# Rectilinear Paradox

The British mathematician and math popularizer Marcus du Sautoy passed around on twitter.com a link to a page at a wonderful Futility closet blog which presents a paradox by Roberto Casati.

Write an infinite list

All the statements below this one are false.

All the statements below this one are false.

All the statements below this one are false.

All the statements below this one are false.

...

If you succeed, the paradox would stem from the mere existence of such a list.

The purpose of this short note is to show that in order to obtain a paradox Roberto Casati has requested too much. It would be sufficient to replace the universal quantifier ∀ (All) with the existential quantifier ∃ (Some of) as in the infinite list:

Some of the statements below this one are false.

Some of the statements below this one are false.

Some of the statements below this one are false.

Some of the statements below this one are false.

...

Roberto Casati has aptly referred to his paradox as a *rectilinear* one; both fall into that category; both rely on the infinitude of the sentences below each of them; and both could be extended to infinity upwards making them even more markedly rectilinear.

Casati argues as follows:

This array manages to be paradoxical without being circular. **The statements can't all be false, because that would make the first one true, a contradiction. But neither can any one of them be true, as a true statement would have to be followed by an infinity of false statements, and the falsity of any one of them implies the truth of some that follow.** "A paradox - but a rectilinear one."

For the "existential" modification, we adapt that reasoning. We argue that *the statements can't all be true, because this would make the first one false, a contradiction. But neither can any one of them be false, as a false statement would have to be followed by an infinity of all true statements, and the truth of any one of them implies the falsity of some that follow.*

- What Is Infinity?
- What Is Finite?
- Infinity As a Limit
- Cardinal Numbers
- Ordinal Numbers
- Surreal Numbers
- Infinitesimals. Non-standard Analysis
- Various Geometric Infinities
- Paradoxes of Infinity

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