0 + 1 = 1 by additive identity, moreover, 0 + 1 = 1 + 0 by commutativity for addition; therefore, 1 + 0 = 1 by identity elimination.
1 + S(0) = S(1 + 0) by Peano's definition of addition, moreover, 1 + 0 = 1 as previously established; therefore, 1 + S(0) = S(1) by identity elimination.
S(1) = 2 by Peano's definition of numbers, moreover, 1 + S(0) = S(1) as previously established; therefore, 1 + S(0) = 2 by identity elimination.
S(0) = 1 by Peano's definition of numbers, moreover, 1 + S(0) = 2 as previously established; therefore, 1 + 1 = 2 by identity elimination.
Assume 1 × 1 = 2, moreover, 1 × 1 = 1 by multiplicative identity; thus 1 = 2 within the assumption by identity elimination. 1 = 1 by identity introduction, moreover, 1 = 2 by the assumption's entailment; thus, 1 + (−1) = 2 + (−1) within the assumption by the subtraction property. 1 + (−1) = 0 by additive inverse, moreover, 1 + (−1) = 2 + (−1) by the assumption's entailment; thus, 0 = 2 + (−1) within the assumption by identity elimination. 1 + 1 = 2 as previously established, moreover, 0 = 2 + (−1) by the assumption's entailment; thus, 0 = (1 + 1) + (−1) within the assumption by identity elimination. (1 + 1) + (−1) = 1 + (1 + (−1)) by associativity for addition, moreover, 0 = (1 + 1) + (−1) by the assumption's entailment; thus, 0 = 1 + (1 + (−1)) within the assumption by identity elimination. 1 + (−1) = 0 by additive inverse, moreover, 0 = 1 + (1 + (−1)) by the assumption's entailment; thus, 0 = 1 + 0 within the assumption by identity elimination. 1 + 0 = 1 as previously established, moreover, 0 = 1 + 0 by the assumption's entailment; thus, 1 = 0 within the assumption by identity elimination. 1 ≠ 0 by multiplicative identity, moreover, 1 = 0 by the assumption's entailment; thus, contradiction within the assumption by negation elimination. Therefore, 1 × 1 ≠ 2 by negation introduction.
S(0) ≠ 0 by Peano's eighth axiom, moreover, 1 = S(0) by Peano's definition of numbers; therefore, 1 ≠ 0.
1 × S(1) = (1 × 1) + 1 by Peano's definition of multiplication, moreover, 2 = S(1) by Peano's definition of numbers; therefore, 1 × 2 = (1 × 1) + 1.
0 + 1 = 1 by additive identity, moreover, 0 + 1 = 1 + 0 by commutativity for addition; therefore, 1 + 0 = 1 by identity elimination.
1 + S(0) = S(1 + 0) by Peano's definition of addition, moreover, 1 + 0 = 1 as previously established; therefore, 1 + S(0) = S(1) by identity elimination.
S(1) = 2 by Peano's definition of numbers, moreover, 1 + S(0) = S(1) as previously established; therefore, 1 + S(0) = 2 by identity elimination.
S(0) = 1 by Peano's definition of numbers, moreover, 1 + S(0) = 2 as previously established; therefore, 1 + 1 = 2 by identity elimination.
(1 × 1) + (1 × 1) = 1 × (1 + 1) by distributive property, moreover, 1 + 1 = 2 as previously established; thus, (1 × 1) + (1 × 1) = 1 × 2 by identity elimination.
0 + 2 = 2 by additive identity, moreover, 0 + 2 = 2 + 0 by commutativity for addition; therefore, 2 + 0 = 2 by identity elimination.
2 + S(0) = S(2 + 0) by Peano's definition of addition, moreover, 2 + 0 = 2 as previously established; therefore, 2 + S(0) = S(2) by identity elimination.
2 + S(0) = S(2 + 0) by Peano's definition of addition, moreover, 2 + 0 = 2 as previously established; therefore, 2 + S(0) = S(2) by identity elimination.
3 = S(2) by Peano's definition of numbers, moreover, 2 + S(0) = S(2) as previously established; therefore, 2 + S(0) = 3 by identity elimination.
1 = S(0) by Peano's definition of numbers, moreover, 2 + S(0) = 3 as previously established; therefore, 2 + 1 = 3 by identity elimination.
2 + S(1) = S(2 + 1) by Peano's definition of numbers, moreover, 2 + 1 = 3 as previously established; therefore, 2 + S(1) = S(3) by identity elimination.
4 = S(3) by Peano's definition of numbers, moreover, 2 + S(1) = S(3) as previously established; therefore, 2 + S(1) = 4 by identity elimination.
