Suppose two things: Time is infinite in the future, we can make a time machine. Now imagine a machine that takes any question as input and runs on it just like a turing machine. If the turing machine halts in any point in the future it has the ability to go back in time to a exactly a second after you pushed the calculate button. If you see a machine that came back in time after a second then the program halts. If you don't see a machine then it doesn't halt.
What do you think?
This would be a halting oracle. This is very powerful - the entire computable complexity hierarchy collapses into O(1). However, it still cannot solve problems higher on the arithmetical hierarchy. For example, it cannot solve "does this TM halt on all inputs". It also of course cannot solve its own halting problem - whether a machine that uses this oracle as a step halts.
Since it's time travel it is also self contradictory. What if I get the machine coming back in time, but turn off the original machine before it can finish computing? This is exciting because by the principle of explosion this means we can now prove anything we want, and all our concepts of truth collapse.
This is exciting because by the principle of explosion this means we can now prove anything we want, and all our concepts of truth collapse.
In a not-so-distant future, Dr. Benjamin Hartley, a maverick physicist and logician, stumbled upon a groundbreaking breakthrough that sent shockwaves through the scientific community. His discovery was a peculiar application of the paradoxical computational model, one that threatened to dismantle our very notions of truth.
Dr. Hartley's innovation was a time-traveling theorem generator, a machine capable of proving any proposition with absolute certainty. The machine, named TRUTH, took inputs in the form of logical statements, and within a fraction of a second, it would produce an airtight proof of the statement's validity. If a proof existed, TRUTH would manifest in the past, exactly one second after the "Calculate" button was pressed, displaying the proof for all to see.
News of TRUTH spread like wildfire, captivating the imaginations of scientists, philosophers, and mathematicians worldwide. They envisioned a future where every conundrum, every mystery, and every philosophical debate could be resolved definitively. However, this vision was accompanied by a sense of unease, for the implications of such a machine were far-reaching and profound.
As TRUTH's proofs began to flood into the world, a paradigm shift ensued. Long-held beliefs, theories, and doctrines were either confirmed or refuted, and humanity found itself grappling with a new reality. The very foundations of scientific inquiry trembled as TRUTH exposed the flaws in well-established theories, debunking ideas that had been regarded as indisputable truths for centuries.
But the consequences went beyond scientific debates. Societies were thrown into disarray as the principles they had built their legal systems upon crumbled. People questioned the legitimacy of laws, ethics, and moral frameworks when every proposition could be rigorously proven or disproven. Chaos ensued as previously upheld notions of justice and fairness were called into question, and anarchy threatened to consume the world.
As the turmoil escalated, Dr. Hartley found himself at the epicenter of a moral and existential crisis. He realized that the pursuit of absolute truth, while seemingly desirable, had unleashed an unforeseen catastrophe. The collapse of subjective truth had eroded the very fabric of human society, leaving only confusion and disillusionment in its wake.
Determined to restore balance and salvage the remnants of truth, Dr. Hartley embarked on a perilous quest. He sought to create a new machine—a counterpart to TRUTH—that would inject uncertainty back into the equation. This machine, codenamed WISDOM, aimed to generate plausible doubts and alternative perspectives, reintroducing the human element of interpretation and subjectivity.
The battle between TRUTH and WISDOM became emblematic of the struggle to find a harmonious coexistence between certainty and uncertainty, proof and doubt. As Dr. Hartley navigated this treacherous terrain, he confronted the ethical implications of wielding such immense power. He grappled with the delicate balance between the pursuit of knowledge and the preservation of the human spirit, recognizing that the essence of truth lay not in absolutes but in the collective understanding and interpretation of a complex world.
As news of the Goldbach Conjecture's definitive proof spread, a wave of excitement and anticipation rippled through the scientific community. Technological breakthroughs are being generated as long as TRUTH exists, and such proofs being proved sends ripples to technological advancements:
Amid the jubilation, a young and ambitious physicist, Dr. Sophia Ramirez, saw an intriguing opportunity to explore uncharted territories.
Inspired by the proof, Dr. Ramirez embarked on a mission to harness the power of prime numbers revealed by the Goldbach Conjecture to revolutionize energy distribution. She envisioned a world where energy grids could operate more efficiently, with reduced waste and increased reliability.
Drawing from the insight that even integers greater than 2 could be expressed as the sum of two primes, Dr. Ramirez developed a groundbreaking algorithm. By mapping energy demands onto the mathematical landscape of prime numbers, her algorithm optimized the allocation and routing of electricity, maximizing the utilization of power sources and minimizing transmission losses.