The nth Iteration

“It's a life's work to see yourself for what you really are and even then you might be wrong. And that is something I don't want to be wrong about.”

Every day brings another chance to nail it down, to get it right. And when it's clear that you've veered too far off course, or that the wheels have come off of the wagon, tomorrow brings another chance to redefine yourself completely.

Or does it?...

“You think when you wake up in the mornin' yesterday don't count. But yesterday is all that does count. What else is there? Your life is made out of the days it's made out of. Nothin' else. You might think you could run away and change your name and I don't know what all. Start over. And then one mornin' you wake up and look at the ceilin' and guess who's layin' there?”

No matter how much forgetting you do, or think you've been able to do, there come those times, maybe in the middle of the night after waking from a deep sleep, or maybe during a break at work when you've allowed your mind to wander, when forward movement all but stops and momentum brings that trail of debris you've been dragging behind you forth and slaps you right in the back with it. And that either lays you out flat or knocks you forward, getting you moving again.

Drag it behind you long enough, and eventually it will wear down, surface layers rubbed off and replaced by slightly less unattractive ones. Gone from your view, but rubbed onto the road. And you're not the only one who has traveled that road, nor is it closed to future travelers. Maybe those future travelers can trace the remnants back to you, maybe they can't. But, either way you slice it, you've been sloppy.

Do sons deserve to bear the sins of their fathers?

"Deserve's got nothin' to do with it..."

There's always going to be a reckoning.

1st two quotes from "No Country For Old Men" by Cormac McCarthy
3rd quote from the film "Unforgiven"

Do I Know You?

In another life, I had a wife, a house and a neurotic dog.

Journeys across the cerebral cortex reveal real connections between the present and the past, but travels back in time always see me entering the life of a different person. Surely, that person must be dead now. It's clear that some part of him was preserved and transported into the future and now occupies a small region of my consciousness. However, control is never an issue and he remains a guest doomed to roam the hallways and corridors only to poke his head out once in awhile from the shadows to remind me of various things. Particularly, of the consequences. Always with the consequences.

It's unbalanced. We can travel back in time and witness and feel, but we can never interact with the past. Yet, our past selves travel to the future all the time and have a direct influence on our present. In fact, if we're to grow as people, we require this. Some of us have found ways to close those portals, but most have to live with the connections.

As for me... lately I've been a bit obsessed with looking back through time and wondering if I didn't just arrive here in the present in some unexplainable fashion, as if there's a disconnect back there somewhere. I see that guy with the wife, the house and the neurotic dog...

But, I don't think we know each other.

Bijections and Finite Sets: Proof by If And Only If

Here's a proof that requires a bit of subtlety...

Prove that a nonempty set $T_{1}$ is finite if and only if there is a bijection from  $T_{1}$ onto a finite set  $T_{2}$.

We can use the following:

• Definition     (a)  The empty set is said to have zero elements.
• (b)  If $n\in \mathbb{N}$,  a set S is said to have n elements if there exists a bijection from the set         $\mathbb{N}_{n}$ := {1, 2, ..., n} onto S.
• (c)  A set S is said to be finite if it is either empty or has n elements for some $n\in \mathbb{N}$.
• (d) A set S is said to be infinite if it is not finite.

Step 1: If
Assume there is a bijection from $T_{1}$ onto $T_{2}$. We're given that $T_{2}$ is finite. We know that by definition, every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set ($T_{1}$). Since $T_{2}$ is finite, $T_{1}$ is finite.

Step 2: Only If
Assume $T_{1}$ is finite set, which is nonempty as described in the original problem statement. This means that by the definition above, $T_{1}$ has n elements for some $n\in \mathbb{N}$ and there exists a bijection from the set $\mathbb{N}_{n}$ := {1, 2, ..., n} onto $T_{1}$. Let's call that bijection f.

We also are given that $T_{2}$ is a finite set. Assuming that $T_{2}$ is not empty, we know by the definition above that there exists a bijection from the set $\mathbb{N}_{n}$ := {1, 2, ..., n} onto $T_{2}$. Let's call that bijection g.

The composition $g\circ f$ would map $T_{1}$ into $T_{2}$, and since the composition of two bijections is a bijection, $g\circ f$ is a bijection from $T_{1}$ onto $T_{2}$.

QED