The Number Line Hypothesis with Respect to Set Theory

The Number Line Hypothesis with Respect
to Set Theory
With respect
to infinite sets (with a fixed starting point), it has been said that the set of all nonnegative integers
(set A) is the same size, ie has the same cardinality, as the set of all
positive integers (set B).
I have said
that set A (the set of all nonnegative integers) has a greater cardinality
than set B (the set of all positive integers). My argument is that set A
consists of and contains all the members of set B AND it has at least one
element that set B does not.
That is the
set comparison method. Members that are the same cancel each other and the
remains are inspected to see if there is any difference that can be discerned
with them.
Numbers are
not arbitrarily assigned positions along the number line. With set sizes, ie
cardinality, the question should be “How many points along the number line does
this set occupy?”. If the answer is finite then you just count. If it is
infinite, then you take a look at the finite because what happens in the finite
can be extended into the infinite (that’s what the ellipsis mean, ie keep
going, following the pattern put in place by the preceding members).
With that in
mind, that numbers are points along the number line and the finite sets the
course for the infinite, with infinite sets you have to consider each set’s
starting point along the line and the interval of its count. Then you check a
chunk (line segment) of each set to see how many points each set occupies (for
the same chunk). The chunk should be big enough to make sure you have truly
captured the pattern of each set being compared.
The set with
the most points along the number line segment has the greater cardinality.
For set A =
{0.5, 1.5, 2.5, 3.5,…} and set B = {1,2,3,4,…}, set A’s cardinality is greater
than or equal to set B. It all depends on when you look.
And it is strange that Patrick would tell me to read a math site and then prove that he doesn't even understand it.
5 Comments:
At 1:50 AM, Unknown said…
All countably infinite sets have the same cardinality. That's just the way it is.
You can attempt to create a new mathematics of the infinite but first you should make an attempt to understand the work that's already been done. Over 100 years of such work.
When you can explain the equivalence of Zorn's Lemma and the Axiom of Choice let me know.
At 7:03 AM, Joe G said…
The way it is can neither be proven nor disproven. Also it appears it doesn't have any practical application which means I am neither correct nor incorrect as it doesn't matter one bit.
At 12:09 PM, socle said…
Joe,
It's still not clear exactly how your JoeC definition works, so it is difficult to pin down contradictions at this point. For example, how would you handle the set {2/1, 3/2, 4/3, 5/4, 6/5, ...}? That is an infinite set, all of whose elements are between 1 and 2. I'm not sure how your number line hypothesis would work with it.
That said, here are a few general points:
1) It's not difficult to come up with a new definition which creates no contradictions, as long as it doesn't have any connections to previously defined concepts. I could define my own cardinality socleC(A), by setting socleC(A) to be equal to 17 for all sets A. That's perfectly fine, as long as I don't make any claims about my definition satisfying "reasonable" properties that we would expect to hold (for example, we normally expect that the cardinality of the empty set should be 0).
2) This statement of yours:
For set A = {0.5, 1.5, 2.5, 3.5,…} and set B = {1,2,3,4,…}, set A’s cardinality is greater than or equal to set B. It all depends on where along the number line you look.
is problematic, because you normally don't want cardinality to depend on "where you look". Just like a biological cell should either have CSI or not, regardless of who is observing where.
3) The notion of countability is actually useful (or should be, for an IDer), because that notion is used in measure theory, which is in turn used in probability theory.
In fact, if you look at The Search for a Search: Measuring the Information Cost of Higher Level Search by Dembski and Marks, you will find a reference to countable sets used in their arguments. See this sentence:
"... note also that such combinations, when restricted to a countable dense subset of Ω, form a countable dense subset of M(Ω) in the weak topology, showing that M(Ω) is itself separable in the weak topology."
At 12:35 PM, Joe G said…
For example, how would you handle the set {2/1, 3/2, 4/3, 5/4, 6/5, ...}?
It's an infinite set that rests between two finite points. And it appears to contain an infinite number of points on the number line. It just has a finer interval gradiation than {0,1,2,3,...}.
As for greater than or equal to, how is that problematic? That is indicative of an injective function.
As for 3, I am OK with an infinite set having countability. I just disagree with the statement tat all countable infinite sets have the same cardinality.
At 4:17 PM, Joe G said…
Corrected the OP to say "when you look" not where...
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