The Infinity Continuation

It had been bright and clear over the university campus, but typical of fall, as the day waned, a light scattering of cloud began decorating the sky. In the early evening, Aidan and Iris approached the university library. On one of the conventional wooden park benches beside the cobbled path, a grey haired old woman sat smoking a cigarette. Aidan watched the woman enjoying her smoke a bit longer than he should have, because as they reached the library doors, he said "You know what? I'll wait for you out here and enjoy this fine evening weather. I've nothing to do and that way I won't be bothering your work. Text me if you need anything?"

She gave him a wry smile. "I thought you were going to quit that nasty habit."

"I am!" He hesitated. "It's just taking longer than I thought."

"I'll give you this one, but all too soon I may need to get tough on you! See you in a bit." She lightly kissed his cheek before turning to the doors and vanishing inside.

Less enthusiastically, Aidan turned to survey the park-like approach to the library. The only other bench was already occupied by three students, speaking in low conspiratorial voices over some pages of notes, their book bags scattered around them. The old woman, on the other hand, leaning heavily on the arm of her bench, seemed lost in her own thoughts. Aidan wandered toward the smoker while he removed a partially flattened pack from his pocket and extracted one slightly bent cylinder. "Good evening! Mind if I share your bench?"

"Not at all!" Smoke curled up lazily in the heavy evening air, already suggesting an ample dew to come. "Are you a student here?" she asked in casual conversation.

"Yes, computer science." Aidan fiddled with his cigarette, not yet lit.

"Need a light?" the woman observed.

"Uh, not yet, thanks! I'm trying to quit."

"Good luck with that! I've been trying for years," she chuckled dryly.

Aidan let the implication fade. The sun was beginning its approach to the horizon, painting the sky and cloud with counter-pointed colour. He hazarded, "Are you a professor here?"

"Me? Oh, no! Since I retired I'm back taking a second degree. Cosmology. My first degree was in pure mathematics!" The woman punctuated that with another puff, watching the sky. "Beautiful evening to be on campus. The shifts of form and hue are quite suggestive." She leaned slightly forward. "I've been contemplating how much our view of the universe has changed since I was your age." She tapped off her ash.

"How so?"

"I remember a professor on the radio talking about the expansion of the universe. Gravity was expected to slow it down. If there was enough mass, it would come to a stop and then come crashing back together. This would mean the geometry of the universe was spherical and space would be finite. Somehow this was preferable to continued expansion." She leaned back.

Aidan frowned, thinking. "Isn't the expansion of the universe accelerating? I'm sure I heard that was confirmed by several independent studies."

"Indeed! That's my point. Such a change! So instead of a spherical universe, it might be Euclidean or hyperbolic, but certainly an infinite universe." The old woman paused in some internal reverie. "It might be the first time science has been forced to accept an actual infinity. Such a thing was always relegated to mathematicians in the past. Not something real." The woman tossed her finished cigarette to the concrete and stubbed it out. Extracted a fresh one. Lit it with a flourish. "But what kind of infinity?" she continued. "History shows us the counting numbers and the fractions made from them are not enough for the universe. Real numbers are needed for the motions of the stars. To me, that means the infinity of the continuum."

A question occurred to Aidan. "Don't the counting numbers provide the smallest infinity, which is the first infinite ordinal number called little omega? I recall the surreal numbers contain the reals and the infinite ordinals and that there is an infinite succession of higher infinities, toward absolute infinity called Omega. Which infinite ordinal do the reals provide?"

The old woman seemed not to hear. "You are a computer scientist. Recall a logician named Kurt Godel, proving an earth-shaking result back around 1932. It was a favourite of John von Neumann and Alan Turing. Do you remember the famous Incompleteness Theorem? It's effectively about where you can reach by building outward with a collection of Lego blocks having only particular shapes, but in the context of arithmetic. Think of axioms as the types of blocks. It turns out there is no set of axioms which can reach out to all of arithmetic. You always miss some portion. Just cannot get to it. Things lost in the gaps. If you change your set, you miss some other portion." She smiled broadly. "It's incomplete! Some people feel this is a disaster. But I think it is wonderful! Arithmetic ends up being composed of different neighbourhoods, each accessible by one set of axioms." She paused, glancing at him appraisingly. "So, what are the boundaries like between these neighbourhoods? Are they far away from each other? Or do they meet crisply or even overlap?" She sat quietly for a moment happily gazing into the fall sunset.

