Proof of the inconsistency of arithmetic

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The proof of the inconsistency of arithmetic is a pseudomathematical proof used by some creationists^[Who?] to prove that 0=1 as a means to prove that something can exist out of nothing. According to believers of this theory, it is fundamental to believe in it because it is a proof that a creator exists. The actual inventor of this proof is Luigi Guido Grandi, and it was developed by him in 1703.

Proof

The proof can be explained in 4 steps:

1. $0=0+0+0+0+\ldots$
2. $0=(1-1)+(1-1)+(1-1)+(1-1)+\ldots$
3. $0=1+(-1+1)+(-1+1)+(-1+1)+\ldots$
4. $0=1~\blacksquare$

Problems with the proof

At step one, it is stated that zero is the sum of an infinite series whose terms are all zero. This is true, although — to be precise — not a statement on arithmetic how it is intended nowadays (see Peano axioms), because it involves infinitely many additions.

Anyway, the error is in step three, where the terms of the infinite series

$(1-1)+(1-1)+(1-1)+(1-1)+\ldots$

are rearranged to get the series

$1+(-1+1)+(-1+1)+(-1+1)+\ldots$

Indeed, the terms of a series can be rearranged so that the series has still the same value only if some conditions are satisfied (for example, absolute convergence). So the real conclusion is not that 0 = 1, but that manipulations of infinite series are a delicate matter.

The series with elements of (1-1) sums to zero if and only if all terms are evaluated before the sum is evaluated. Removing the parentheses, then creates a series with the sum alternating between 1 for an odd number of terms and 0 for an even number of them. So evaluating the series as finally stated as zero is simply and obviously incorrect. The sum of the series is indeterminate. There are many indeterminate expressions, a common one is 0/0. Any real number is a solution to x = 0/0. Many fallacious proofs depend on slipping in an unrecognized division by zero. This "proof" is similar in that normally one can remove parenthesis without changing the value of an expression. But a series is not a simple expression. The parentheses actually indicate a requirement to evaluate what is within them first, and there are procedures where sequence matters. The series without the parentheses does not converge, so evaluating it by comparing it with a series that does converge is misleading.

Problems for mathematics

Needless to say, if 0 really were equal to 1, there would be far bigger concerns than the fact that something can come from nothing. Since any equation can have 0 added to it and keep its value, then it follows that any number is equal to itself plus one. And since 0 = ½ follows from 0 = 1, as well as any other fraction, suddenly every number is equal to every other number.

Eventually, absolutely anything becomes true.

Problems for sanity

Apart from the issue of why nobody else can harness this power of bringing something from nothing, it also raises the concern that nothing can easily arise from something; that is, everything around you could, at a moment's notice, suddenly cease to exist.

The fact that anyone might take this seriously is very concerning.

Extra notes

The series at step 2 has been named as Grandi's series in his honor. Though Niels Henrik Abel would have probably thought of him as a Satan worshipper.

Serious attempt to prove the inconsistency of arithmetic

The mathematician Edward Nelson did some serious mathematics trying to prove the inconsistency of Peano arithmetic, motivated by some deep philosophical objections to the principle on induction. In 2011, he claimed to have proved the inconsistency of Peano arithmetic, but later an error was found in the proof, and he retracted his claim.