A fake proof

Prooftoys does not fall into mathematical fallacies, and will prevent you from doing so when you use it. In cases like this one, the steps are all allowable, but the end result only superficially looks like the one you may have imagined!

Here is a Prooftoys version of a fake “proof” that 2 = 1 based on fallacious reasoning. This one seems to have originated in Michael Spivak’s Calculus (1967). It is customarily described as a sequence of proof steps applied to two arbitrary nonzero, equal numbers x and y. Although Prooftoys reaches something resembling the incorrect conclusion, the end result is based on assumptions that are false. Prooftoys keeps track of them, and they turn out to be important to the meaning of the proof.

To view the proof, press the “Solve” button in the proof editor.

About the steps

The step both sides of an equation are both divided by x - y, is usually pointed out as the incorrect step because it uses division by zero. The Prooftoys rendition of it simply introduces an assumption x - y != 0, which is inconsistent with the assumption that x = y.

The proof here postpones showing that the assumptions cannot all be true until after transforming the conclusion into 2 = 1, but that part could be done immediately after dividing by x - y. All of the reasoning is still correct in both versions — just pointless, because false implies anything. The final step is an instance of the tautology F => a — one typical result of trying fallacious reasoning in Prooftoys.

Remember, when viewing a simplification step, if you click on the word “simplify”, the display will show step by step how the simplification is done.

Playing around

Feel free to play around with the proof editor, below.

If you like, you can prove the final step above in just a step or two. When nothing is selected in the editor, the basic menu lets you \check a tautology. Selecting that menu item and entering \F => a will give you a general form, and you can then substitute for a. Or if you select \check a tautology instance, you could enter a more specific formula immediately such as \F => 2 = 1, proving the final result above in a single step.