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Does infinity exist?

There are a number of interesting possible consequences. For example, there may not be infinitely many numbers:

Are there infinitely many numbers?

Are the integers bounded? Let's imagine that they are. Then there'd be a number N, let's call it, that's the biggest. But, we can produce the number N+1, which is bigger. So, this N can't be the biggest number, even though we said it was, so, by contradiction, we conclude that there can't be an upper bound to numbers, so, the integers are infinite.

At least, that's the conventional proof. If you're familiar with dialetheism, you'll see the flaw in the proof - it assumes the law of non-contradiction.

Taking dialethism into account, we could re-visit this proof. It concludes that there's a contradiction between N being the biggest number and N+1 being bigger than that. Let's imagine that both of those statements were true. What would it mean?

It would mean that, when you get to the biggest of all numbers, N, you find that N + 1 is not greater than it. So, we could deduce from this that the integers are finite, and there's a very special integer - we don't know what it is, that has the property that adding 1 to it doesn't make it any bigger.

Well, is that more difficult to swallow than the idea that there are infinitely many numbers? I'm not sure. Actually, if you use one of the old pocket calculators and type in the biggest number possible 9999999 and add 1 to it, you usually got 999999, so there's an intuitive sort of logic there.

There is a Universal set

Russell's Paradox said:

for any formula φ(x) and any set A, there exists another set

   {x ∈ A | φ(x)}

that contains exactly those elements x of A that satisfy φ. If a universal set U existed and the axiom of comprehension could be applied to it, then there would also exist another set {x ∈ U | x ∉ x}, which give the contradiction that U ∈ U ⋀ U ∉ U => if p = U ∈ U then ~p = U ∉ U. So we have p ⋀ ~p. Which, in conventional logic => ∄ U.

This is only a problem if you believe in the law of the excluded middle. Otherwise we can say that in the case where x = U, and only in that case, yes, both p and ~p, and, with this, we can be happy that ∃U. There is a Universal set.

What does this mean for mathematics?

Bad news - for pessimistic mathematicians. Suddenly many proofs thought valid and treated as such are unsafe. So some mathematical results that are relied upon may not be reliable.

It's not that bad because we may have a mathematical sentence m that's only known to be part of Tc - but it might have gained some credibility by establishing proofs that are consistent with other proofs that don't rely on sentences that are part of Tc. A sort of backwards fitting of a sentence into Tp. It'd need to be more formal than that, but, any much relied upon proof that's been useful isn't likely to be discovered to be a secret dialethia - some might be, though.

It's good news for the optimistic mathematician because there are now much more work to be done. All m ∈ Tc ⋁ Tn must be re-examined to see if a proof can be found that puts them safely into Tp, or, in some cases, Tg, or maybe a safe To. That's a lot of work.

Part of the reason for the suggestion that there is no infinity, at the top of this page, is not really to put it in doubt, but, rather, to see that the common proof is unsafe and it's worth finding a better proof that's securely in Tp. That's for integers. For real numbers the fact that there are is a real number between any two real numbers is quite a convincing demonstration that R is infinite so that's part of Tp - maybe a proof could be formed back from the situation for R to that for I.

It also means that there might be new areas of mathematics waiting to be discovered. It was the doubting of the fifth postulate of Euclid's geometry that gave rise to hyperbolic and elliptic geometry, a huge step forward for geometry. It's at least conceivable that a similar discovery in mathematics based on a previously unsuspected dialethia could open up new fields of maths.


Then there's the solution to many paradoxes, some trivial, entertainina ones, like the Cretan liar who said 'All Cretans are liars' and some really important, historically, like Russell's paradox.

Dialetheism recognises most of these as dialethias, thus removing their paradoxical nature. It simply is the case that the statement 'This statement is false' is both false and true.

It's a stronger solution than just that. Once the sting of paradox is taken from meta-statements, they cease to be a problem for maths, so the whole taboo surrounding metastatements can be forgotten and they can deliver value in understanding, instead of being treated as suspect.

Peter.Brooks (talk)