#388 improving the explanation for why an infinite set must contain at least one element that is infinite itself ; new textbook: "Mathematical-Physics (p-adic primer) for students of age 6 onwards"

Archimedes Plutonium wrote:

Now I can do a better job than the above, and perhaps it will take me
several posts
to get it in words and ideas that are clear to even a layperson who
hates math or logic.

The important thing is that I am here on this issue now and will not
stop until it is ironed
out perfectly.

Earlier tonight by serendipity, David wrote about set supremum and
infinimum of set theory.

Now, what I am going to say and do is that we can combine Set theory
with Number theory
to where one is the same as the other and where neither is
distinguishable from the other.

Let me give an example. Suppose I have the set {2,3,4} and that is a
set with three elements
all three of which are Real Numbers and finite numbers. Now instead of
a set of that just listed
suppose I combine the elements and form the number 234 of a finite
Real Number of 234.000.....

So in this sense I have combined Set theory with Number theory and we
can get more specific
and say the number 234.0000...... is the set of 10^2 place value and
10^1 place value and
10^0 place value such as this set {2 x 10^2, 3 x 10^1 , 4 x 10^0}

What I am getting at is that Set theory is interchangeable with Number
theory where we replace
membership of Set theory with Place Value in Number theory.

Now we tackle the question of can you have an Infinite Set composed
solely of finite numbers?

And the answer is obviously not. Because if all the elements of a set
are finite elements then they
compose another finite element.

So the error of Heckman's thinking is that he says his set has only
finite elements stretched over
infinity. But because they are all finite elements means that they can
be strung into a single number
which is finite itself.

So there. If you want an infinite set, then at least one member or one
element itself has to be
infinite. If all the elements of a set are finite, no matter how far
stretched they are, they cannot
be infinite.


The more I improve this, the clearer it is going to be to everyone
reading this.

I used the words "no matter how far stretched" the set of finite
members

Well I can replace stretched to infinity with a actual number itself
and that is
9999....99999 which is an infinite integer.

So the argument that Heckman is trying to maintain is that he is
saying that if you
have a set of only finite integers of 1,2,3,4,..... and that you add
them all together
that this summation would equal 9999....9999999. But that is obviously
false
because adding numbers that are solely finite can never sum to
1111....11111 or
2222.....22222 or 99999....999999

So here again. If you have a set that is composed of only finite
numbers means the
set is not infinite for the summation of the entire set has to be an
infinite-integer.

And another pictorial of this situation. An infinite set itself can be
turned into a Infinite
Integer such as 99999.....99999999 where we replace a member with a
Place Value.
So if all the members are finite then there is no hope or way of
filling infinite place value.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

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