Re: Review of Mueckenheims book.

On 3 Mrz., 15:14, "William Hughes" <wpihug...@xxxxxxxxxxx> wrote:
On Mar 3, 7:18 am, mueck...@xxxxxxxxxxxxxxxxx wrote:

i: Every finite number has finite size

ii: The set of natural numbers only includes finite numbers.

iii: The set of natural numbers has finite size.

Which bit of "i and ii never imply iii" do you fail to understand?

Saying that "there are a finite number of kumquats" is not the same
thing as saying "every kumquat is finite".


i: Every finite tree has finite number of nodes.

ii: The set of finite trees only includes trees with a finite number
of nodes.

iii: The set of finite trees has finite size.

Which bit of "i and ii never imply iii" do you fail to understand?

Saying that "there are a finite number of kumquats" is not the same
thing as saying "every kumquat has a finite number of nodes".



(Recall: the infinite union of finite numbers does not have a fixed
maximum,

Recall. The infinite union of finite trees does not have a fixed
maximum,

However, every finite segment has a fixed maximum. Something that has
a fixed
maximum cannot contain something that does not have a fixed maximum)

However every finite segment of trees has a fixed maximum of nodes.

We know

A: the infinite union of natural numbers contains only finite
numbers


We know
A: the infinite union of finite trees contains only trees with finite
numbers of nodes.


B: there is no single finite segment, L1, that contains the
infinite
union of natural numbers.

B: there is no single finite segment, L1, that contains the infinite
union of all finite trees.


despite the fact that each natural number is finite there
are infinitely many of them

despite the fact that each tree is finite there are infinitely many of
them.

We need infinitely many nodes to make infinitely
many finite trees (note: every finite tree has a
different size)

We need infinitely many units to make infinitely many finite numers
(note: every finite number has a different size).

Let C be the collection of finite trees.
Let D be the nodes in the collection of finite trees.
If D was finite then we could only make a finite number of finite
trees.
However, there are an infinite number of finite trees,
Therefore D is infinite.
The union of all finite trees must contain every node in D.
Therefore the union of all finite trees contains an infinite
number of nodes.

Let C be the collection of finite numbers.
Let D be the units in the collection of finite numbers.
If D was finite then we could only make a finite number of finite
numbers.
However, there are an infinite number of finite numbers,
Therefore D is infinite.
The union of all finite numbers must contain every unit in D.
Therefore the union of all finite numbers contains an infinite number
of units.

What is a finite number that contains an infinite number of units?

Regards, WM

.

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