B Can Infinity Be Considered Even?

[Chris Miller]

Since every point on a circle has exactly one other point (opposite its diameter) whose tangent is parallel, can it be said (proven?) that a circle is composed of an even number of points? It's messing with my head to think of infinity as even. I realize one-to-one mappings in infinite sets don't prove this, but here it's just too symmetrical. There can be no "odd" point.

 

[phinds]

Since every point on a circle has exactly one other point (opposite its diameter) whose tangent is parallel, can it be said (proven?) that a circle is composed of an even number of points? It's messing with my head to think of infinity as even. I realize one-to-one mappings in infinite sets don't prove this, but here it's just too symmetrical. There can be no "odd" point.

Sorry, but all that is just a meaningless argument. Infinity is not even or odd, it's not even a number like 3 or 4.

 

[jbriggs444]

Since every point on a circle has exactly one other point (opposite its diameter) whose tangent is parallel, can it be said (proven?) that a circle is composed of an even number of points? It's messing with my head to think of infinity as even. I realize one-to-one mappings in infinite sets don't prove this, but here it's just too symmetrical. There can be no "odd" point.

How do you define "even"? Does the cardinality of the set of points on a circle qualify to be considered?

How do you define "odd"? Might the cardinality of the set of points on a circle also qualify as odd?

 

[Chris Miller]

I realize infinity isn't a number, but more a function or algorithm. But as the number of sides of a regular polygon approaches infinity, the polygon approaches a circle. Every side of a polygon with an even number of sides will have another that is parallel. A regular polygon with an odd number of sides will have no parallel sides. If one considers a point to be a side of an infinite-sided regular polygon (i.e., circle), then it must have an even number of "sides."

If n&1==0 then n is even, else odd.
No, I would not have thought any infinite set could qualify as either, until now. And I see no evidence of the set of points on a circle as being odd, only even.

 

[NFuller]

If infinity was odd then $2\cdot\infty=\infty$ should be even, which is a contradiction. If infinity was even then $\infty+1=\infty$ should be odd, which is a contradiction. We must therefore conclude, that infinity is neither even nor odd.

 

[CWatters]

Likewise a point on a circle will have 2 other points that are 120 degrees either side of it making sets of 3 points. That also gets to infinity faster than sets of two point. It's all nonsense.

 

[Chris Miller]

If the expressions given were valid then you could also prove 1 == 2. So they cannot be used to conclude anything. They are the equivalent of division by zero.

An even number can be divided evenly by 3. Although, this does raise other interesting (to me) ideas (like that the "number" of points is a perfect factorial).

 

[phinds]

An even number can be divided evenly by 3. Although, this does raise other interesting (to me) ideas (like that the "number" of points is a perfect factorial).

You're just playing with numerology here and it will get you nowhere. As Cwatters said, it's all nonsense.

 

[jbriggs444]

I realize infinity isn't a number

Then you have to stop right there. If it is not a number then you cannot meaningfully speak of the last digit in its binary expansion. That would be needed in order to evaluate n&1 == 0.

That is a very strong hint that your definition of "even" is inadequate to deal with infinite cardinalities.

but more a function or algorithm.

That is a pre-Cantorian intuition. It is not correct.

 

[Mark44]

An even number can be divided evenly by 3.

Was the 3 here a typo? Some even numbers are divisible by 3 (such as 6), but not all even numbers are divisible by 3.

 

[Chris Miller]

Sorry, Mark. I meant some even numbers. E.g., all factorials are even.

Agree n&1 is meaningless applied to infinity, jbriggs. Maybe even could be defined as a perfect one to one mapping... For every set element there is one and only one other corresponding element. I get it's "nonsense." Just as I see the cosmological notion that the mass/size of the universe is infinite to be nonsense (another thread). I guess the step from a regular polygon whose number of sides approaches infinity and a circle is just too big.

 

[jbriggs444]

all factorials are even

I can think of two obvious counter-examples.

Yes, one could define a generalized notion corresponding to evenness:

Let S be a set and n be an strictly positive integer.

Definition: set S "is a multiple" of n if and only if there exist n disjoint subsets $S_1 ... S_n$ such that for every pair of integers (i, j) with 1 <= i <= n and 1 <= j <= n there exists a bijection between $S_i$ and $S_j$ and S is the union of all the subsets $S_1$ through $S_n$.

It is immediately obvious that every set whose cardinality is that of the continuum "is a multiple" of every strictly positive integer. Which is to say that this definition is pretty much pointless.

We could go on in this vein:

Definition: a set S is "even" if and only if it "is a multiple" of 2

Definition: a set S is "odd" if and only if there exists an element s in S such that the set difference, S - {s} is "even".

It follows quickly that all sets whose cardinality is that of the continuum are both odd and even.

Edit: and @TeethWhitener has exhibited a demonstration of this.

 

[TeethWhitener]

Since every point on a circle has exactly one other point (opposite its diameter) whose tangent is parallel, can it be said (proven?) that a circle is composed of an even number of points? It's messing with my head to think of infinity as even. I realize one-to-one mappings in infinite sets don't prove this, but here it's just too symmetrical. There can be no "odd" point.

Mappings can get weird and counterintuitive. For instance, let's assume for now that you're right and the number of points in a circle is "even." (Call this set A) Then let's consider adding a single "odd" point at the center of the circle. (Call the set with the circle plus point at center B) The problem with saying that one set has an even number of points and the other has an odd number is that you can make a bijection between the two sets: For simplicity, consider the set A as the unit circle in the complex plane. Take the center "odd" point ($0 \in \mathbb{C}$) in B and map it onto the 0° point in A ($1=e^{2\pi i}$). Then take the 0° point in B ($e^{2\pi i}$) and map it onto the 180° point in A ($e^{\pi i}$). Continuing, we get a map:
$$
f(x) =
\begin{cases}
e^{2\pi i} & \text{if } x =0 \\
e^{2\pi i/2^{n+1}} & \text{if } x = e^{2\pi i/2^n}\\
x & \text{otherwise}
\end{cases}
$$
So there is a bijection between an "even" set and an "odd" set: they have the same number of points.

 

[PeroK]

Since every point on a circle has exactly one other point (opposite its diameter) whose tangent is parallel, can it be said (proven?) that a circle is composed of an even number of points? It's messing with my head to think of infinity as even. I realize one-to-one mappings in infinite sets don't prove this, but here it's just too symmetrical. There can be no "odd" point.

Definition: a non-empty set $X$ is "even" if it can be expressed as a disjoint union of two sets $X = X_1 \cup X_2$, with $X_1 \cap X_2 = \emptyset$, where there exists a one-to-one mapping from $X_1$ onto $X_2$.

Proposition: all infinite sets are even.

Can you prove that?

Note: by definition we could define the empty set to be even.

With a similar definition of "odd" you could prove that all infinite sets are odd. For example:

Definition: a non-empty set $Y$ is odd if for some $y \in Y$, the set $Y - \{y\}$ is even.

PS can you also prove that these definitions of "even" and "odd" are equivalent to the normal ones for finite sets?

 

[Mark44]

Not being a number (or a function or algorithm), infinity is neither odd nor even.
Thread closed.