Re: #523 a proof that Peano Axioms are supposedly self contradictory and set of all finite integers is finite

Archimedes Plutonium wrote:
In other words, you cannot reach infinity if every element of a set is a mere
finite element.

Given a set that contains a mix of finite integers and infinite-
integers then what is this set overall? Well it can be either finite
or infinite overall. Now let us correspond that answer with
even verus odd. Example {1,2,3,....., 3333....33333} has both finite
and infinite so it is infinite but this set {11111.....11111,3333.....33333}
contains two infinite integers but is overall finite.

So let's take that first set {1, 2, 3, ..., 333...333}, which we'll
call F.
We remove all of the finite integers from it and put them into a
second separate set S, so S = {1, 2, 3, ...}. Then according to you,
set S is finite, because it contains merely finite elements.

Then we take what's left over from set F, i.e., the elements we
did not remove from it, and put them into a third set T = {333...333},
which contains only the infinite integer from set F. Obviously, set T
has only one element, an infinite element, so set T is a finite set.

So what you've got now is two sets S and T, both of which you say
are finite sets. Set S is finite according to you because it contains
only finite elements. Set T is finite because it contains only one
element.

But the union of the two finite sets S and T is set F, which is an
infinite set. That seems quite strange, that we can combine two
finite sets and get an infinite set, don't you think?


Let's look at it from another angle. Start with an empty set, and
add the finite integers to it (1, 2, 3, etc.) until it contains all of
the
finite integers, and we'll call this set E. According to you, set E
is finite because it does not contain any infinite elements.

Next we take the set I = {333...333}, which is a finite set containing
a single infinite integer (the same one you mentioned above).

Then we combine the two finite sets E and I to get set D, so
D = union(E, I) = {1, 2, 3, ..., 333...333}. And by your logic, set D
is
an infinite set (because it's the same as your set F above).

Again, it seems very strange that combining two finite sets, both
with a finite number of elements, can produce a set with an infinite
number of elements.

Since set I has only one element, does this mean that set E has
one less than infinity elements? That the size of set E is almost
but not quite infinite, and all it takes is adding one more element to
this finite set to get an infinite set? And that likewise, we can
take
an infinite set and remove just one element from it to get a finite
set?

All of this sounds like an inconsistent system of set logic.
.

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