[FOM] PA inconsistencies

[José Manuel Rodriguez Caballero]

>
> Tim wrote:
> One way to interpret his appeal to Goedel's theorem is as a blanket
> objection to all consistency proofs.  But if we do that, then the
> criticism applies equally to Voevodsky's own proposed foundations.


I completely agree: Voevodsky was not able to provide a foundation of
mathematics in which the consistency of arithmetic was not an issue. He
tried to use homotopy type as fundamental concept rather than set, but his
actual system, known as UniMath, was not radical enough to overcome his own
criticisms of mathematics. Nevertheless, the failure of UniMath to overcome
the problem of consistency of arithmetic does not prevent more radical
homotopy theoretic foundations of mathematics capable to reach this goal it
in another way. As Voevodsky said: homotopy theory is as fundamental as
arithmetic.

I recall that  Voevodsky's goal in his controversial lecture was to defend
the point (3) among the following possibilities:

(1)  If we somehow know that the first order arithmetic is consistent then
we should transform this knowledge into a proof and then the second
incompleteness is false as stated.

(2) Admit a possibility of transcendental provably unprovable knowledge.

(3) Admit that the sensation of knowing in this case is an illusion and
that the first order arithmetic may be inconsistent.

Voevodsky was criticized as follows:

In your own papers, you have used axioms/theorems that imply (dwarf is
> perhaps a better word) the consistency of PA, and far stronger
> systems.  Why single out the consistency of PA? Why is the
> latter questionable, while the rest of math that implies it is left alone?


I do not see any contradiction between the above-mentioned criticism and
the point (3) in Voevodsky's trichotomy. Indeed, if a contradiction in PA
is found, the rest of mathematics that it implies will be affected. For
example, if PA is inconsistent, then (some formulation of) the
Bolzano-Weiestrass theorem is false. Ok, according to (3),
Bolzano-Weiestrass theorem was another illusion of knowledge.

I am not claiming that PA is inconsistent, I am just trying to clarify, in
particular to myself, what Voevodsky's said.

For example, I do not see any way in which an inconsistency in PA may
affect the operation VERY as defined by Jean Benabou:
https://www.sciencedirect.com/science/article/pii/0022404994900450

Kind Regards,
Jose M.
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