Re: Logarithm of transfinite numbers

Tony Orlow wrote:
Jonathan Hoyle said:
No, it inductively proves it for all set sizes which are finite
natural numbers. It is not proven (nor is it even true) for
sets of infinite size.
No, it proves for every natural number in the set that the set
up to and including that natural number cannot be infinite.
So, the set never is anything but finite, as long as you are
adding elements which are never anything but finite. See?
You are failing to see your own proof here. You have indeed proven
that sets of the form {1,2,...,n} are finite, for each and every
natural number n. That is what your induction proof demonstrated. It
does not however say anything about the set {1,2,...}.

It says soemthign about each and every n in that set, which are all the points in the space of that set, that that space is not infinite at any one of those points.

At none of those additions of an incremented value does the
value reach an infinite point OR the set reach an infinite size,
because those two numbers are always exactly the same,
being incremented in tandem.
True, but again, this applies only to sets of the form {1,2,...,n}.
Your induction proof says nothing about unbounded sets.

It doesn't have to. It says that as the set increases without finite bound the element values increase without bound, but that as long as the element values are finite, the set size is equally finite.

No, it says that as long as the maximum value (i.e. the "n" that is being
considered) is finite, the set size is equally finite.


But, if all of the elements in the set are finite, and in positions
within the set equal to their finite value, then there exists no
element in the set which marks anything but a finite set, ...
<snip>

Correct.

...and there exists nothing else in the set besides its elements
which contribute to the size of the set.
Set size may or may not have any relation to the value of its elements.
Occassionally, sets do have such a relationship, such as sets of the
form {1,2,...,n}. The vast majority though, such as { 17, 101, 10^10
}, do not.

Acutally, there is a universal rule about this. If a set has a minumum finite difference between any two elements, or more generally, has a finite average difference between successive elements, then it can only have a finite number of elements in any finite range. That is, if no two elements are infinitely different, then no two elements can have an infinite number of elements between them, and no subset of the set is infinite in that sense. If two elements are infinitely different, on the other hand, one or another of them must have an infinite value, since the difference between any two finite values is a finite value.


What does it mean for two elements to be "infinitely different"?

Now, a set may have an infinite number of elements ina finite range, provided that is has at least one point of condensation, where the density is an infinite number of elements per finite unit of difference. In other words, if the average difference between successive elements is infinitesimal, then you may have an infinite set within a finite range such that no two elements are infinitely different. Does anything about this rule sound off to you?


Other than the fact that you don't state here exactly what you mean by an
infinitesimal difference, and that I still don't know what it means to say that
"two elements are infinitely different"?

You do not have an infinite set of naturals in the quantitiative
sense until you have infinite quantities in the set.
This is the basic assumption you are circularly making. You have not
proven it (nor can you since it is false).

It is only false in standard set theory, but froma quantitative standpoint, it is true, which indicates problems in the set theoretic approach which i have specifically pointed out many times.

It's only true if you either (i) assume it as an axiom or (ii) prove it based
on other axioms. In either case, you will need axioms not present in the
standard set theory. If you choose approach (i), you will probably want to
specify other axioms so that you can actually conclude something other than
just this statement. If you choose approach (ii), on the other hand, you need
to provide a specific list of the axioms used to prove the statement, and you
will also need to prove that the group of these axioms is logically consistent.

t


When you use "finite" in the definition of a set, that is a weak
definition, because finiteness imposes a restriction that is not
well defined. There is no clear point where the finite becomes
infinite, so you have what appears to be some kind of hazy
infinity at the end which doesn't quite go on forever.
Infiniteness is not a quality that happens at a "point". It happens at
the end of all points.

Wording like, "at the end of all points" would get me excoriated. I am not so particular about that wording, as it's more a matter of political convention than anything. Still, I will disagree, and say that infinity occurs, not at the end of all points, but within the progression. There is no single "point" where this occurs. It occurs over the course of an infinite number of points.

What you call "points" is analogous to what set
theorists call "successor cardinals", that is, cardinals which have an
immediate predecessors. Limit cardinals, such as Aleph_0, have no such
immediate predecessor (nor can it). That doesn't make it a "weak
definition"; it merely means that it does not contain properties which
you assume it does.

I have become quite familiar with the notion of limit ordinals and their associated limit cardinals, and in my opinion, the von Neumann ordinals upon which they are modeled is not a good model of the naturals. To say that each natural is the set of all its predecessors says that this natural is one greater than the max value in the equivalent set. Is omega one greater than the maximum finite? Uh, no, that would lead to a contradiction. There is a logical lapse there. To say the naturals start at 0, and therefore the set of all naturals less than n is the same as n, is nonsensical, expecially when starting the naturals at 1 and saying that n is the max value of the first n naturals directly contradicts the notion that the set size is larger than any finite natural. To me, we start at 1, and the set size is inductively provable to be precisely equal to the largest value in the set, which we all agree for the finite naturals is a finite natural.

Between any two finite naturals is a finite number of naturals...
Correct.

...so there is no infinite sequence within the set of finite naturals.
Incorrect. Your conclusion does not follow.

If there is not an infinite number of elements between ANY pair of elements, then how do you get an infinite number of elements in the set? By adding those two elements and making a finite set infinite by the addition of two? No, that doesn't work. Between any two distinct reals, there are an infinite number of other reals. That's why any finite interval denoted by such a pair of distinct reals contains an actually infinite number of real numbers. If you think my conclusion doesn't follow, please explain why, and leave out aleph_0 if you can. It's not a number, or even a valid concept.

You get an infinite number of elements in the set by showing that no matter
how many times you remove a single element, there are always more left.

(Yes, I know that this is the Dedekind infinity that you don't like, but it
is hard for me to pin down exactly what you mean when you use the word
"infinity", let alone to decide whether it is logically coherent; so I'm doing
what I can.)

Formally speaking, let A be a set, and define a sequence (A_i) of sets with

A_0 = A
A_i+1 = A_i - {x_i}, where x_i is an element of A_i

If there exists a set A_k in this sequence which is empty, then the set is
finite; otherwise it's infinite.

In this case, no matter how many times you remove a number from the set of
naturals, there will always be at least one left (namely, its successor).
Thus, there are an infinite number of naturals, even though there is never
an infinite distance (whatever *that* means) between any two specified
elements.

Matt


Regards,

Jonathan



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