2 = S(1), by Peano's definition of numbers, moreover, 2 + S(1) = 4 as previously established; therefore, 2 + 2 = 4 by identity elimination.
0 + 3 = 3 by additive identity, moreover, 0 + 3 = 3 + 0 by commutativity for addition; therefore, 3 + 0 = 3 by identity elimination.
3 + S(0) = S(3 + 0) by Peano's definition of addition, moreover, 3 + 0 = 3 as previously established; therefore, 3 + S(0) = S(3) by identity elimination.
4 = S(3) by Peano's definition of numbers, moreover, 3 + S(0) = S(3) as previously established; therefore, 3 + S(0) = 4 by identity elimination.
1 = S(0) by Peano's definition of numbers, moreover, 3 + S(0) = 4 as previously established; therefore, 3 + 1 = 4 by identity elimination.
3 + 1 = 1 + 3 by commutativity for addition, moreover, 3 + 1 = 4 as previously established; therefore, 1 + 3 = 4 by identity elimination.
Assume 1 × 1 = 2, moreover, (1 × 1) + (1 × 1) = 1 × 2 as previously established; thus, 2 + 2 = 1 × 2 within the assumption by identity elimination. 2 + 2 = 4 as previously established, moreover, 2 + 2 = 1 × 2 by the assumption's entailment; thus, 4 = 1 × 2 within the assumption by identity elimination. 1 × 2 = (1 × 1) + 1 as previously established, moreover, 1 × 1 = 2 by the assumption; thus, 1 × 2 = 2 + 1 within the assumption by identity elimination. 2 + 1 = 3 as previously established, moreover, 1 × 2 = 2 + 1 by the assumption's entailment; thus, 1 × 2 = 3 within the assumption by identity elimination. 4 = 1 × 2 by the assumption's entailment, moreover, 1 × 2 = 3 by the assumption's entailment; thus, 4 = 3 within the assumption by identity elimination. 4 = 3 within the assumption's entailment, moreover, 3 = 3 by identity introduction; thus, 4 + (−3) = 3 + (−3) within the assumption by the subtraction property. 3 + (−3) = 0 by additive inverse, moreover, 4 + (−3) = 3 + (−3) by the assumption's entailment; thus, 4 + (−3) = 0 within the assumption by identity elimination. 1 + 3 = 4 as previously established, moreover, 4 + (−3) = 0 by the assumption's entailment; thus, (1 + 3) + (−3) = 0 within the assumption by identity elimination. (1 + 3) + (−3) = 1 + (3 + (−3)) by associativity for addition, moreover, (1 + 3) + (−3) = 0 by the assumption's entailment; thus, 1 + (3 + (−3)) = 0 within the assumption by identity elimination. 3 + (−3) = 0 by the additive inverse, moreover, 1 + (3 + (−3)) = 0 by the assumption's entailment; thus, 1 + 0 = 0 within the assumption by identity elimination. 1 + 0 = 1 as previously established, moreover, 1 + 0 = 0 within the assumption's entailment; thus 1 = 0 within the assumption by identity elimination. 1 ≠ 0 as previously established, moreover, 1 = 0 by the assumption's entailment; thus, contradiction within the assumption by negation elimination. Therefore, 1 × 1 ≠ 2 by negation introduction.
Here is a variant of The Game, which will be referred to as The Game henceforth. The set of gods is a subset of the set of agents for brevity. For all agents
You have lost this variant of The Game now.
Assume it is not the case that either the Christian god is informed about The Game or the Christian god is not informed about The Game. Further assume the Christian god is informed about The Game; hence, either the Christian god is informed about The Game or the Christian god is not informed about The Game within the second assumption. It is not the case that either the Christian god is informed about The Game or the Christian god is not informed about The Game by the first assumption, moreover, either the Christian god is informed about The Game or the Christian god is not informed about The Game by the second assumption entailment; hence, contradiction within the second assumption. Thus, the Christian god is informed about The Game within the first assumption. The Christian god is informed about The Game by the first assumption's entailment; thus, either the Christian god is informed about The Game or the Christian god is not informed about The Game within the first assumption. It is not the case that either the Christian god is informed about The Game or the Christian god is not informed about The Game by the first assumption, moreover, either the Christian god is informed about The Game or the Christian god is not informed about The Game by the first assumption's entailment; thus, contradiction within the first assumption. Therefore, either the Christian god is informed about The Game or the Christian god is not informed about The Game.
For all gods, a god is informed about The Game only if a god lost a game as stipulated by the game; therefore, the Christian god is informed about The Game only if The Christian god lost a game.
