[FOM] PA inconsistencies

[José Manuel Rodriguez Caballero]

First, I will focus on the operation VERY, as defined by Jean Benabou,
because I consider that this is the first step to properly understand
Voevodsky's discourse. I recall that Voevodsky's
main reference was Grothendieck's Esquisse d'un Programme. So, his way of
thinking was holistic, whereas most mathematicians think in an analytic
way, because set theory is more popular than category theory.

Sam Sanders wrote: So I skimmed that paper, but could not really figure out
where to find VERY.  Could you perhaps tell me where to find it?

paper: https://www.sciencedirect.com/science/article/pii/0022404994900450

 In his lecture about VERY, Jean Benabou said that VERY is the translation
t in his paper, because t is the first letter of the French word TRES =
VERY. Benabou's lecture: https://www.youtube.com/watch?v=_7uONqXQvp8&t=3929s

Sam Sanders wrote: If VERY has vague properties (like small and large etc),
then I would point to Nonstandard Analysis, in which vague properties may
be formalised.

If we use Nonstandard Analysis for defining VERY, then a bounded number of
VERY will be operative and an unbounded number of VERY will be inoperative.
Notice that in the expression

VERY, VERY good job

the word VERY is operative, whereas in the expression

VERY, VERY, VERY, VERY, VERY, VERY, VERY, VERY, VERY, VERY, VERY good job

the word VERY is not operative. It do not think that 11 VERYs is an
unbounded number.

Now, I will focus on the infinitely large strictly decreasing sequence of
ordinals.

Sam Sanders wrote: What would the significance be of such a sequence, when
it plays a role in some theory of say physics?

As I already mentioned, such a sequence is very similar to the sorites
paradox: https://plato.stanford.edu/entries/sorites-paradox/

So, such a sequence could be used in order to model emergent properties.
Even if such a sequence does not exist in reality, real numbers neither and
they are useful to express something that almost behave like then. The word
VERY is another example of a non-ZFC concept with applications in reality.

Concerning the statement: In your own papers [Voevodsky's 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?

Kronecker worked in analysis too, beside his criticism to real numbers.
Was Kronecker's creative mathematical work consistent with his criticism of
foundations of mathematics? No, and Voevodsky's creative mathematical
work neither. Nevertheless, I recall that Voevodsky was developing
synthetic homotopy theory, i.e., homotopy theory without real numbers. So,
statements like

BW (Bolzano Weierstrass). Every sequence of real numbers from [0,1]
has an infinite 2^-n convergent subsequence.

may be nonsense in such a foundations of mathematics without real
numbers. Also, Jean Benabou liked the topology without points, because,
after him, the actual space has not points:
https://www.youtube.com/watch?v=biQKcTmsIwY

Kind Regards,
Jose M.
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