Re: Logarithm of transfinite numbers

Isn't it inductively provable that if we increment the max value
of a finite set and simultaneously increment the set size, if
they are equal, they will remain equal through any number of
simulatneous increments?

Yes.

If this is the case, can one achieve infinity while the other
remains finite?

I'm not sure exactly what you mean here. However, if I think I
understand where you are going with this, you cannot reach infinity
incrementally. Perhaps this is a better example:

Consider:
Step 1: 3/10 = 0.3 < 1/3
Step 2: 33/100 = 0.33 < 1/3
Step 3: 333/1000 = 0.333 < 1/3
....
Step n: 33...3/10^n = 0.33...3 < 1/3
....
It can be easily shown that each of the statements above is true for
each and every finite n. The set of decimal positions with a 3 in Step
#1 is simply {1}. In Step #2, the set is {1,2}. In Step #n it's
{1,2,...n}. And thus this proposition holds true for all sets of the
form {1,2,...,n} for all n.

However, it fails for the set {1,2,...} since that is the decimal value
0.333... which is *not* less than 1/3.

This is correct reasoning. It is essentially a proof as to
why any two finite natural numbers is only a finite
distance apart.

Right, and if there are only finite sequences within the set
from any element to any other, then where do infinite
sequences in the set come from?

From sets with no last element. Those with a last element can subtract
the first element and show that it is finite. This is essentially an
outline for the proof that any set of natural numbers with a largest
member must be a set of finite size.

To me, the question is whether there are actually an
infinite NUMBER of steps, and if every element is a
finite number of steps from every other, there can't be.

Why not? Just because each individual one is a finite distance away
from the beginning does not mean there are not an infinite number of
them. I think you are confusing the finiteness of the individual
element with the finiteness of set size. With an infinite number of
finite numbers, all the elements are finite, but the set is infinite.

If we put this in spatial terms, can a 3D (for instance)
space be infinite in volume, if every point in that space
is within a finite distance of every other? I don't think so.

Sure it can. Consider a cylinder with a base on the x-y plane in the
form of a circle x^2 + y^2 = 1. Now imagine this cylinder moving up
along the positive z axis containing all finite values of z (but no
infinite z). The only points enclosed in this cylinder are those of
the form (x,y,z) where x,y,z are all finite and x^2+y^2<1 and z>0.
Yet, this cylinder is infinite in volume.

If each successive element is one greater than the last,
and you have an infinite number of such successions, isn't
that equivalent to performing an infinite number of
increments on a finite value?

Yes, it is.

Doesn't that mean it is now an infinite number of units
away from that finite value...

Ah ah, caught you. What is the "it" you are referring to here? Your
language is ambiguous, and you are again confusing the *number of
iterations* with *a specific value*. There is no "it* that is an
infinite number of units away. Each value is a finite distance away,
all (infinitely many) of them. Better stated: there are infinitely
many successions, with each succession being a finite distance away.

I'm not sure what the relevance to real numbers are to
this discussion.

I am just pointing out that, in any set I consider infinite,
there are elements which are infinitely far apart, in terms
of the number of intermediate elements.

Using real numbers for this example is a poor one, as it does not prove
anything. Worse still, real numbers have the opposite situation: any
two has an infinite number of others between them. Are you suggesting
that this must be true with naturals? I would presume not.

Reals and Naturals have an opposite polarity with regard to denseness.
Reals are everywhere dense (that is, given any two reals, no matter how
close together, they are necessarily separated by an infinite number of
elements between them). Naturals are nowhere dense (that is, given any
two naturals, no matter how far apart, they are necessarily separated
by at most a finite number of elements between them).

With the unit increments, one unit of value for each element,
there is an identity relationship between element count and
value which makes it impossible for the set to be infinite
and the elements not to be infinite, if infinite is a word used
at all consistently. If the element values can't become
infinite by increments, then neither can the set size. They
are the same number as the set increases without bound.

This is true. What is not true is extrapolating this to infinite sets.
Finite sets "increasing without bound" are still each (individually)
finite: {1}, {1,2}, ... , {1,2,...,n}, .... Your proposition holds for
an infinite *number* of finite sets (those of the form {1,2,...,n}),
but does not hold for infinite sets, such as {1,2,...}. Your confusion
with the distinction between the two is entirely the area of dispute.

This isn't a matter of "religion". Sometimes it seems like
that with those defending the status quo in this area, and
to be honest, the methods in transfinite set theory seem
rather like hocus pocus and not mathemtics to me.

If it's any consolation, a number of math students have similar
difficulties the first time they are exposed to these truths. But the
"hocus pocus" part goes away once these students rigorously go through
the "assumptions-imply-results" proofs. Change the assumptions, and
you get different theorems, and counter-intuitive results are just an
occupational hazard.

The closest to your way of believing is a mathematical philosophy which
(I think) is called Finitism. If I recall correctly, Finitists do not
accept the Axiom of Infinity, so {1,2,3,...} is not even a set in their
axiomatic schema. Therefore, all sets are in fact finite by their
definition. I do not know how they handle real numbers (not well, I
imagine), so they stay fairly within the realm of Number Theory. There
may be some lip service paid to "potential infinity", but I am not
familiar enough with their philosophy to speak with any authority on
it.

This is a matter of finding consistency between
different mathemtical treatments of infinity, and finding
that this theory doesn't play well with others.

I think they play well with others, they are just different. But I
agree, that is an annoying aspect of the infinite. We would like to
have just a "shared" infinite number across mathematics which behaves
according to our intuition. Unfortunately, no such infinity exists.
With regard to cardinality (the most natural application for infinity,
in my mind), we find an unsatisfying desert to work with: aleph_0 and
its powerset C. Everything else is either extremely rare or
undecidable. The ordinal infinite is a bit more graspable, but
bizarrely breaks communitivity. Non-standard infinite numbers have the
opposite problem in that they are *too much* like finite numbers. Only
within the external theory can we even see that they are infinite;
within the internal theory, they are indistinguishable from finite
numbers. The idealized point of infinity (used in analysis for
topological equivalence) is so spartan in definition and use, it's
barely a number at all.

The problem is: which of these theories is any less valid than the
others? None of them, really. If there was an obvious one, it would
have been discovered and embraced by the ancients. The
"contradictions" of infinity found by mathematicians of old, are no
more than misapplication of assumptions. It wasn't until true rigor
was added to mathematics in the 19th century by Cantor, Weierstrass and
Bolzano, could we even consistently have these discussions.

Happy Hunting,

Jonathan Hoyle

.

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