For all gods, a god lost a game only if a god is not perfect; therefore, the Christian god lost a game only if the Christian god is not perfect.
For all gods, a god is not informed about The Game only if a god is ignorant by definition; therefore, the Christian god is not informed about The Game only if the Christian god is ignorant.
For all gods, a god is ignorant only if a god is not perfect; therefore, the Christian god is ignorant only if the Christian god is not perfect.
Either the Christian god is informed about The Game or the Christian God is not informed about The Game as previously established. Assume the Christian god is informed about The Game, moreover, the Christian god is informed about The Game only if the Christian God lost a game as previously established; thus, the Christian God lost a game within the assumption. The Christian god lost a game only if the Christian god is not perfect as previously established, moreover, the Christian God lost a game by the first assumption's entailment; thus, the Christian god is not perfect within the first assumption. Separately assume the Christian god is not informed about The Game, moreover, the Christian god is not informed about The Game only if the Christian god is ignorant as previously established; thus, the Christian god is ignorant within the second assumption. The Christian god is ignorant only if the Christian god is not perfect as previously established, moreover, the Christian god is ignorant by the second assumption's entailment; thus, the Christian god is not perfect within the second assumption. Therefore, the Christian God is not perfect.
The Christian God is not perfect as previously established; therefore, for all gods, a god is not perfect.
Assume the initial choice is identical to the correct choice; thus both the initial choice is identical to the correct choice and the initial choice is identical to the correct choice within the assumption. The initial choice is identical to the correct choice within the assumption given that both the initial choice is identical to the correct choice and the initial choice is identical to the correct choice by the assumption's entailment. Separately assume the initial choice is not identical to the correct choice, moreover, the initial choice is not identical to the correct choice only if swapping is identical to the correct choice as stipulated by the problem; thus, swapping is identical to the correct choice within the second assumption. Therefore, the probability that either swapping is identical to the correct choice or the initial choice is identical to the correct choice is equal to the probability that swapping is identical to the correct choice plus the probability that the initial choice is identical to the correct choice.
Assume it is not the case that either swapping is identical to the correct choice or the initial choice is identical to the correct choice. Further assume the initial choice is identical to the correct choice; hence, either swapping is identical to the correct choice or the initial choice is identical to the correct choice. It is not the case that either swapping is identical to the correct choice or the initial choice is identical to the correct choice by the first assumption, moreover, either swapping is identical to the correct choice or the initial choice is identical to the correct choice by the second assumption's entailment; hence, contradiction. Thus, the initial choice is not the correct choice within the first assumption. The initial choice is not identical to the correct choice only if swapping is identical to the correct choice as stipulated by the problem, moreover, the initial choice is not the correct choice by the first assumption's entailment; thus, swapping is identical to the correct choice within the first assumption. Either swapping is identical to the correct choice or the initial choice is identical to the correct choice within the first assumption given that swapping is identical to the correct choice by the first assumption's entailment. It is not the case that either swapping is identical to the correct choice or the initial choice is identical to the correct choice by the first assumption, moreover, either swapping is identical to the correct choice or the initial choice is identical to the correct choice by the first assumption's entailment; thus, contradiction. Therefore, either swapping is identical to the correct choice or the initial choice is identical to the correct choice.
Accordingly, the probability that either swapping is identical to the correct choice or the initial choice is identical to the correct choice is equal to one.
The probability that either swapping is identical to the correct choice or the initial choice is identical to the correct choice is equal to the probability that swapping is identical to the correct choice plus the probability that the initial choice is identical to the correct choice as previously established, moreover, the probability that either swapping is identical to the correct choice or the initial choice is identical to the correct choice is equal to one as previously established; therefore, one is equal to the probability that swapping is identical to the correct choice plus the probability that the initial choice is identical to the correct choice.
One is equal to the probability that swapping is identical to the correct choice plus the probability that the initial choice is identical to the correct choice as previously established, moreover, the probability that the initial choice is identical to the correct choice is equal to one-third as stipulated in the problem; therefore, one is equal to the probability that swapping is identical to the correct choice plus one-third.
Accordingly, three times one is equal to the result of the probability that swapping is identical to the correct choice plus one-third then times by three.
The result of the probability that swapping is identical to the correct choice plus one-third then times by three is equal to three times the probability that swapping is identical to the correct choice plus three times one-third, moreover, three times one is equal to the result of the probability that swapping is identical to the correct choice plus one-third then times by three as previously established; therefore, three times one is equal to three times the probability that swapping is identical to the correct choice plus three times one-third.
Three times one-third is equal to one, moreover, three times one is equal to three times the probability that swapping is identical to the correct choice plus three times one-third as previously established; therefore, three times one is equal to three times the probability that swapping is identical to the correct choice plus one.
