Re: Logarithm of transfinite numbers

Randy Poe said:

Tony Orlow wrote:
Randy Poe said:


sum(n=1,...) 2^(2n+1) is not the same as

sum(n=1,...M) 2^(2n+1) for any value of M.

Right. The first one doesn't have a specific answer. the second allows for some
analysis.

The point is that because your analysis of the second one
does not ever consider the first one, your conclusions about
the second one do not bear on the first.

No more than analyzing dogs exclusively allows you to
draw conclusions about cats.

The second is sfficient to draw a concluson about the elements which has
bearing on the set size, as I showed.

You did not "show" any such thing. It is a logical principle
you keep claiming, that you keep relying on, but no
such logical principle exists. Stating over and over
that your data on dogs implies the same thing holds
for cats does not make it true. It's just wishful thinking,
no matter how hard and how often you wish it.

So, the fact that the size of the set is finite up to and including each and
every element in the set has no bearing on the set size? Stop calling half of
the apples oranges, and you will have a lot less problem.


Do any of your naturals achieve any infinite values?

No. And no sum ending at M represents the sum over the
entire set, or the value of the string I am discussing.

Again we have your usual logic:

1. Here I have a collection of things (strings that
have a leftmost 1) that all have property P (their
value is finite).

2. Here I have an object (the string which has a 1
in all odd positions) that we all agree is not in the
collection.

3. Therefore I conclude my object has property P.

Yes it's true that every finite string with 1's in odd bits
has a finite value. Now can we get back to discussing
the value of the non-terminating string?

I am getting tired of this, Randy, and may stop responding to this nonsense if
you don't pay better attention,

I'm paying plenty of attention, that's why I can keep identifying
the same pattern.

That's why you keep thinking something happens in the set of elements when
there is no element in the set at which it can possibly happen. Very attentive.

The set is neither: (a) a particular element, nor (b) a proper
subset of itself.

The set is a collection of elements which share the property, due to their
finiteness and finite differences, that the set up to each and every point in
the set is finite.


There is absolutely no reason why the set as a whole has
to share a property with a collection of some of its proper
subsets. The phrase "at which it can possibly happen" is
a red herring. By considering the sequence of proper
subsets you focus on, your tunnel vision avoids thinking
not only about the set as a whole, but about infinitely many
other proper subsets.

You are the one focusing on the subsets. I am talking about a property of the
elements which impacts the size of the set.


and stop accusing me of being illogical on
points you can't refute, or even grasp.

My refutation remains that #1 and #2 do not imply #3.

Your 1, 2 and 3 are yours.

They are merely abstractions of yours.

Not very faithful renditions of anything I have said. Why don't you try
responding to what I've actually said, instead of responding to some
"abstracted" version of it?


Your statement: Every subset of the form {1,2,...,n} is
finite for any n in N.

My statement: For every n in N, the set of elements up to and including n is
finite. Get it straight.

My characterization: Here is a collection of objects
which all have property P. Statement #1.

Rather vague. Your version of my original statement is not even correctly
expressed.


Is there some way my statement differs from yours?

I don't see any statement of mine, but yes, your abstraction loses a little of
the detail being discussed.

The collection of objects is the collection of subsets of
the form {1,2,...,n}. Property P is "is finite".

No, the collection of objects is the set of natural numbers, each of which has
the property of having a finite number of predecessors. There is no infinite
number of elements in your set.


Your statement: There is no point at which {1,2,...,n}
becomes N.

My statement: There is no point at which any finite natural has an infinite
number of predecessors.


My restatement: N is not a member of the collection of
subsets of the form {1,2,...,n}.

In your collection of subsets, which need not be proper, there is no set with
infinite size.


Do you disagree with this paraphrase?
See above.

Statement #2. Here I have an object which we agree is
not in the collection.

Do you really object to that? Why?

Because my collection of "objects" ar enot "intiial sequences of the naturals".
They ARE the finite naturals. All of them.


Your statement: Therefore N is finite.

There is no finite natural in the set at which point the set is infinite, and
there is nothing else in the set, so there is no point at which adding such
finite naturals creates an infinite set.


Statement #3: Therefore this object has property P.

Are you really saying that N doesn't have property P?

At which point has my abstraction differed from what you're
saying?

see above


- Randy



--
Smiles,

Tony
.

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