What is wrong with this deduction of $\text{ZF} \vdash \text{Cons ZF}$

Question

I realize from the answer to this post that the fallacy in my "proof" of "ZF is inconsistent" was that I was not considering that there are models with non-standard integers. However now I think I developed an actual deduction of $T \vdash \text{Cons} T$ for any sufficiently powerful theory $T$ thus implying by Godel's Second Incompleteness Theorem that $T$ is inconsistent.

As before, let $\text{Prb}_T \sigma$ represent $T \vdash \sigma$ and $\text{Cons} T$ be the sentence $\neg \text{Prb}_T (0=1)$. By the fixed-point lemma we have the existence of a sentence $\sigma$ such that:

$$T \vdash (\sigma \leftrightarrow (\text{Prb}_T \sigma \rightarrow \text{Cons} T))$$

By reflection we have:

$(1) \; T \vdash \text{Prb}_T(\sigma \rightarrow (\text{Prb}_T \sigma \rightarrow \text{Cons} T))$

By formalized modus ponens we have:

$(2) \; T \vdash (\text{Prb}_T \sigma \rightarrow \text{Prb}_T(\text{Prb}_T \sigma \rightarrow \text{Cons} T))$

By formalized modus ponens again we have:

$(3) \; T \vdash (\text{Prb}_T \sigma \rightarrow (\text{Prb}_T \text{Prb}_T \sigma \rightarrow \text{Prb}_T \text{Cons} T))$

Now formalized reflection is $T \vdash (\text{Prb}_T \sigma \rightarrow \text{Prb}_T \text{Prb}_T \sigma)$ so from the last step and sentential logic we have:

$(4) \; T \vdash (\text{Prb}_T \sigma \rightarrow \text{Prb}_T \text{Cons} T))$

Now, Godel's Second Incompleteness Theorem formalized is: $T \vdash (\text{Prb}_T \text{Cons} T \rightarrow \neg \text{Cons} T)$. Since anything follows from a contradiction, we have $T \vdash (\neg \text{Cons} T \rightarrow \tau)$. Replacing $\tau$ with $\text{Cons} T$ and following this chain of implications, line (4) implies:

$(5) \; T \vdash (\text{Prb}_T \sigma \rightarrow \text{Cons} T)$

By our choice of $\sigma$ we now have $T \vdash \sigma$ which by reflection yields $T \vdash \text{Prb}_T \sigma$. From (5) therefore we have $T \vdash \text{Cons} T$.

Where am I going wrong here? The only thing I can think of is that I formalized Godel's Second Incompleteness Theorem incorrectly, but then how is it formalized?

Answer 1

Your mistake is the sentence

Since anything follows from a contradiction, we have $T\vdash(\neg Cons(T)\rightarrow\tau)$ [for any $\tau$].

This is not the case. $T$ does prove that, if $T$ is consistent, then $T$ proves everything; that is, $$T\vdash \neg Cons(T)\rightarrow Pr_T(\tau)$$ for all $\tau$, but this is a far cry from what you claim.

Indeed, think about it this way: in a model $M$ of $T$ in which $T$ appears inconsistent, the sentence $$\mbox{"$0=1$"}$$ will definitely not be true (since $M\models T$ and $T\vdash 0\not=1$. So such a model will satisfy $$\mbox{"$\neg Cons(T)\wedge \neg(0=1)$."}$$ But this means that "$\neg Cons(T)\rightarrow (0=1)$" is not true in every model of $T$! So, by Soundness, $T\not\vdash \neg Cons(T)\rightarrow (0=1)$.

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Another take on the same point: There are two senses in which "anything follows from a contradiction": the external version e.g. $$T\vdash 0=1\rightarrow \tau,$$ and the internal version e.g. $$T\vdash Pr_T(0=1)\rightarrow Pr_T(\tau).$$ The former, stronger case applies if the hypothesis is a contradiction, that is, a statement $T$ disproves; the latter applies if the hypothesis is merely the assertion that a contradiction occurs.

The issue with claiming $$T\vdash Pr_T(0=1)\rightarrow \tau,$$ however, is that $Pr_T(0=1)$ is not a contradiction! This is the whole thrust of Godel's incompleteness theorem.

Answer 2

Just to sum up Noah's answer, the lengthy comment exchange which ensued (from which you must have learned a lot), as well as briefly touch on the philosophical point you raised in your final comment, I'll say the following:

Your approach as presented in your question itself is flawed because as Noah pointed out we do not have $T \vdash (\neg Cons T \rightarrow \tau)$, but rather $T \vdash (\neg Cons T \rightarrow Prb_T \tau)$.

In addition, even your approach as outlined in the comments is wrong. While Godel's Second Incompleteness Theorem (i.e. $T \vdash Cons T \implies \text{ "T is inconsistent"}$) and its representation within $T$ (i.e. $T \vdash (Prb_T Cons T \rightarrow \neg Cons T)$) are true, it is not the case that $T \vdash (Prb_T (\neg Cons T) \rightarrow \neg Cons T)$ and therefore we also do not have the external claim of $T \vdash \neg Cons T \implies \text{"T is inconsistent"}$.

This is due to the simple fact as pointed out in this answer to another one of your questions that when $T$ is consistent we have $T \nvdash Cons T$ so we can add $\neg Cons T$ to $T$ without affecting the consistency of the resulting theory. This new theory $R$ can prove $R \vdash Prb_T \tau \rightarrow Prb_R \tau$, so we can get $R \vdash Prb_R(0=1)$.

As to the philosophical issue about your tension over the use of semantic vs. syntactic proofs (which should be dispelled by the Soundness and Completeness Theorems) as well as your anxiety over all mathematical reasoning "being trapped inside a powerful theory $T$" (which by Godel's Second Incompleteness Theorem would provide itself with proofs that it is thinking consistently precisely when it is not) it occurred to me that you may want to try to think about it this way:

Mathematical reasoning, like all human thought, is based upon certain assumptions and value judgements (e.g. "Humans ought to be able to think deductively", "Mathematicians should avoid circular reasoning and contradictions", etc.) so it is more closely related to modal logic rather than first-order logic. Thus, mathematicians first make the assumption that they are capable of thinking deductively and then move on from there to prove theorems (e.g. the Soundness Theorem, the Completeness Theorem, etc.) The amazing ability of mathematics to help us comprehend the physical world in which we live as well as its deep aesthetic value (at least for those not afflicted with "mathphobia") testify to its worthiness of being pursued. As such, there is no reason to have an "existential crisis" over the possibility of a syntactic proof of $T \vdash Cons T$ for a powerful theory $T$ since this can be ruled out by the fact that we have models for these theories. (although you would undoubtedly counter that by completeness we have $\text{(Mathematical Reasoning)} \vdash \text{(Soundness Theorem)}$ so if there is somehow "a deep flaw" in mathematical reasoning itself it is only a matter of time until we develop a proof of $\neg \text{(Soundness Theorem)}$, ...). Life's simply too short to worry about such things.