One times three is equal to three, moreover, three times one is equal to one times three; therefore, three times one is equal to three.
Three times one is equal to three times the probability that swapping is identical to the correct choice plus one as previously established, moreover, three times one is equal to three as previously established; therefore, three is equal to three times the probability that swapping is identical to the correct choice plus one.
One is equal to one, moreover, three is equal to three times the probability that swapping is identical to the correct choice plus one as previously established; therefore, three plus negative one is equal to three times the probability that swapping is identical to the correct choice plus one plus negative one.
One plus negative one is equal to zero, moreover, three plus negative one is equal to three times the probability that swapping is identical to the correct choice plus one plus negative one as previously established; therefore, three plus negative one is equal to three times the probability that swapping is identical to the correct choice plus zero.
Zero plus the probability that swapping is identical to the correct choice is equal to the probability that swapping is identical to the correct choice, moreover, zero plus the probability that swapping is identical to the correct choice is equal to the probability that swapping is identical to the correct choice plus zero; therefore, the probability that swapping is identical to the correct choice plus zero is equal to the probability that swapping is identical to the correct choice.
Three plus negative one is equal to three times the probability that swapping is identical to the correct choice plus zero as previously established, moreover, the probability that swapping is identical to the correct choice plus zero is equal to the probability that swapping is identical to the correct choice as previously established; therefore, three plus negative one is equal to three times the probability that swapping is identical to the correct choice.
Zero plus two is equal to two, moreover, two plus zero is equal to zero plus two; therefore, two plus zero is equal to two.
Two plus the successor of zero is equal to the successor of two plus zero, moreover, two plus zero is equal to two as previously established; therefore, two plus the successor of zero is equal to the successor of two.
The successor of two is equal to three, moreover, two plus the successor of zero is equal to the successor of two as previously established; therefore, two plus the successor of zero is equal to three.
The successor of zero is equal to one, moreover, two plus the successor of zero is equal three as previously established; therefore, two plus one is equal to three.
Three plus negative one is equal to three times the probability that swapping is identical to the correct choice as previously established, moreover, two plus one is equal to three as previously established; therefore, two plus one plus negative one is equal to three times the probability that swapping is identical to the correct choice.
One plus negative one is equal to zero, moreover, two plus one plus negative one is equal to three times the probability that swapping is identical to the correct choice as previously established; therefore, two plus zero is equal to three times the probability that swapping is identical to the correct choice.
Two plus zero is equal to two as previously established, moreover, two plus zero is equal to three times the probability that swapping is identical to the correct choice as previously established; therefore, two is equal to three times the probability that swapping is identical to the correct choice.
Three times the probability that swapping is identical to the correct choice is equal to the probability that swapping is identical to the correct choice times three, moreover, two is equal to three times the probability that swapping is identical to the correct choice as previously established; therefore, two is equal to the probability that swapping is identical to the correct choice times three.
The successor of two is not equal to zero, moreover, the successor of two is equal to three; therefore, three is not equal to zero.
Two is equal to the probability that swapping is identical to the correct choice times three as previously established, moreover, three is not equal to zero as previously established; therefore, two-thirds is equal to the result of the probability that swapping is identical to the correct choice times three then over three.
The result of three over three then times the probability that swapping is identical to the correct choice is equal to the result of the probability that swapping is identical to the correct choice times three then over three, moreover, two-thirds is equal to the result of the probability that swapping is identical to the correct choice times three then over three as previously established; therefore, two-thirds is equal to the result of three over three then times the probability that swapping is identical to the correct choice.
Three over three is equal to one, moreover, two-thirds is equal to the result of three over three then times the probability that swapping is identical to the correct choice as previously established; therefore, two-thirds is equal to one times the probability that swapping is identical to the correct choice.
One times the probability that swapping is identical to the correct choice is equal to the probability that swapping is identical to the correct choice, moreover, the probability that swapping is identical to the correct choice times one is equal to one times the probability that swapping is identical to the correct choice; therefore, the probability that swapping is identical to the correct choice times one is equal to the probability that swapping is identical to the correct choice.
Two-thirds is equal to one times the probability that swapping is identical to the correct choice as previously established, moreover, the probability that swapping is identical to the correct choice times one is equal to the probability that swapping is identical to the correct choice as previously established; therefore, two-thirds is equal to the probability that swapping is identical to the correct choice.
Accordingly, the probability that swapping is identical to the correct choice is equal to two-thirds.
The purpose of this site is to jot down somewhat formal argumentation to some proposition, which I would find either interesting or satisfying.
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Set theory