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Proof of the inconsistency of arithmetics

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The proof of the inconsistency of aritmetics is a pseudomathematical proof used by some creationists 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 was developed by him in 1703. [1].


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[edit]

At step one it is assumed that zero can be written as an infinitely converging row of values. In arithmetic, such a statement is extremely problematic as you cannot know the outcome of infinity multiplied by zero at face value.

It's clear to see that step three arbitrarily adds one to the beginning of the series without countering that effect. If it were written as

 0=1+(-1+1)+ \ldots +(-1+1)-1

then the end result is 0=0 (a true statement).

Problems for mathematics[edit]

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[edit]

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[edit]

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


  2. N. H. Abel wrote "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever" (Gardner 1984, p. 171; Hoffman 1998, p. 218).