"Is this part of cosmology?" Aidan twiddled his un-lit cigarette.

"Well, what if you could embed arithmetic into our universe? I mean, the universe is infinitely bigger, so why not? Plenty of room. Figure out a way to represent all of it. Encode it, if you will. Do you buy that?"

"Maybe," Aidan said skeptically. "Suppose I say yes."

She clapped her hands together, a spark flying. "Ok. Now, along comes someone who claims to have a grand unified theory of the universe, a so-called GUTS theory. A theory of everything!" The twinkle in the old woman's eyes was bright in the setting sun. "Such a theory can build the universe, describing everything. Being able to describe everything, it can describe the embedded arithmetic too, right?"

The cigarette spun around the top of Aidan's thumb. Once. Twice. "I guess that would have to be true." He examined the tobacco protruding from the end, tapped it back in place.

"So, the grand unified theory is able to describe and thus reach all of arithmetic. It is effectively an axiom system for arithmetic. But Godel proved that was impossible!" She spread her up-turned hands apart, "No GUTS theory. Not possible in an infinite universe."

"That seems somewhat disappointing." Aidan inserted the cigarette back into its pack.

"Why? It makes the universe wonderful!" the old woman exclaimed. "Just like with arithmetic, the entire universe decomposes into an infinite collection of neighbourhoods, each with its own axiom system. That means each with its own model of physical reality!"

Aidan thought for a bit. Where was this argument going? "But what kind of physical realities might be supported?"

"I was wondering about that," the old woman admitted. "I think that the key requirement to keep in mind is self-consistency. As long as the reality you want to describe is consistent with itself, then it should be a perfectly good candidate to actually exist as a neighbourhood in this universe." Chuckling, the woman added, "Anything you can imagine, as long as it is self-consistent will be in this universe."

Perhaps it was time to test this notion, Aidan decided. "What about seeming opposites? Are you suggesting religion and science can coexist?"

"Ha! Religion! Science! These are just words. There is not that much difference between them. Both chase their definitions back to the edge of the infinite regress. Even science finally draws a line saying we take these as self-evident truths. Axioms again. Nothing escapes the infinite regress. Truth is not finitely definable! Some people, however, would say the interesting questions are those which cannot be expressed in language. But these are both systems which indeed can be self-consistent and as such have their own separate neighbourhoods in the infinite universe. Separate neighbourhoods! That means there is never any conflict, between any of them." The old woman realized she had become over agitated and visibly forced herself to calm. Another cigarette was extracted and lit.

Was she finished? Aidan felt it would be rude to interrupt. A familiar tone warned him of an incoming text message. He fumbled for his phone. Iris was packing up.

Noticing, the old woman offered, "My apologies. Sorry to lecture you like that. I hope I haven't spoiled your evening or offended you in any way." Pausing briefly, she added, "I realized this a few years ago, but since then I continue to wonder about the boundaries. What should we expect if two of these neighbourhoods in the universe, these disparate realities, were to overlap in some way. With our tragically limited senses, what is it that we just fail to notice? I wonder about that a lot," and she pulled again at her cigarette, the burning coal bright in the rapidly fading light.

Starting to get up, Aidan said, "This has been pleasant and I would like to thank you for a very stimulating conversation, but I have to go meet my friend," he explained.

"It has been nothing short of a pleasure. Good night to you," speaking while continuing to puff tobacco smoke.

As Aidan began walking toward the library doors, he waved behind to the odd old woman. Ahead Iris had exited the library already. As he reached her, she was looking at him curiously. "Who were you waving to just now?" she asked.

"I spent the entire time absorbing a most singular story," he explained. "And you know what? I didn't even smoke my cigarette! I met a quite interesting character sitting on the bench over there." He turned to point, but in the lamp light it was clear that the bench was vacant. "Oh. We had a very long talk and it's getting quite dark. I guess she left for home."

Iris encircled his waist as they walked. "I'm proud you prevailed against the demon tobacco," she murmured into his jacket.

He smiled at that, but it faded as they walked past the now familiar bench. Curiously, on the clean concrete before it, not a single butt or speck of ash was to be seen.