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All numbers are the same

Pf: First I will show that for any countable set of numbers Z, all the numbers in Z are the same. Let A = {a, b, c, ..., d} be any nonempty subset of Z. Then a = b= c= ... = d as I shall show by induction on the number of elements in A. Notice that for all singletons, A = {a}, the statement is vacuously true. Now assume that the statement is true for any subset that contains n elements {a1, a2, ..., an} and let A = {a1, a2, ..., a(n+1)} with n + 1 elements. Then the set B= {a1, a2, ..., an} has n elements and so by assumption a1 = a2 = ... = an. Likewise the set C = {a2, a3, ..., a(n+1)} has n elements and so by assumption a2 = a3 = ... = a(n+1). Combining these two sets of equalities, a1 = a2 = a3 = ... = a(n+1). So all the elements of A are the same. Hence, by induction, all numbers in Z are the same.

Next I will show that all numbers are the same. By the first part, all integers are the same. Suppose there is a number which is not the same as the integers, call it w. Then the set W = {w} union the integers is a countable set of numbers and so by the first part they are all the same, contradicting the statement that w is different. Therefore, all numbers are the same.

This same argument can be used to prove that all horses are the same color. And since Washington rode a white horse, we know that all horses are white. Since horses have two hindlegs in back and forelegs in front, that makes 6 legs so they have an even number of legs. However, for a mammal, 6 is an odd number of legs and so they have a number of legs which is both even and odd. Since no number is both even and odd, they must have infinitely many legs. We know that in the Triple Crown of 1973, Secretariat raced on all three legs, but that's a horse of another color.

Pf: First I will show that for any countable set of numbers Z, all the numbers in Z are the same. Let A = {a, b, c, ..., d} be any nonempty subset of Z. Then a = b= c= ... = d as I shall show by induction on the number of elements in A. Notice that for all singletons, A = {a}, the statement is vacuously true. Now assume that the statement is true for any subset that contains n elements {a1, a2, ..., an} and let A = {a1, a2, ..., a(n+1)} with n + 1 elements. Then the set B= {a1, a2, ..., an} has n elements and so by assumption a1 = a2 = ... = an. Likewise the set C = {a2, a3, ..., a(n+1)} has n elements and so by assumption a2 = a3 = ... = a(n+1). Combining these two sets of equalities, a1 = a2 = a3 = ... = a(n+1). So all the elements of A are the same. Hence, by induction, all numbers in Z are the same.

Next I will show that all numbers are the same. By the first part, all integers are the same. Suppose there is a number which is not the same as the integers, call it w. Then the set W = {w} union the integers is a countable set of numbers and so by the first part they are all the same, contradicting the statement that w is different. Therefore, all numbers are the same.

This same argument can be used to prove that all horses are the same color. And since Washington rode a white horse, we know that all horses are white. Since horses have two hindlegs in back and forelegs in front, that makes 6 legs so they have an even number of legs. However, for a mammal, 6 is an odd number of legs and so they have a number of legs which is both even and odd. Since no number is both even and odd, they must have infinitely many legs. We know that in the Triple Crown of 1973, Secretariat raced on all three legs, but that's a horse of another color.

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