Does there exist a finite false proposition $P$ such that assuming $P$ we can derive any given false proposition?

Question

I'm unsure if this is an appropriate question to ask, in some sense it seems closer to philosophy, but it's something I've wondered for a while. If it is not an appropriate question say so and I'll take it down.

1. Does there exist a false proposition $P$ which can be written in a finite number of characters such that assuming $P$ we can derive any given false proposition?

2. Does there exist a false proposition $P$ which can be written in a finite number of characters such that assuming $P$ we can derive any given proposition?

(note this may include contradictions such as interchangeably being able to use $1<2$ and $1>2$).

My thoughts were initially that for $1.$, it ought to be the same as the answer to the question

"Does there exist a true proposition $P$ which can be written in a finite number of characters such that assuming $P$ we can derive any given true proposition?" but then here to determine that such a $P$ is true we must derive it from axioms, and so this question is equivalent to asking whether or not all true statements are provable, which I think I'm right in saying that the answer to this is 'no'.

However, I think the answer to the original question $(1.)$ needn't be 'no' as we have the usual axioms to work with in addition to a new set of results to use derived from $P$ together with usual axioms, and so I simply don't know.

Thanks and feel free to let me know if the question should be taken down.

Answer 1

There's a subtlety in "ex falso quodlibet" which is often overlooked: it not true that from a false statement, we can deduce anything, but rather that from a known false statement (that is, a contradiction) we can deduce anything.

To illustrate the difference, take a theory like Peano arithmetic. PA is (let's say for the sake of argument) consistent, but by Godel's second incompleteness theorem this means that PA+"PA is inconsistent" is consistent. "PA is inconsistent" is obviously false, so why can't we prove everything from PA+"PA is inconsistent"? Well, the reason is that - although "PA is inconsistent" is false - PA doesn't know that "PA is inconsistent" is false (that is, PA can't prove its own consistency), so EFQ doesn't apply here.

In particular, this means that the theory consisting only of the sentence "PA is inconsistent," and no other axioms, is consistent, i.e. doesn't prove everything.

What you're looking for is a logical contradiction - a sentence which isn't true in any theory. Equivalently, a sentence $p$ such that $p$ alone proves every sentence. These sentences exist - e.g. "$\exists x(x\not=x)$" is such a sentence. Moreover, these sentences are false (of course), so any sentence proving all false sentences proves these, and hence proves all sentences - so your (1) and (2) are in fact equivalent.