fairly trivial procedure exists to find its largest
element max(F), eventually leading to a contradiction when
attempting max(N).

But when asking how many natural numbers there are,


There are exactly W natural numbers, where W is the last natural number.

Contemplate this statement.

I have contemplated it for many years, and although it is largely
incomprehensible to our finite oriented/limited minds, I think that we
MUST include the idea of a "last number" in any complete and functional
system of mathematics; otherwise, we cannot even mathematically explain
something as simple as walking out of a room. But I would like to hear
YOUR thoughts about the matter, even if most or all of those thoughts
came from some book (which is obviously NOT the case for me ;-).

It is trivial to prove that for a finite set there is a
largest element. In fact, I've laid most of the groundwork
already; if one assumes enumS() exists, and that there is
a total ordering of S's elements (in other words for any
a and b in S, one has one of a < b, a > b, or a = b), then
the following procedure finds the largest element of S:

max(S) {
m = nil;
for(s in enumS())
{
if(m == nil || m < s)
m = s;
}
return s;
}

(The term "nil" is not in S, and there are some minor
coding issues; one could either do the "nil" or leave it
undefined and add another local variable which indicates
whether we've seen a set element in S or not. These are
corner cases.)

Therefore N, since it is finite, has a largest element.
What is that element?

And once you've given me that element, what is to prevent
me from adding 1 to it, producing an item that cannot
possibly be in N (since it's larger than N's largest
element) yet shows every indication of being a natural
number?


Phil



mathematicians
unconsciously SHIFT from induction and the premise of potential
infinity, to the premise of actual infinity, and under that premise, the
lack of an upper bound DOES equal infinitely many numbers. Of course,
when actual infinity is combined with the lack of an upper bound for the
number of marks in the sets used to represent the natural numbers, or
the number of non-zero leading places in the usual decimal number
system, the result is infinitely many marks or non-zero leading places,
which equals INFINITE numbers, but what the heck? Why not use potential
infinity to determine the maximum size of numbers, then SHIFT to a
different set of premises, and use actual infinity to determine the
number of numbers? And why stop there? Why not use actual infinity to
create infinitely large numbers, and potential infinity to insure that
there are less than infinitely many numbers? Doing so will NOT make us
more stupid that we already are; it will just make us more obviously stupid.

While it is clear that the set of natural numbers that humanity will
be able to write is clearly finite (there's only so much paper on
Earth!), that doesn't mean much in the mathematical realm. :-)


Now THAT is true!

Phil





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