# Closure of a theory under consistency

Godel's incompleteness theorem tells us that there cannot be any complete consistent theory $$T$$ at least as strong as Peano arithmetic, because we can't prove $$\text{Con}(T)$$ within $$T$$.

But suppose that we have a theory $$T_0$$ that we intuitively trust, for example Peano arithimetic or ZFC. We won't be able to prove $$\text{Con}(T_0)$$, yet we know that it's "true", so we also trust $$T_1 = T_0 + \text{Con}(T_0)$$. Similarly, we trust $$T_2 = T_1 + \text{Con}(T_1)$$, despite not being able to prove it, and so on.

This means that if we trust $$T_0$$, we'll trust $$T_0, T_1, T_2, ...$$ We can keep going for every ordinal, letting $$T_\lambda = \sum_{\alpha < \lambda} T_\alpha$$ for limit ordinals $$T_\lambda$$. This eventually gives us $$T_\infty = \sum_{\text{countable ordinal }\alpha} T_\alpha$$. But this still isn't the strongest theory we trust. Even though we ran out of ordinals, there's still $$T_\infty + \text{Con}(T_\infty)$$ that we trust.

It seems to be impossible to define what the actual strongest theory we trust, since whenever we have a theory $$T$$ we trust, there will always be a stronger one $$T + \text{Con}(T)$$ that we trust. But intuitively, there is a "thing" $$T$$ which is the closure of $$T_0$$ under consistency.

What's the explanation for this? Godel's theorem tells us there is no complete consistent theory, but the sentence it builds is exactly the one that we intuitively know to be true.

• @DougSpoonwood Those are logics, not theories; the OP is talking about theories, and the incompleteness theorem kicks in there (note that the (in)completeness of a logic is totally different from the (in)completeness of a theory - there's annoying terminology overload). Commented Jun 12, 2020 at 22:02
• "We can keep going for every ordinal" Actually, when we dig into the details we can't - you have to show how to express the relevant consistency principles in the relevant language, and eventually you run into ordinals which can't be defined in that language. Even with ordinals you can define the choice of definition matters: different expressions for the same ordinal can yield different theories. This has come up variously on MSE and MO: 1, 2, 3. Commented Jun 12, 2020 at 22:04
• @Stefan First of all, note that in general we should not be as confident in the consistency of $T+Con(T)$ as we are in the consistency of $T$: a theory can prove its own inconsistency while being consistent, and if $S$ is such a theory then $S$ is consistent but $S+Con(S)$ is inconsistent. Commented Jun 12, 2020 at 22:09
• I have my doubt about the construction you describe. The language of $T$, if it is even half way reasonable, will have only so many formulas, but there are arbitrarily many ordinals, so you can't be adding a new formula at every stage. You would have to hit a stage in your recursion where the process simply stalls out somehow, wouldn't you? Commented Jun 12, 2020 at 22:11
• @MaliceVidrine Indeed, per my comment second comment above you'd hit a problem long before cardinality became an issue. Commented Jun 12, 2020 at 22:12