In the quaint study of Professor Hercule Poirot, a serene silence filled the air, save for the ticking of the mantelpiece clock. The renowned detective, his mustache impeccably groomed and his gray cells poised for action, sat at his desk, surrounded by an assortment of leather-bound tomes and scrolls. His eyes flickered with a hint of anticipation as he embarked upon a seemingly innocent pursuit—a proof that 1+1 equals 2, akin to the mystery he had yet to unravel.
Poirot's gaze fell upon the foundational Peano axioms, a set of principles upon which the entire realm of arithmetic was built. Ah, such elegant simplicity, he thought, as he began his voyage into the world of logical deductions.
Firstly, he invoked the axiom of identity, which declared that for any natural number 'n,' n equals n. In this case, let us consider '1.' Therefore, 1 = 1.
To proceed further, Poirot required the concept of the successor function—a transformative operation that takes a natural number and yields the next one in the sequence. The axiom of induction, another jewel in the Peano crown, bestowed upon him the knowledge that if a property holds for 1 and also for the successor of any number for which it holds, then it holds for all numbers. It was with this axiom that Poirot's deductive abilities truly shone.
Poirot contemplated the implications of the axiom of induction, for it offered a path to the solution. He observed that the successor of 1, denoted as 'S(1),' must exist and be distinct from 1. Intuitively, Poirot understood that the successor of 1, by definition, would be 2. But he needed airtight reasoning to satisfy the rigors of mathematical inquiry.
His scrutiny turned to the successor function itself. As per the axiom of induction, if a property holds for 1, it must hold for the successor of 1 as well. Thus, applying this axiom to our investigation, if 1+1 were indeed equal to 2, then it must also be true that S(1) + 1 = S(2).
Yet, Poirot, ever the discerning detective, did not leap to conclusions without meticulous analysis. He invoked another axiom of Peano—the axiom of order—which dictated that if two numbers, 'a' and 'b,' are distinct, then their successors, S(a) and S(b), must also be distinct.
Applying this axiom to our scenario, Poirot observed that S(1) and 1 were indeed distinct, given the nature of the successor function. Hence, according to the axiom of order, S(1) and S(2) must also be distinct—critical evidence pointing to the conclusion that 1+1 cannot be equal to S(1)+1.
Ah, the detective's mind buzzed with anticipation, for he knew he was nearing the moment of truth. To complete the final piece of the puzzle, Poirot scrutinized the property of equality—its transitivity, to be precise. If 'a' equaled 'b' and 'b' equaled 'c,' then it followed that 'a' equaled 'c.' The enigmatic formula was laid bare, and Poirot's mustache twitched in delight.
Combining the axiom of identity (1 = 1) with the transitivity of equality, Poirot deduced that if 1 = 1 and S(1) + 1 ≠ S(2) + 1, then it must be that S(1) + 1 ≠ 2.
Finally, Poirot's logical mastery reached its crescendo as he unveiled the last piece of the puzzle. With a flourish, he brought forth the axiom of addition—the cornerstone upon which the concept of arithmetic rested.
The axiom of addition declared that if 'a' and 'b' were any two natural numbers, then their sum, denoted as 'a + b,' must be a natural number as well. Ah, such elegance in its simplicity!
Now, let us consider the equation at hand: 1 + 1. Poirot, with his gaze fixed upon the axioms, noted that '1' and '1' were both natural numbers, and by the axiom of addition, their sum, '1 + 1,' must also be a natural number. In other words, '1 + 1' is equal to some natural number, which we shall denote as 'n.'
But wait! Poirot's keen sense of deduction alerted him to a contradiction, lurking in the shadows of the equation. On one hand, he had determined that S(1) + 1 could not be equal to '2.' On the other hand, if 'n' were to represent '1 + 1,' it must equal '2' according to our earlier reasoning.
The great detective's eyes gleamed with the satisfaction of a mystery solved. With a resounding triumph, he declared that the contradiction exposed the falsehood of the assumption that 'n' was equal to '1 + 1.' Therefore, it followed logically that '1 + 1' could not be anything other than '2.'
Poirot had employed the meticulous logic of the Peano axioms to unveil the truth—a truth that had confounded the minds of both the learned and the curious for ages. The equation '1 + 1 = 2' stood firm, supported by the foundational principles of arithmetic.
As the silence settled upon the study once more, Poirot leaned back in his chair, his mission accomplished. The proof lay before him, an intricate web of deductions woven with the precision of a master detective. And with that, he returned to his realm of enigmatic mysteries, leaving behind a trail of illuminated minds, convinced of the eternal truth that 1+1 shall forever equal 2.
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